THE AUSLANDER-REITEN CONJECTURE
AND AN EXPLICIT COUNTEREXAMPLE
============================================================================
A plain-text rewrite.
Written so that a curious high-school reader can follow the story,
and so that a specialist can check every step.
Author: OpenAI
Date: September 23, 2026
TABLE OF CONTENTS
============================================================================
PART 0 A STORY FOR A CURIOUS READER
0.1 What is a conjecture
0.2 Algebras over fields
0.3 Modules
0.4 Projective modules
0.5 Ext groups
0.6 The Auslander-Reiten conjecture
0.7 What this paper proves
0.8 How the proof works (one page)
PART I PRECISE STATEMENTS
1.1 The main theorem
1.2 Gorenstein-projective modules
1.3 Terms and notation
PART II THE TWO ENGINEERING IDEAS
2.1 Stable maps and complete resolutions
2.2 The conversion principle
PART III THE ALGEBRAS C AND T
3.1 The ten-dimensional algebra C
3.2 The trivial extension T = C ext DC
3.3 The explicit Hochschild cocycle
PART IV SELF-EXTENSIONS OF THE SIMPLE MODULE
4.1 A resolution of s over C
4.2 Ext_T(s, s) = kk[tau], tau of degree 3
4.3 The tensor square: X over A = T tensor T
PART V THE TWO CONES
5.1 Stable duality for symmetric algebras
5.2 The two-cone bimodule Ccal
5.3 The Hom profile: only degrees 0 and 3 survive
PART VI THE TWO TWISTS AND THE FINITE FIBER
6.1 The twisted bimodules U_H and the lift f_H
6.2 The finite side-projective fiber F
6.3 The twist eigenvalues: invertibility of delta^a
PART VII CONCLUSION
7.1 The radical, noncommutativity, and the eight simples
7.2 Persistence under field extensions
APPENDIX A. THE FINITE COCHAIN TABLE
APPENDIX B. GLOSSARY OF SYMBOLS
PART 0. A STORY FOR A CURIOUS READER
============================================================================
This part explains what problem this paper solves. It assumes only
high-school algebra and a little linear algebra. If you already know
homological algebra, skip to Part I; nothing here is needed later.
0.1 WHAT IS A CONJECTURE
------------------------------------------------------------------------
A conjecture is a precise mathematical statement nobody has yet proven
true or false. A counterexample is a single example that shows the
conjecture is false: one counterexample is enough to sink the
conjecture for good. This paper is a construction of one such
counterexample, for the Auslander-Reiten conjecture.
0.2 ALGEBRAS OVER FIELDS
------------------------------------------------------------------------
A field K is a set with +, -, *, and division by anything nonzero.
Examples: the real numbers R, the rational numbers Q, or the finite
field F_p (the integers mod a prime p) where p + 1 = 0.
A vector space over K is a set V where you can add, subtract, and scale
by field elements, with the natural rules.
An algebra A over K is a set that is a vector space AND has a
multiplication A x A -> A, written (a, b) |-> ab, satisfying:
(ab)c = a(bc) (associativity)
a(b + c) = ab + ac (distributivity on each side)
(a + b)c = ac + bc
(lambda a)b = a(lambda b) = lambda(ab) for lambda in K
there is an identity 1 with 1a = a1 = a
Example: the real 2 x 2 matrices M_2(R) form an algebra. Matrix
multiplication is associative but not commutative; AB is usually
different from BA. The identity is the 2 x 2 identity matrix.
Throughout this paper the field has two special features.
(i) The characteristic is 2. This means 1 + 1 = 0, so -1 = 1.
All signs flip or die, and many formulas simplify enormously.
(ii) It is the field
k = F_2(q, H_1, H_2)
of rational functions in three "transcendental symbols" q, H_1,
H_2 over the two-element field F_2. An element of k is a ratio
of two polynomials in q, H_1, H_2 with coefficients in {0, 1}.
"Transcendental" means no nonzero polynomial in F_2[X] vanishes
at q. In particular q^m != 1 for m > 0, and H_1^m != H_2^n
unless m = n = 0.
The transcendental symbols q, H_1, H_2 are the "knobs" that make certain
determinants nonzero throughout the paper.
0.3 MODULES
------------------------------------------------------------------------
Fix an algebra A over a field. A left A-module is a vector space M
together with an action A x M -> M, written (a, m) |-> am, satisfying
(ab)m = a(bm), 1m = m,
a(m + m') = am + am', (a + a')m = am + a'm,
(lambda a)m = a(lambda m) for lambda in k.
Matrix example: A = M_n(K) acts on K^n (column vectors) by A v = the
matrix product. So K^n is a left M_n(K)-module.
A submodule is a subspace closed under the action. A module is simple
if it has no nonzero proper submodules.
A module map f : M -> N is a k-linear map with f(am) = a f(m). The
k-vector space of all such maps is Hom_A(M, N). We will write Hom or
Hom_A depending on clarity.
0.4 PROJECTIVE MODULES
------------------------------------------------------------------------
The algebra A acts on itself by multiplication. As a left A-module
this is the regular (or free, rank one) module. Direct sums A^n give
the free modules of any rank.
A module P is projective if it is a direct summand of a free module.
Equivalently, P satisfies the LIFTING PROPERTY: for every surjection
f : M ->> N and every map g : P -> N, there is a map h : P -> M with
f o h = g:
P ---g---> N
\ ^
\ /
h f
\ /
v /
M
The lift h always exists when P is free: send each free basis vector
to a preimage. Summands of free modules inherit the same property.
Every vector space is projective (over a field). But over a general
algebra many modules are not projective. For example, take the
algebra A = k[x] / (x^2). The module s = A / (x) is k with x acting
by 0. It sits in a short exact sequence 0 -> rad(A) -> A -> s -> 0,
which does not split, so s is not a direct summand of A. Hence s is
not projective. That kind of "failure" is what the conjecture is
about: some modules are forced to be projective, and some are not.
0.5 EXT GROUPS
------------------------------------------------------------------------
Let M, N be left A-modules. An extension of M by N is a short exact
sequence
0 --> N --> E --> M --> 0.
Extensions form a set up to isomorphism of E fixing the ends. Two
extensions are equivalent if the obvious diagram commutes. The split
extension 0 -> N -> M (+) N -> M -> 0 is the trivial one. The set of
equivalence classes of extensions is a k-vector space Ext^1_A(M, N).
Intuition: Ext^1_A(M, N) = 0 means every extension of M by N is a
direct sum. That is, you cannot glue N onto M in any nontrivial way.
For higher i there is a similar definition: Ext^i_A(M, N)
classifies i-fold extensions
0 -> N -> E_i -> E_{i-1} -> ... -> E_1 -> M -> 0,
or equivalently is the i-th cohomology of a projective resolution of
M mapped into N. Two useful facts:
Ext^0_A(M, N) = Hom_A(M, N).
If M or N is projective, then Ext^i_A(M, N) = 0 for every i >= 1.
The last fact says: projective modules do not have higher extensions
out of themselves. It is the easy direction of the conjecture.
0.6 THE AUSLANDER-REITEN CONJECTURE
------------------------------------------------------------------------
Fix a finite-dimensional algebra A over a field (an Artin algebra,
more generally). Take a finite-dimensional left A-module M.
CONJECTURE (Auslander and Reiten, 1975). Suppose
Ext^i_A(M, M (+) A) = 0 for every integer i >= 1.
Then M is projective.
Since Ext is additive in each slot, the hypothesis means
Ext^i_A(M, M) = 0 and Ext^i_A(M, A) = 0 for every i >= 1.
So the conjecture says: if M has no self-extensions and no extensions
into the free module A, then M must be projective.
The trivial direction is easy: if M is projective, both vanishings
hold because Ext^i_A(P, -) = 0 for any projective P and i > 0.
The hard direction is the converse, and it is false as stated. This
paper is the counterexample.
The companion paper of this release also constructs a symmetric
counterexample (a case where the algebra is symmetric) by a different
route; see the references at the end. The two constructions are
independent.
0.7 WHAT THIS PAPER PROVES
------------------------------------------------------------------------
THM. Let k = F_2(q, H_1, H_2). There exist a finite-dimensional
k-algebra Lambda and a finite-dimensional left Lambda-module Z with:
(a) Ext^i_Lambda(Z, Z) = 0 for every i >= 1,
(b) Ext^i_Lambda(Z, Lambda) = 0 for every i >= 1,
(c) Z is not projective,
(d) Z is Gorenstein-projective,
(e) Lambda is split with Lambda / rad Lambda = k^8 (eight simple
modules), and rad(Lambda)^4 != 0,
(f) all these properties persist under any field extension K of k.
In particular, taking K the algebraic closure of k gives a
counterexample over any algebraically closed field of
characteristic two.
Consequence: the Auslander-Reiten conjecture (module form) is false
for Artin algebras. So is the Gorenstein-projective conjecture, since
Z is Gorenstein-projective with the same vanishings. Both fail as
stated even over algebraically closed fields.
The algebra Lambda is NONCOMMUTATIVE and triangular:
Lambda = ( A 0 )
( F A )
where A = T (x)_k T for a symmetric algebra T, and F is a finite
bimodule over A, projective on both sides. The final module Z is built
by the conversion principle below.
0.8 HOW THE PROOF WORKS (ONE PAGE)
------------------------------------------------------------------------
The proof has three physical parts. If you take nothing else from this
paper, remember these three boxes.
BOX 1. THE CONVERSION PRINCIPLE.
Suppose A is a symmetric algebra (has a nondegenerate symmetric
trace). Suppose F is a finite A-bimodule that is projective as a
left A-module AND as a right A-module. Suppose X is a finite left
A-module, and v : X -> F (x)_A X is a "stable" map (defined in
Part II) satisfying two conditions on certain delta maps. Then the
triangular algebra
Lambda = ( A 0 )
( F A )
has a finite left module Z = (X, Y, iota) built from X, F, and v,
and Z has the Ext vanishings and nonprojectivity we want.
This machine lets us "pre-assemble" a module Z with clean self-
extension algebra, provided we have the ingredients (A, F, X, v).
BOX 2. THE INGREDIENTS.
Step 2a. Build a very specific k-algebra C of dimension 10 by an
explicit multiplication table. C has a simple module s of
dimension 1. The important calculation: Ext_C^i(s, s) = 0 for i > 0
(so s has no self-extensions over C itself), but Ext_C^2(s, C) is
nonzero, one-dimensional, and equal to s as a right module.
Step 2b. Take the trivial extension T = C (x) DC of the algebra by
its dual bimodule DC. This T is symmetric (dim 20). The miracle of
this construction: over T, the self-extension algebra of s becomes
Ext_T^*(s, s) = kk[tau], degree(tau) = 3.
That is, s has self-extensions in degrees 0, 3, 6, 9, ... and nowhere
else, and it is a polynomial ring on one variable tau of degree 3.
Step 2c. Take A = T (x)_k T (a symmetric algebra of dimension 400)
and X = s (x) s. Then
Ext_A^*(X, X) = kk[tau_1, tau_2], degree(tau_1) = degree(tau_2) = 3.
So X has self-extensions in degrees that are multiples of 3.
BOX 3. THE ELIMINATION (TWO CONES + TWO TWISTS).
Step 3a. Form a bimodule Ccal by taking two cones of the classes
tau_1, tau_2. The key property: the stable groups
Hom_A(X, Ccal (x)_A X [a])
vanish unless a is 0 or 3. That is, Ccal "swallows" every self-
extension of X except the identity in degree 0 and one scalar in
degree 3.
Step 3b. For each H in k*, twist A by an automorphism h_H of T
that fixes C and scales DC by H. Take U_H = A with this twist on
the right. Under the canonical identification U_H (x)_A X = X, the
twist acts on Ext^{3m}(X, X) as multiplication by H^{-m}.
Step 3c. Using a Casimir element and a homotopy on the bar
resolution, construct two stable bimodule maps f_H : U_H -> Ccal[3]
with a prescribed top projection. Choose H_1 and H_2.
Step 3d. Both maps have the SAME image in the top cell (up to
rescaling). Take the finite fiber F of the two maps into Ccal[3].
The diagonal map v : X -> F (x)_A X takes the identity to the pair
(id_X, id_X).
Step 3e. The delta maps delta^a : H^a (+) H^a -> V^a (V^a is the
stable Hom of X into F (x) X [a]) have a 2x2 matrix in each degree
3m:
delta^{3m} = ( H_1^{-m} 1 )
( H_2^{-m} 1 )
This matrix is invertible for every m > 0 because H_1^m != H_2^m
(transcendence of H_1, H_2 over F_2(q)). So delta^a is an
isomorphism in every positive degree. In degree 0 the map is
surjective with a one-dimensional kernel (the diagonal line in k^2).
Step 3f. Apply the conversion principle. Wrap everything into
Lambda = (A 0; F A). Get the module Z. Check its structural
properties (eight simples, rad^4 != 0, noncommutative). Done.
The rest of the paper makes each step precise and verifies every
computation. Nothing is assumed: every claim is proved from the
multiplication tables given in Parts III and IV, plus the standard
homological machinery in Parts I and II.
0.9 HOW TO READ THE REST OF THE PAPER
------------------------------------------------------------------------
The paper is organised so that a reader can follow the proof in
roughly this order:
1. Part I. The precise statements and the standard definitions.
2. Part II. Stable categories and complete resolutions — the
mechanics used everywhere. The conversion principle is the
central statement here.
3. Parts III and IV. The specific algebra C, its trivial extension
T, the resolution of s, and the polynomial Ext algebra kk[tau].
4. Part V. The two cones Ccal. The Hom profile.
5. Part VI. The construction of the lift f_H, the finite fiber F,
and the twist eigenvalues.
6. Part VII. Wrap up: apply the conversion principle, check the
properties of Lambda, and prove base change.
7. Appendix A. The finite cochain table for the class tau.
8. Appendix B. Glossary of symbols.
If you get lost, look at the box score in 0.8. Every subsequent
section fleshes out exactly one box. Every subsequent section is also
self-contained: it starts with a statement of what it uses from
previous sections, does the calculation, and lists what it hands to
the next section.
End of Part 0.
PART I. PRECISE STATEMENTS
============================================================================
This part states the main theorem exactly, defines Gorenstein-projective
modules, and fixes notation used throughout. It does not prove
anything deep; every proof is in later parts.
1.1 THE MAIN THEOREM
------------------------------------------------------------------------
Let k = F_2(q, H_1, H_2) with q, H_1, H_2 algebraically independent
over F_2.
THEOREM 1.1. There exist a finite-dimensional k-algebra Lambda and a
finite-dimensional left Lambda-module Z with:
(1) Ext^i_Lambda(Z, Z) = 0 for every integer i >= 1.
(2) Ext^i_Lambda(Z, Lambda) = 0 for every integer i >= 1.
(3) Z is not projective.
(4) Z is Gorenstein-projective.
(5) Lambda is split with Lambda / rad(Lambda) = k^8 (so eight
isomorphism classes of simple modules) and rad(Lambda)^4 != 0.
(6) The same conclusions hold after any field extension K / k,
with Lambda_K = K (x)_k Lambda and Z_K = K (x)_k Z.
Taking K the algebraic closure of k gives the counterexample over an
algebraically closed field. Taking the algebraic closure in general is
what "over an algebraically closed field of characteristic two" means
in the abstract.
The algebra Lambda has the explicit triangular form
Lambda = ( A 0 )
( F A )
of dim_k Lambda = 2 dim_k A + dim_k F = 800 + dim_k F,
where A = T (x)_k T with dim_k T = 20 (so dim_k A = 400),
and F is the finite bimodule of Part VI, projective as a left
A-module and as a right A-module separately.
The module Z is the triangular module Z = (X, Y, iota) built from X,
F, and the stable map v of Proposition 6.7, via the conversion principle
of Proposition 2.4.
1.2 GORENSTEIN-PROJECTIVE MODULES
------------------------------------------------------------------------
For the reader who has not seen this definition, we recall it here.
A COMPLEX over a ring is a sequence of modules and maps
... -> P_{-1} -> P_0 -> P_1 -> ...
(possibly infinite in both directions) with the property that the
composite of two consecutive maps is zero: d_i d_{i+1} = 0. The
complex is EXACT if the kernel of each d_{i+1} equals the image of d_i
at every degree.
A COMPLETE PROJECTIVE RESOLUTION of an R-module M is an exact complex
P of finite projective R-modules, indexed in both directions, with
M = coker(P_{-1} -> P_0)
i.e. M is the cokernel of the map at degree 0, and the complex extends
downwards to negative indices.
A finite R-module M is GORENSTEIN-PROJECTIVE if:
(a) M = coker(P_{-1} -> P_0) for some doubly infinite exact complex P
of finite projective R-modules, and
(b) the complex Hom_R(P, R) is also exact (where R is the
"coefficient module").
Condition (b) is a "dual exactness" condition: not only must P be
exact as a complex of modules, its Hom into the regular module must
be exact as well. This is the property "TOTALLY ACYCLIC".
Over a symmetric algebra (defined in 1.3), every finite module embeds
in a finite projective module, so complete resolutions always exist.
In that case the two conditions (a), (b) can be merged into one: M is
Gorenstein-projective iff M is the cokernel of some totally acyclic
complex of finite projectives.
Important remark: an algebra is self-injective if finite projective
modules and finite injective modules coincide. A finite-dimensional
symmetric algebra over a field (in the sense of Part I.3) is
self-injective. Over a self-injective algebra, "Gorenstein-projective"
and "projective" coincide only when every module has a complete
projective resolution that also works as a complete injective one;
in general Gorenstein-projective modules form a strictly larger class.
The Gorenstein-projective conjecture (Luo and Huang, 2008) asks:
If M is a finite Gorenstein-projective module with
Ext^i_R(M, M) = 0 for all i > 0, must M be projective?
It is precisely the restriction of the Auslander-Reiten conjecture to
Gorenstein-projective modules, because total acyclicity implies
Ext^i_R(M, R) = 0 for every i > 0. The example Z disproves this
conjecture too.
1.3 TERMS AND NOTATION
------------------------------------------------------------------------
Field k. Throughout, k = F_2(q, H_1, H_2). The characteristic is 2,
so -1 = 1, and every sign in every formula is plus. The parameters
q, H_1, H_2 are algebraically independent over F_2.
Algebras and modules. All algebras are finite-dimensional over k with
identity. All modules are finite-dimensional and left unless otherwise
stated. dx denotes the k-dimension. "Artin algebra" means a
finite-dimensional algebra over a field, or more generally a finite
algebra over a commutative Artinian central base ring; every finite
algebra over a field is an Artin algebra.
Symmetric algebra. A finite-dimensional k-algebra A is SYMMETRIC if
there is a k-linear form t : A -> k with:
t(ab) = t(ba) for all a, b in A,
the pairing (a, b) |-> t(ab) is nondegenerate.
The resulting A-bimodule isomorphism A ~= DA (where DA = Hom_k(A, k))
shows that finite projective A-modules are also finite injective.
This is what makes complete resolutions of any module exist.
For the algebras C, T, A of Parts III and IV, the symmetrising trace
is explicit: it is the trace on the trivial extension followed by
projection to k, and it is compatible with tensor products.
Opposite algebra and enveloping algebra. For an algebra R,
R^op is R with multiplication a .op b := ba,
R^e = R (x)_k R^op is the ENVELOPING ALGEBRA,
an R-bimodule M is the same thing as a left R^e-module.
Right and left duals. DV = Hom_k(V, k) for a finite k-vector space.
For a bimodule M the dual DM has actions
(a phi)(m) = phi(m a), (phi a)(m) = phi(a m).
Tensor product. (x) and (x)_k mean tensor product over k unless a
subscript is given. F (x)_A X means tensor product over A; if F is
projective as a right A-module, this functor preserves exactness.
Direct sums. M (+) N is the direct sum, also written M x N (not to be
confused with tensor product). Direct sum of modules M_1, ..., M_r is
the product of the underlying vector spaces with componentwise action.
Hom and stable Hom.
Hom_A(M, N) is the k-vector space of A-linear maps.
stHom_A(M, N) = Hom_A(M, N) / {maps factoring through a finite
projective}.
Similarly End_A(M) and stEnd_A(M).
Shift. For a complex P, the shift P[a] has (P[a])_i = P_{i-a}. For a
module M in the stable category of a symmetric algebra:
M[-1] = Omega M = kernel of a projective cover P_0 -> M.
M[1] = cokernel of a projective embedding M -> I_0.
These are well defined and inverse to each other in the stable
category. The identification
stHom_A(M, N[a]) = Ext^a_A(M, N) for a > 0
is standard. Shift signs are plus because the characteristic is 2.
Cone. For a map u : M -> N of complexes, cone(u) is the complex
cone(u)_i = N_i (+) M_{i-1},
with differential d(x, y) = (du(x), -u(x) + dm(y)); in characteristic
two the sign disappears. There is a standard triangle
M -> N -> cone(u) -> M[1].
Trivial extension. For a k-algebra B with a bimodule L, the trivial
extension B (x) L is the algebra with underlying space B (+) L and
product (b, l)(b', l') = (bb', b l' + l b'). We will use this in
Part III with B = C and L = DC.
Radical and semisimple. rad(R) is the Jacobson radical: the intersection
of all maximal left ideals. For finite-dimensional algebras over a
field, rad(R) is the largest nilpotent two-sided ideal such that R /
rad(R) is semisimple. "Split" means R / rad(R) is a product k x k x ...
x k.
References for the general theory. For the classical module theory
referenced here, see Auslander-Reiten (1975) for the conjecture; for
total acyclicity, see Veliche (2006); for the triangular algebra
theory, see Eshraghi et al. (2014); for the stable category and
Tate duality, see Linckelmann (2013) and the classical references
cited there. Every foundational fact used in this paper is stated
explicitly at its point of use; the reader does not need to consult
external sources to verify the proof.
PART II. STABLE MAPS AND THE CONVERSION PRINCIPLE
============================================================================
This part develops the two tools used throughout. Section 2.1 sets up
the stable category over a symmetric algebra and proves the comparison
lemma that computes stable Hom via complete resolutions. Section 2.2
proves the conversion principle: a suitable stable map over a symmetric
algebra produces a finite module over a triangular algebra with
prescribed self-extensions.
Throughout, R is a finite-dimensional k-algebra with char k = 2, and all
modules are finite left R-modules.
2.1 STABLE MAPS AND COMPLETE RESOLUTIONS
------------------------------------------------------------------------
DEFINITION 2.1. For finite left modules M, N,
stHom_R(M, N) = Hom_R(M, N) / IF,
IF = { f : M -> N that factor through a finite projective }.
A map in IF is a PROJECTIVE FACTORIZATION. Composition passes to stHom,
so finite modules form the STABLE CATEGORY stmod(R).
SYMMETRIC ALGEBRAS. An algebra R is SYMMETRIC if it carries a linear
form t : R -> k with t(ab) = t(ba) and nondegenerate pairing
(a, b) |-> t(ab). Then R ~= DR as bimodules; finite projectives equal
finite injectives; every finite M embeds in a finite projective and is
a quotient of one. Given this:
M[-1] = Omega M = ker(P_0 -> M), P_0 -> M a projective cover;
M[1] = coker(M -> I_0), M -> I_0 a projective envelope.
These are inverse functors on stmod(R); any f : M -> N has shifted
versions f[-1], f[1]. In char 2, signs vanish, so shifting a cone gives
the cone of the shift. For a symmetric R and finite M, N,
stHom_R(M, N[a]) = Ext_R^a(M, N) (a > 0),
compatibly with composition (Yoneda product).
EXACT TRIANGLES. Let u : M -> N be a map and i : M -> I a projective
embedding. The cone is cone(u) = (N (+) I) / {(u(m), i(m)) : m in M}.
There is a stable triangle M --u--> N -> cone(u) -> M[1], and its
associated long exact sequence of stHom groups runs in every integer
degree.
COMPLETE RESOLUTIONS. A COMPLETE PROJECTIVE RESOLUTION of M over
symmetric R is a doubly infinite exact complex
P : ... -> P_{-1} -> P_0 -> P_1 -> ...
of finite projectives with M = coker(P_{-1} -> P_0). One exists by
splicing a projective resolution with a projective-injective coresolution.
TOTAL ACYCLICITY. A complex P of finite projectives is TOTALLY ACYCLIC
if P and Hom_R(P, R) are both exact. Complete resolutions over symmetric
R are totally acyclic. Total acyclicity into R implies total acyclicity
into every finite projective (finite sums and summands of R), so we can
test against any projective target. The class of Gorenstein-projective
modules is exactly the class of cokernels of totally acyclic complexes.
FULL HOM COMPLEX. For complexes P, Q of left R-modules,
Hom_R^a(P, Q) = PROD_j Hom_R(P_j, Q_{j-a}),
(d f)_j = d^Q_{j-a} f_j + f_{j-1} d^P_j.
The product (not sum) permits maps with infinitely many components; see
Veliche (2006). Degree-a cocycles are chain maps P -> Q[a]; degree-a
boundaries are nullhomotopic chain maps.
SIDE-PROJECTIVE BIMODULES. A finite A-bimodule F is SIDE-PROJECTIVE if
it is projective as a left A-module AND as a right A-module (weaker
than projective over A^e). Then M |-> F (x)_A M is exact on finite left
modules, preserves finite projectives, and induces a shift-compatible
endofunctor of stmod(A). Write FM = F (x)_A M.
LEMMA 2.2 (COMPARISON). Let P be a totally acyclic complex of finite
projective left R-modules and Q an exact complex of finite projective
left R-modules. Put M = coker(P_1 -> P_0) and N = coker(Q_1 -> Q_0).
Then there are natural, composition-compatible identifications
H^a Hom_R(P, Q) = stHom_R(C_a(P), N) = stHom_R(M, C_{-a}(Q))
for every integer a, with C_a(P) = coker(P_{a+1} -> P_a). For a > 0
these groups are naturally Ext_R^a(M, N).
PROOF. Total acyclicity gives the EXTENSION PROPERTY: any map
phi : im d_j^P -> T into a finite projective T extends to P_{j-1},
because phi d_j^P is a cocycle of Hom_R(P, T) trivialized by
psi : P_{j-1} -> T with psi d_j^P = phi d_j^P.
Given u : M -> N, lift u . epsilon_P to f_0 : P_0 -> Q_0, then iterate
the standard recursion (projectivity of P_{j+1} lifts the descent
d_j^Q f_j through im d_{j+1}^P upward; the extension property lifts
the descent of d_0^Q f_0 through im d_0^P to f_{-1} downward, and
iterates to f_{-2}, ...). This constructs a chain comparison f : P -> Q.
A nullhomotopy pushes to coker as (boundary) + (factor through P_{-1}),
giving a projective factorization. Conversely, given a factorization
M -> T -> N with T finite projective, extend M -> T over P_{-1} and
lift T -> N to Q_0; subtracting the resulting homotopy boundary leaves
a chain map f' inducing zero on coker.
For f' inducing zero, take h_{-1} = 0. Standard lifting (upward through
d_{j+1}^Q onto im d_{j+1}^Q, downward through im d_j^P and then the
extension property) fixes one component per step and only adjusts
adjacent equations, giving a full nullhomotopy. Hence
H^0 Hom_R(P, Q) = stHom_R(M, N) naturally in the second variable, hence
composition-compatible.
Apply to P[-a] (or Q[a]) for the shifted version. For a > 0, C_a(P)
is the a-th syzygy of M, and Ext_R^a(M, N) is Hom_R(C_a(P), N) mod maps
extending across P_{a-1}; by the extension property these are exactly
the projective factorizations. So the quotients agree (a = 1 included).
QED.
Lemma 2.2 needs R symmetric only to guarantee complete resolutions and
to identify finite projectives with finite injectives, hence to read
stHom(M, N[a]) as Ext for a > 0. Parts III and IV satisfy this.
2.2 THE CONVERSION PRINCIPLE
------------------------------------------------------------------------
This subsection takes a symmetric algebra A, a side-projective bimodule
F, a finite left A-module X, and a stable map v : X -> F X, and produces
a finite left module Z over the triangular algebra
Lambda = Tri(A; F) = ( A 0 )
( F A )
with (a 0; f b)(a' 0; f' b') = (aa' 0; af' + fb', bb').
TRIANGULAR MODULES. A finite left Lambda-module is a triple (M, N, eta)
with M, N finite left A-modules and eta : F (x)_A M -> N A-linear; the
action is (a 0; f b)(m, n) = (a m, eta(f (x) m) + b n). A morphism
(alpha, beta) satisfies beta eta = eta' (F (x) alpha).
COLUMN FUNCTORS. Define
E(M) = (M, FM, id), J(N) = (0, N, 0).
E is exact (right projectivity of F), J is exact, and both preserve
finite projectives: E(A) and J(A) are the two projective columns of
Lambda and Lambda = E(A) (+) J(A) as a left module.
LEMMA 2.3 (FOUR HOM GROUPS). For finite left A-modules M, M', N, N',
Hom_Lambda(E(M), E(M')) = Hom_A(M, M')
Hom_Lambda(J(N), J(N')) = Hom_A(N, N')
Hom_Lambda(E(M), J(N)) = 0
Hom_Lambda(J(N), E(M)) = Hom_A(N, FM').
PROOF. Write a map as (alpha, beta). For E(M) -> E(M') the condition
is beta = F (x) alpha, so beta is determined by alpha. For E(M) ->
J(N), beta = 0. For J(N) -> E(M'), alpha = 0 and no condition on beta
beyond A-linearity. The fourth case has beta arbitrary in Hom_A(N, N').
Naturality is immediate. QED.
CONVERSION SETUP. Let A be symmetric of char 2, F a finite
side-projective A-bimodule, X a finite left A-module, v in
stHom_A(X, FX). Set for each integer a
H^a = stHom_A(X, X[a]), K^a = stHom_A(X, (FX)[a]).
Since F is right projective, F (X[a]) = (FX)[a] compatibly. Define
delta^a : H^a (+) H^a -> K^a, (g, j) |-> F(g) v + v[a] j,
where v[a] : X[a] -> (FX)[a] is v shifted and F(g) is g tensored with F.
PROPOSITION 2.4 (CONVERSION PRINCIPLE). Let A, F, X, v be as above,
and assume
(C1) delta^0 is surjective with nonzero kernel;
(C2) delta^a is an isomorphism for every integer a > 0.
Choose v_0 : X -> FX representing v and a projective embedding
i : X -> Q into a finite projective A-module. Define
Y = (FX (+) Q) / {(v_0(x), i(x)) : x in X}, iota : FX -> Y,
Z = (X, Y, iota), Lambda = Tri(A; F) = (A 0; F A).
Then Z is a finite nonprojective left Lambda-module that is
Gorenstein-projective, and
Ext^a_Lambda(Z, Z) = 0 = Ext^a_Lambda(Z, Lambda) (a >= 1).
FINITENESS. Since i is injective, iota is injective and there is an
exact sequence 0 -> FX -> Y -> Q/i(X) -> 0 of finite A-modules. So Z
is finite. Also dim_k Lambda = 2 dim_k A + dim_k F.
COMPLETE RESOLUTIONS OF X. Since A is symmetric, X has a complete
resolution P with P_{-1} = Q and d_0 = i epsilon_P. Put
G_0 = E(P), G_1 = J(P).
Both are exact complexes of finite projective Lambda-modules. By the
four Hom computations,
Hom_Lambda(G_0, Lambda) = Hom_A(P, A),
Hom_Lambda(G_1, Lambda) = Hom_A(P, F) (+) Hom_A(P, A),
and both are exact (F and A are projective-injective over the symmetric
algebra A). So G_0 and G_1 are totally acyclic. Also F (x)_A P is a
complete resolution of FX: right projectivity of F for exactness, left
projectivity of F for projectivity of terms, self-injectivity of A for
Hom exactness.
THE CONE COMPLEX. Lift v_0 : X -> FX to a chain map V : G_1 -> G_0 by
the comparison construction. Set
P_Z = Cone(V) : (P_Z)_j = (G_0)_j (+) (G_1)_{j-1},
d_j^{P_Z} = ( d_j^{G_0} V_{j-1} ; 0 d_j^{G_1} ).
The cone of a chain map between totally acyclic complexes is totally
acyclic, so P_Z is totally acyclic with finite projectives.
COKERNEL IS Z. The degree-0 cokernel has upper component coker(P_1 ->
P_0) = X. The lower component is coker(F (x) P_1 -> F (x) P_0) = FX
together with the cross-term relations v_0(x) ~ i(x), which are exactly
the relations defining Y. So coker d_1^{P_Z} = Z, and P_Z is a complete
resolution of a concrete finite module.
SELF-EXTENSIONS. A graded endomorphism of P_Z has two diagonal blocks
(on G_0 and G_1[1]) and one off-diagonal block G_1[1] -> G_0; the
opposite off-diagonal vanishes by Lemma 2.3. Thus
0 -> Hom_Lambda(G_1[1], G_0) -> End_Lambda(P_Z)
-> End_Lambda(G_0) (+) End_Lambda(G_1[1]) -> 0
is a short exact sequence of cochain complexes, split in every graded
degree (the splitting need not commute with differentials). Reindexing
the full Hom gives H^a Hom_Lambda(G_1[1], G_0) = K^{a-1}. The
connecting map from diagonal degree a to off-diagonal degree a+1 is
exactly delta^{a-1}: a diagonal lift has cross terms obtained by
composing with V on each side, with the E-diagonal acting by F(g) and
the J-diagonal by precomposition with j, and comparison compatibility
identifies them with F(g) v + v[a] j. So the long exact sequence reads,
for every a,
0 -> coker(delta^{a-1}) -> H^a(End_Lambda(P_Z))
-> ker(delta^a) -> 0.
For a = 1, surjectivity of delta^0 and injectivity of delta^1 kill both
outer terms. For a >= 2, (C2) kills both. So
H^a(End_Lambda(P_Z)) = 0 for every a >= 1. Lemma 2.2 identifies this
cohomology with Ext^a_Lambda(Z, Z); total acyclicity of P_Z suffices
(Lambda need not be symmetric).
NONPROJECTIVITY. At degree 0 the same sequence shows
H^0(End_Lambda(P_Z)) surjects onto ker delta^0, which is nonzero by
(C1). By comparison this is the stable endomorphism ring of Z. A
projective module has zero stable endomorphism ring (every endomorphism
factors through itself). Hence Z is not projective.
EXT INTO LAMBDA. Since P_Z is totally acyclic,
Hom_Lambda(P_Z, Lambda) is exact, and its positive-degree cohomology
computes Ext^a_Lambda(Z, Lambda) from the nonnegative tail. So
Ext^a_Lambda(Z, Lambda) = 0 for every a >= 1.
GORENSTEIN-PROJECTIVE. Total acyclicity of P_Z and Z = coker d_1^{P_Z}
= im d_0^{P_Z} = ker d_{-1}^{P_Z} show Z is a cycle of a totally
acyclic complex, hence Gorenstein-projective. QED.
REMARK 2.5. (C1) is a genuine additional condition: without it, the
a = 1 term can have Ext^1_Lambda(Z, Z) != 0. In Parts III-VI, (C1) is
proved in Proposition 6.7 and (C2) is proved by a 2 x 2 determinant
using only transcendence of H_1, H_2. That concludes the theory;
Parts III-VI construct the input (A, F, X, v).
PART III. THE ALGEBRAS C AND T
============================================================================
This part constructs the algebraic input. Section 3.1 defines a ten-
dimensional algebra C over k = F_2(q, H_1, H_2) by an explicit
multiplication table and proves it is associative with a prescribed
radical. Section 3.2 takes the trivial extension T = C (x) DC and
proves T is symmetric of dimension 20, exhibiting its simple module s
and the normalized bar resolution used later. Section 3.3 introduces
the explicit degree-three Hochschild cocycle whose evaluation on s
generates the polynomial self-extension algebra kk[tau].
Throughout: q is transcendental over F_2, so q is nonzero and no
polynomial in q with F_2 coefficients vanishes at q. In particular
q != 0 and 1 + q != 0. All tensor products without a subscript are
over k.
3.1 THE TEN-DIMENSIONAL ALGEBRA C
------------------------------------------------------------------------
As a k-vector space, C has the following ten basis vectors, arranged
by corners. The notation "iCj" means "jC i as a bimodule", i.e. the
left idempotent is listed first. So an element of iCj is a vector that
left-multiplies by i and right-multiplies by j.
corner basis
-----------+-------------------------
eCe e, x, y, z
eCf u, v
fCe t, j
fCf f, n
Here e and f are orthogonal idempotents with e + f = 1. The action of
e and f on the corners is the natural one: left-multiplication by e
kills eCf and fCf, keeps eCe and fCe; right-multiplication by f kills
eCe and fCe, keeps eCf and fCf. The complete list of nonzero products
of non-idempotent basis vectors is:
xy = q z, yx = z,
x u = y u = v,
t x = j, t y = q^2 j,
u t = y + q x, v t = q z,
u j = z,
t u = (1 + q) n, n t = q j,
u n = v.
All other products of two non-idempotent basis letters are zero. So for
example x^2 = 0, y^2 = 0, u^2 = 0, t^2 = 0, and every product whose
left and right corners do not match is zero.
LEMMA 3.1 (C IS AN ASSOCIATIVE ALGEBRA). The rules above define an
associative unital k-algebra C of dimension 10. Its radical is the
k-span of the eight non-idempotent basis vectors, and
(rad C)^5 = 0, and (u t)(u t) = q (1 + q) z != 0.
In particular rad(C)^4 is nonzero, and C is noncommutative, because
e u = u but u e = 0.
PROOF OF LEMMA 3.1. Assign a degree to each non-idempotent letter so
that every listed product is homogeneous:
degree(u) = degree(t) = 1,
degree(x) = degree(y) = degree(n) = 2,
degree(v) = degree(j) = 3, degree(z) = 4,
degree(e) = degree(f) = 0.
Every listed product has total degree equal to the sum of the degrees
of its factors, so the table is homogeneous.
Associativity involving an idempotent follows from the corner
convention: for example, (a e) b = a (e b) reduces to whether b
belongs to the appropriate corner, and the table was built corner by
corner. For three non-idempotent basis letters, either the corners do
not line up (both bracketings are zero) or the total degree exceeds 4
and both bracketings vanish.
The remaining possibilities are checked directly. The complete table
of nontrivial cases is (compare the original):
abc | utu tut xut yut ntu unt txu tyu utx uty tun
(ab)c=a(bc) | (1+q)v q(1+q)j qz qz 0 qz 0 0 z q^2 z 0
Substituting the multiplication table on each side gives the identical
entry. Covering all three-letter words by these cases, associativity
follows by trilinearity.
For the radical: let N be the k-span of the eight non-idempotent
letters, i.e. (x, y, z, u, v, t, j, n). It is a two-sided ideal by the
table, it is nilpotent with N^5 = 0 by the grading, and the quotient
C / N = k e x k f = k x k is semisimple. A nilpotent two-sided ideal
whose quotient is semisimple equals the radical, so N = rad(C).
The fourth radical power contains the nonzero product
(ut)(ut) = (y + qx)(y + qx). Expanding, using y^2 = 0, x^2 = 0,
yx = z, and xy = qz:
(y + qx)(y + qx) = y^2 + q y x + q x y + q^2 x^2
= 0 + q z + q(q z) + 0
= q z + q^2 z
= q (1 + q) z.
Because q is transcendental over F_2, q != 0 and 1 + q != 0, so
q (1 + q) != 0. The element z is a basis vector, hence nonzero, so
(ut)^2 != 0. Since ut lies in N, the fourth power N^4 is nonzero.
QED.
REMARK 3.2 (ANTECEDENT IN SCHULZ). The corner eCe is the quantum
exterior algebra
eCe = k<x, y> / (x^2, y^2, x y - q y x), with z = y x.
The right module fCe has basis (t, j) and structure
t x = j, t y = q^2 j, j x = j y = 0.
Put R = eCe. The map
t |-> x + q^{-1} y, j |-> q^{-1} z
identifies fCe with the right ideal (x + q^{-1} y) R of R. In Schulz
(1995, Example 7) the parameter is q^{-1} and this ideal is the first
syzygy of (x + y) R. The interactions of fCe with the other corners of
C are fixed by the complete multiplication table above; Schulz used
only the corner fCe, so his example sits inside C.
3.2 THE TRIVIAL EXTENSION T = C (x) DC
------------------------------------------------------------------------
Let L = DC = Hom_k(C, k), equipped with the dual bimodule actions
(a phi)(c) = phi(c a), (phi a)(c) = phi(a c) for a,c in C,
phi in L.
Then L is a C-bimodule, and the trivial extension
T = C (x) L = C (x) DC,
(a, phi)(b, psi) = (a b, a psi + phi b)
is an associative k-algebra of dimension 20. Trivial extensions and
their cohomology are studied in Cibils-Marcos-Redondo-Solotar (2003).
CAPITAL-LETTER NOTATION. For each lower-case basis letter a of C, let
a* denote the corresponding dual basis element of L. So e* is dual to
e, x* is dual to x, and so on; the notation J = j* and F = f* is used
for the two dual basis elements tied to the simple module s. We also
set (a*)* = a, so the star involution is a bijection between the
lower-case and upper-case alphabets of the basis.
The 20 basis vectors of T are the ten lower-case letters (e, x, y, z,
u, v, t, j, f, n) and the ten upper-case (E, X, Y, Z, U, V, T, J, F,
N), where uppercase W = w*.
LEMMA 3.3 (T IS SYMMETRIC OF DIMENSION 20). The algebra T is symmetric
of dimension 20. A symmetrising trace is
lambda(a, phi) = phi(1),
and the trace-dual of each basis letter is its starred letter, i.e.
lambda(a b*) = delta_{a, b} for lower-case basis letters a, b. The
radical r = rad(T) is the k-span of all upper-case and non-idempotent
lower-case basis letters (i.e. all basis letters except e and f). It
satisfies
r^6 = 0, T / r = k x k.
PROOF OF LEMMA 3.3. First, the trivial extension of an associative
algebra by a bimodule is associative (standard, direct check). The
dual actions above satisfy the bimodule identities by associativity of
C.
Symmetry of the trace: compute
lambda((a, phi)(b, psi)) = lambda(ab, a psi + phi b) = (a psi + phi b)(1) = psi(a) + phi(b).
The right-hand side is symmetric in the pair ((a, phi), (b, psi)) up to
swapping a and b, phi and psi, which is exactly what "symmetric" means:
lambda(xy) = lambda(yx) for all x, y in T. Nondegeneracy: if
lambda((a, phi)(b, 0)) = 0 for all b, then phi(b) = 0 for all b, so
phi = 0; similarly lambda((0, psi)(0, psi')) = 0 forces psi = 0. So
the trace gives a nondegenerate symmetric bilinear form T x T -> k.
Trace-dual basis letter: lambda(a b*) = lambda((a, 0)(0, b*)) = b*(a) = delta_{a, b}. This is the content that determines the "trace-dual"
of each basis letter.
Radical: extend the C-grading by assigning
degree(a*) = 5 - degree(a).
Then formula (dual actions) shows every product in T is homogeneous
with respect to this extended grading, and the only basis letters of
degree zero are e and f. All other basis letters have positive degree,
and no degree exceeds 5, so their span r is a nilpotent two-sided
ideal with r^6 = 0 and T / r = k e x k f = k x k, which is semisimple.
A nilpotent ideal with semisimple quotient is the radical. QED.
THE SIMPLE MODULE s. Let s denote the one-dimensional left T-module on
which f acts by 1 and every other basis letter of C (including e, x,
and the upper-case letters) acts by 0. Equivalently, s is the left
simple inflating the f-simple of C. Similarly write s^r for the
one-dimensional RIGHT T-module defined by the same character (f acts
by 1, all others by 0 on the right). The left and right simple
characters coincide.
THE BAR RESOLUTION. Put I = k e (+) k f = k x k, which is the
semisimple subalgebra spanned by the two idempotents. The normalized
relative bar resolution of T with respect to I has terms
P_n = T (x)_I r^{(x)_I n} (x)_I T for n >= 0,
P_n = 0 for n < 0.
Write a_0 [a_1|...|a_n] a_{n+1} for the elementary tensor
a_0 (x) a_1 (x) ... (x) a_{n+1} projected to the bar tensor over I. The
differential d : P_n -> P_{n-1} multiplies each adjacent pair of
factors in turn and sums the results (with alternating signs from the
bar complex; in characteristic two all signs equal plus):
d(a_0 [a_1|...|a_n] a_{n+1}) =
SUM_{j=0}^{n} a_0 [a_1|...|a_j a_{j+1}|...|a_n] a_{n+1}.
In particular d(a_0 [a_1] a_2) = a_0 a_1 [ ] a_2 + a_0 [ ] a_1 a_2 in
bar notation. The augmentation P_0 = T (x)_I T -> T is multiplication.
EXACTNESS BY CONTRACTION. The bar resolution is exact. Here is an
explicit contraction sigma. Splitting by idempotents expresses each
P_n as a finite sum of projective bimodules of the form T i (x) j T
with i, j in {e, f}. On such a summand, send a_0 (x) a_1 (x) ... to its
"projection to r" in the first coefficient and add a new first bar:
sigma(a_0 [a_1|...|a_n] a_{n+1})
= (a_0 - pi_I(a_0)) [1 | a_1 | ... | a_n | a_{n+1}],
where pi_I : T -> I is projection to the k e (+) k f component and 1
in the new first bar is the identity of T. In augmented degree
(a_0 alone), send a to 1 [ ] a. Then d sigma + sigma d equals the
identity on P_n for every n >= 0 and on the augmentation target. The
argument is: d sigma inserts a new first bar (or the identity), sigma d
removes one multiplication, and the two cancel pairwise except for the
leftover "I-component" of a_0 balanced over I. This contraction is
right T-linear by construction (which will matter: applying (x)_T to
a right module preserves exactness of the resolution).
Consequently P (x)_T s is a projective resolution of s over T (it is
exact and all terms are projective as left T-modules).
WEIGHT GRADING FOR THE COCYCLE. On T, give a lower-case basis letter
weight 0 and an upper-case letter weight 1. So the weight counts the
number of DC factors. This weight is distinct from the positive
grading of Lemma 3.3. The explicit cochain of Section 3.3 will lower
the weight by exactly one.
End of Sections 3.1 and 3.2. Section 3.3 supplies the Hochschild
cocycle and its boundary identity, which drive the polynomial Ext
calculation in Part IV.
3.3 THE EXPLICIT HOCHSCHILD COCYCLE
------------------------------------------------------------------------
This subsection introduces a specific Hochschild 3-cocycle on T. Its
evaluation on the simple module s produces the polynomial generator
tau of degree three in Ext_T(s, s) that drives all later computations.
The full table of the cocycle is in Appendix A; here we state exactly
the finite properties that are used, all of which are checked against
that table.
THE COCHAIN. A Hochschild 3-cochain on T relative to I is an I-bimodule
map
p : r (x)_I r (x)_I r -> T,
p(a, b, c) in T,
zero on the augmentation ideal of small degrees, extended I-bilinearly.
Appendix A gives the complete table of coefficients: for each ordered
triple (a, b, c) of radical basis letters with composable corners, the
table lists the coefficient p(a,b,c)^v of each radical basis letter v
appearing in p(a, b, c). Coefficients not listed are zero. There are
179 nonzero entries. The example to keep in mind is
p(t, y, J) = q^3 f, p(t, x, J) = 0.
WEIGHT. Recall the weight from Section 3.2: lower-case letters have
weight 0, upper-case letters have weight 1. The cochain p lowers the
total weight by exactly one: in every nonzero entry, the number of
capital (upper-case) input letters is one more than the number of
capital output letters. This is the "weight -1" property. In
particular the output of p is a lower-case-weighted element of T,
compatible with its use as a coefficient map.
LEMMA 3.4 (THE COCYCLE AND ITS BOUNDARY IDENTITY). The cochain p is a
Hochschild 3-cocycle of T relative to I. Its associated cohomology
class, evaluated on the simple module s, is a nonzero element of
Ext_T^3(s, s). Moreover, for every H in k*, the following boundary
identity holds. Let h = h_H be the automorphism of T that fixes C and
multiplies DC by H. On the radical basis define
z_H(a) = H q a for a in {u, v, t, E, X, Y, Z, U, V},
z_H(a) = 0 otherwise,
and extend linearly. Then for all a, b in r,
SUM_{w in B_r} p(a, b, h(w)) w^* = a z_H(b) + z_H(ab) + z_H(a) h^{-1}(b),
where B_r is the radical basis of T (the 18 basis letters other than e
and f), and w^* is the trace-dual of w.
PROOF OF LEMMA 3.4. The endpoint, weight, and cocycle properties, as
well as the boundary identity, are finite checks against the table of
Appendix A: the cocycle identity is the coefficient of each v in the
normalized Hochschild differential of p, which involves only four
radical basis letters and can be checked on the 18^4 = 104976 radical
four-words (with most vanishing by the corners; exactly 15250 are
composable). The boundary identity is the coefficient of each v in
the quantity above, checked for the 18^2 = 324 input pairs. All of
these are polynomial identities in F_2[q], and are verified in
Appendix A. We prove the nonzero evaluation here.
EXPLICIT CYCLE. In the two-sided bar complex s^r (x)_T P (x)_T s,
consider
zeta = q^2 [t | x | J] + [t | y | J].
Both words have endpoint f on the left and on the right: t x = j has
right endpoint f, t y = q^2 j likewise, and J is the dual of j, whose
left endpoint is f. All outer radical actions on the endpoint simples
vanish. Applying the differential and using the multiplication table
together with the dual actions gives
t x = j, t y = q^2 j,
x J = T, y J = q^2 T, j J = F,
where the last set of identities is the defining value of the dual
basis letters on the basis. Then
d zeta = q^2 ( [j | J] + [t | T] ) + q^2 [j | J] + q^2 [t | T]
= 0,
where the four terms cancel in pairs: q^2 [j|J] + q^2 [j|J] = 0 and
q^2 [t|T] + q^2 [t|T] = 0 (in characteristic two, 2 = 0). So zeta is a
cycle.
EVALUATION ON zeta. The table gives p(t, x, J) = 0 and
p(t, y, J) = q^3 f. So the cochain p pairs with zeta to
q^2 . p(t, x, J) + p(t, y, J) = q^2 . 0 + q^3 f = q^3 f.
Because q is transcendental over F_2, q^3 != 0. The element f is a
basis vector, hence a nonzero element of T acting nonzero on s (f
acts by 1 on s). So the evaluated class is nonzero in
Hom_T(T (x)_T P (x)_T s, s) = Hom_k(s, s) = k.
A coboundary pairs to zero with every cycle; hence the class of p in
H^3 is nonzero on s. Every T-linear map from P (x)_T s to s factors
through the left simple quotient s (which is a summand of the
augmentation image), so the "two-sided-simple" bar complex
s^r (x)_T P (x)_T s computes the correct Ext group. This proves the
nonzero evaluation. QED.
THE COMPARISON MAP p : P -> P[3]. For later use (the resolution
calculation in Part IV and the lift construction in Part VI), we
extend the cochain to a chain map of the bar resolution. Define, for
n >= 0,
p_n : P_n -> P_{n-3},
a_0 [a_1|...|a_n] a_{n+1} |-> a_0 [a_1|...|a_{n-3}]
. p(a_{n-2}, a_{n-1}, a_n) . a_{n+1}
for n >= 3, and p_n = 0 for n = 0, 1, 2. The target P_{n-3} is
indexed so that P[-3]_n = P_{n-3}; so p : P -> P[-3]. We identify
P[-3] with P[3] using the opposite sifting convention, in which a
"map of degree 3" lowers the bar index by three.
This p is a chain map of the bar complex: the differential d on P_n
multiplies adjacent bar letters in turn, and the Hochschild cocycle
equation is exactly the statement that p_n d = d p_n for every n. More
explicitly, computing p d on the four adjacent multiplications at the
end of the bar string, the first two (multiplying a_{n-3} into the
first slot of p, or a_{n-2} a_{n-1} and a_{n-1} a_n) vanish termwise
after cancellation; the remaining two equations are precisely the
cocycle equation on p. So p is a genuine chain map P -> P[-3].
At the bottom index (n = 3) the target is P_0; the augmentation on
P_0 is the multiplication and p_3 composed with the augmentation
represents the class [p] in H^3 = Ext_T^3(s, s). Composing further
with the evaluation on s gives the nonzero class q^3 f above. So the
comparison map p is a bare-cocycle-level representative of a nonzero
degree-three self-extension of s.
That finishes Section 3.3. Part IV uses p to compute the full
self-extension algebra kk[tau] over T, then the tensor square
A = T (x) T and X = s (x) s.
PART IV. SELF-EXTENSIONS OF THE SIMPLE MODULE
============================================================================
This part computes the full self-extension algebra of the simple
module s over T. Section 4.1 computes a projective resolution of s
over C, the base algebra (Lemma 4.1). Section 4.2 passes to the
trivial extension T and shows Ext_T^*(s, s) = kk[tau] with tau of
degree three (Lemma 4.2). Section 4.3 takes the tensor square
A = T (x)_k T and the simple X = s (x)_k s, and shows
Ext_A^*(X, X) = kk[tau_1, tau_2], both generators of degree three
(Lemma 4.3).
Throughout this part, "s^r" denotes the right f-simple of C (or T)
of dimension one; it is the right module dual to the left simple s.
4.1 A RESOLUTION OF s OVER C
------------------------------------------------------------------------
Write ell_i = x + q^i y for every integer i >= 0. Since q is
transcendental over F_2, q^i != 0 and 1 + q^m != 0 for every positive
integer m. We will use the second condition for arbitrarily large
indices.
LEMMA 4.1 (THE RESOLUTION OVER C). A projective resolution R -> s over
C has terms
R_0 = C f, R_i = C e (i >= 1),
and differentials given by right multiplication:
... -> C e --(. ell_1)--> C e --(. ell_0)--> C e --(. u)--> C f -> s -> 0.
More precisely, d_{i+2} is right multiplication by ell_i for every
i >= 0. For every integer i >= 0:
Ext_C^i(s, s) = k if i = 0, 0 if i > 0,
Ext_C^i(s, C) = s^r if i = 2, 0 if i != 2 (i = 0, 1, 3, 4, ...).
The second identification is an identification of right C-modules.
PROOF. The left ideals C e and C f have respective bases
(e, x, y, z, t, j) and (f, u, v, n).
The multiplication table gives the following formulas, valid for every
i >= 0, on the basis of C when right-multiplied:
a | e x y z t j
-------------+------------------------------------
a u | u v v 0 (1 + q) n 0
a ell_i | ell_i q^{i+1} z z 0 (1 + q^{i+2}) j 0
Consequently, right multiplication by u has image equal to
<u, v, n> = rad(C f) and kernel equal to <ell_0, z, j> (where
ell_0 = x + y). Right multiplication by ell_i has image
<ell_i, z, j> and kernel <ell_{i+1}, z, j>. In the latter, a kernel
vector has zero e- and t-coefficients, and its x, y coefficients
satisfy a_y = q^{i+1} a_x, which is exactly the form a_{i+1} for the
scalar a_x. These identities prove exactness at every index.
Because Hom_C(C e, s) = e s = 0 and Hom_C(C f, s) = f s = k, and every
map R_i -> s for i >= 1 is zero on the projective module C e, the
self-extension groups over C are Ext_C^0(s, s) = k and Ext_C^i(s, s) =
0 for i > 0.
For Ext_C^*(s, C), identify Hom_C(C e, C) = e C and Hom_C(C f, C) = f C
by evaluation at the idempotent. Precomposition with right
multiplication by a is then given by LEFT multiplication by that same a
on e C or f C. So the Hom complex is
f C --(u .)--> e C --(ell_0 .)--> e C --(ell_1 .)--> e C -> ...
Left multiplication by u sends the basis (f, t, j, n) of f C to
(u, y + qx, z, v). These four vectors are independent: u, y + qx are in
degrees 1, 2, z is degree 4, v is degree 3, and each lives in a
distinct degree or corner combination. On the basis (e, x, y, z, u, v)
of e C, left multiplication by ell_i is
a | e x y z u v
-------------+------------------------------------
ell_i a | ell_i q^i z q z 0 (1 + q^i) v 0
The index i = 0 is exceptional because 1 + q^0 = 0 in F_2 (the
characteristic is two, so 1 + 1 = 0). In that case
ker(ell_0 .) = <q x + y, z, u, v> = im(u .),
im(ell_0 .) = <ell_0, z>.
For every i >= 1, the coefficient 1 + q^i is nonzero, and
ker(ell_i .) = <ell_{i-1}, z, v>,
im(ell_i .) = <ell_i, z, v>.
Cohomology in degrees 0 and 1 therefore vanishes (degree 0 vanishes
because the augmentation onto s is split by evaluation at f and the
first map is injective; degree 1 vanishes because ker(ell_0 .) =
im(u .)). In degree 2 the cohomology is the line represented by v; in
degree 3 and above the displayed kernel equals the preceding image, so
the cohomology vanishes.
The line represented by v is a right C-module: v f = v, v e = 0, and
the only possibly nonzero right multiplication is v t = q z, which is a
boundary (<z, j> lies in the image of ell_0). So the surviving line is
precisely s^r as a right C-module. This proves the identifications.
QED.
4.2 Ext_T(s, s) = kk[tau], tau of degree 3
------------------------------------------------------------------------
Write L = DC. The duality calculation of Lemma 4.1 determines
L (x)_C^{L} s, the derived tensor over C. We exploit the short exact
sequence of (T, C)-bimodules 0 -> L -> T -> C -> 0 and its connecting
morphism over T.
LEMMA 4.2 (TENSOR POLYNOMIAL). For the algebra T = C (x) DC and its
left f-simple s,
Ext_T^*(s, s) = kk[tau], |tau| = 3,
as a graded k-algebra, where tau = p(s) is the evaluation of the
cocycle p of Lemma 3.4 on s. Every power tau^m is nonzero and spans
Ext_T^{3m}(s, s); all other positive degrees vanish.
PROOF. For a finite projective left C-module V, evaluation gives a
natural isomorphism of left C-modules
DC (x)_C V -> D Hom_C(V, C), lambda (x) v |-> (phi |-> lambda(phi(v))).
Apply this termwise to the resolution R of Lemma 4.1. Exactness of
k-dualisation (D is exact on finite k-vector spaces) shows that
L (x)_C R = DC (x)_C R has homology only in degree two, where it is
D Ext_C^2(s, C) = D(s^r) = s. Moreover, L acts by zero on the entire
complex because L^2 = 0 in the trivial extension. Good truncation, i.e.
replacing the complex by its cohomology in degrees where cohomology is
concentrated, gives a T-linear derived identification
L (x)_C R ~ s[2] in D^b(T-mod).
Now regard 0 -> L -> T -> C -> 0 as a short exact sequence of
(T, C)-bimodules. Tensoring it (over C) termwise with R gives a short
exact sequence of complexes of left T-modules:
0 -> L (x)_C R -> T (x)_C R -> R -> 0.
The last complex is inflated along the algebra map T -> C. Exactness
uses only the fact that each left C-module R_i is free (hence
flat/projective) viewed as a right C-module; it does not require
flatness of L or of T as right C-modules. Put I = T (x)_C R.
We obtain the fundamental derived triangle over T:
s[2] --> I --> s --delta--> s[3].
Here delta is the connecting morphism of the short exact sequence. The
middle complex I has homology s in degrees zero and two, and no other
homology. Its terms are finite-dimensional, so good truncation makes
it a bounded complex of finite T-modules. Every term of I is a finite
projective T-module, since T (x)_C R_i is projective over T. Because I
is bounded below homologically, it computes its own derived maps: a map
from I to an acyclic complex vanishes up to homotopy by successive
projective lifts starting at its bottom degree. So for every integer i,
B^i := Hom_{D^b(T-mod)}(I, s[i])
= H^i Hom_C(R, s)
= k if i = 0,
0 if i != 0.
Set E^i := Hom_{D^b(T-mod)}(s, s[i]). These are ordinary Ext groups
Ext_T^i(s, s) for i >= 0, and vanish for i < 0 (s is bounded). Apply
the exact functor Hom(-, s[i]) to the triangle s[2] -> I -> s -> s[3].
The resulting exact segment is
B^{i-1} --> E^{i-3} --delta^*--> E^i --> B^i.
For i = 1, the groups E^{-2} and B^1 vanish, so E^1 = 0. For every
i >= 2, both displayed B-groups vanish, so delta^* : E^{i-3} -> E^i is
an isomorphism. This includes E^2 = 0 (from E^{-1} = 0), and at i = 3
it sends id_s to the nonzero class delta in E^3.
More precisely, if beta : s -> s[i-3] is a class, its image under this
connecting map is delta^*(beta) = beta[3] . delta. Thus the recurrence
above is multiplication by delta in the Yoneda algebra. Every power
delta^m is nonzero (since delta^*(delta^{m-1}) = delta^m, and E^{3m-1} = 0
for m >= 1, so no power can be zero and yet arise as an image of a
nonzero class), and spans E^{3m}. All other nonnegative degrees vanish.
We have therefore proved Ext_T^*(s, s) = kk[delta] as an algebra.
Lemma 3.4 shows that p(s) is nonzero in degree three: the simple-valued
cochain evaluates to q^3 != 0 on the cycle zeta = q^2[t|x|J] +
[t|y|J]. Because E^3 is one-dimensional, p(s) is a nonzero scalar
multiple of delta, hence is itself a polynomial generator. We set
tau = p(s). This gives the stated normalisation without a priori
choosing an isomorphism L (x)_C R ~ s[2] with a prescribed scalar. QED.
4.3 THE TENSOR SQUARE: X OVER A = T (x) T
------------------------------------------------------------------------
We now pass to the symmetric algebra and the simple used in the rest
of the paper:
A = T (x)_k T, X = s (x)_k s.
The tensor product of the symmetrising traces makes A symmetric, and
dim_k A = 20 . 20 = 400. Let p_1 and p_2 be the external copies of the
bimodule class p on the two tensor factors of A, and write
tau_i = p_i(X) for the evaluations on X.
LEMMA 4.3 (TENSOR SQUARE). The simple left A-module X is nonprojective,
and
(DIRECT SUM)_{a >= 0} stHom_A(X, X[a]) = kk[tau_1, tau_2],
|tau_1| = |tau_2| = 3.
The multiplication is composition with shifts, and its positive-degree
part agrees with the ordinary Ext algebra Ext_A^*(X, X). In degree zero,
stHom_A(X, X) = End_A(X) = k.
PROOF. Tensor two projective T-resolutions of s over k. Explicitly,
the total tensor product P (x)_k P of the bar resolution P (of Part III)
with itself: in each total degree the direct sum is finite (terms P_i
(x) P_j with i + j = n), so this defines a resolution of X by finite
projective A-modules. The Hom complex over A is the total tensor
product over k of the two Hom complexes over T:
Hom_A(P (x) P, P (x) P) = Hom_T(P, P) (x)_k Hom_T(P, P).
Splitting complexes of vector spaces gives the Kunneth isomorphism.
Tensoring the comparison maps identifies multiplication as well; the
external maps commute in characteristic two (no Koszul sign). Lemma 4.2
therefore gives
Ext_A^*(X, X) = Ext_T^*(s, s) (x)_k Ext_T^*(s, s)
= kk[tau_1, tau_2].
In particular X has a nonzero positive self-extension (tau_1 of degree
three), so X is not projective. Its ordinary endomorphism ring is
End_A(X) = k, because X is one-dimensional over k: the only A-linear
maps X -> X are scalar multiples of the identity.
The identity of X cannot factor through a projective A-module: such a
factorization would split the identity as X -> P -> X with P finite
projective, making X a direct summand of P, hence projective, contrary
to what we just proved. So the stable endomorphism ring of X is also k.
Positive stable Hom agrees with Ext over the symmetric algebra A
(Section 2.1), giving the stated identification. QED.
We have obtained two polynomial classes tau_1, tau_2 acting on X, both
of degree three, together with specific bimodule-level representatives
for them (the maps p_1 and p_2). Part V now uses the two cones of these
classes to build a bimodule with only two surviving stable Hom groups.
PART V. THE TWO CONES
============================================================================
This part uses the two degree-three classes tau_1, tau_2 of Part IV to
construct an A-bimodule Ccal, projective on each side, with the
following striking property: the stable groups stHom_A(X, Ccal (x)_A
X[a]) vanish for every integer a except a = 0 and a = 3, where each is
one-dimensional. Section 5.1 records the stable duality used to
control negative degrees. Section 5.2 builds Ccal as the tail of an
ordinary cone complex and exhibits a finite side-projective
representative. Section 5.3 computes the Hom profile and identifies
the degree-three group with a specific line (the "top projection").
SETUP. Put
A = T (x)_k T, X = s (x)_k s, R = A^e = A (x)_k A^op.
The algebras A and R are symmetric: the symmetrising forms are the
tensor products of the form lambda of Lemma 3.3. Write
H^a = stHom_A(X, X[a])
for the stable self-extensions of X. Lemma 4.3 gives
H^{>=0} = kk[tau_1, tau_2], |tau_1| = |tau_2| = 3.
In particular H^0 = End_A(X) = k, and the degree-three pieces are
one-dimensional. Since X is one-dimensional over k, its stable
endomorphism ring is exactly k (the identity is nonzero in the stable
category because X is nonprojective, Lemma 4.3).
5.1 STABLE DUALITY FOR SYMMETRIC ALGEBRAS
------------------------------------------------------------------------
Let G be a finite-dimensional symmetric k-algebra with char k = 2.
Let D = Hom_k(-, k) be the k-dual. For finite left G-modules M, N the
following duality of stable categories holds, naturally in the second
variable N (Linckelmann 2013):
D stHom_G(M, N) = stHom_G(N, M[-1]). (DUALITY)
PROOF. Choose a complete resolution Q of M over the symmetric algebra
G. So M = coker(Q_1 -> Q_0) and Q is an exact complex of finite
projectives. For a finite projective Q_i there is an isomorphism
D Hom_G(Q_i, N) = Hom_G(N, Q_i).
It suffices to check this for Q_i = G, where it is the isomorphism
D G = G (trace-dual pairing, nondegeneracy of the symmetrising form),
and then to pass to finite sums and summands. The isomorphism is
natural in N ∈ stmod(G).
Consequently, dualising the cochain complex Hom_G(Q, N) gives the
chain complex Hom_G(N, Q). Its cohomology at degree zero is maps into
ker(Q_0 -> Q_{-1}) = Omega M, modulo maps through Q_1 = the image of
Q_1 in Q_0. The projective factorizations are exactly the maps from
Omega M to N that lift through Q_1 -> Omega M (a surjection of finite
modules). Vector-space duality is exact on finite k-vector spaces, so
the identification descends to the stable category. This proves
(DUALITY). QED.
APPLICATION TO X. Substituting M = X and N = X[a] gives
D H^a = D stHom_A(X, X[a])
= stHom_A(X[a], X[-1])
= stHom_A(X, X[-1 - a])
= H^{-1-a}.
So D H^a = H^{-a-1} for every integer a. In particular the negative
stable self-extensions of X are exactly the k-duals of the nonnegative
ones.
GRADED STRUCTURE. Let P_m denote the degree-m part of the polynomial
ring kk[tau_1, tau_2], where each variable has degree one. So
P_m = 0 for m < 0, P_0 = k, P_1 = k tau_1 (+) k tau_2, and so on,
with dim_k P_m = m + 1. Combining with D H^a = H^{-a-1} and the
computation H^{3m} = P_m:
H^{3m} = P_m for every m >= 0,
H^{-3m-1} = D P_m for every m >= 0,
all other H^a = 0.
ACTION OF THE CLASSES ON NEGATIVE DEGREES. Postcomposition with
tau_i[a] : X[a] -> X[a+3] gives multiplication on the right:
H^a -> H^{a+3}, g |-> g tau_i[a].
By naturality of (DUALITY), the transpose of this map, after shifting
the source back to X, is a map
H^{-a-4} -> H^{-a-1}, g |-> g[3] tau_i,
i.e. left multiplication by tau_i. When a <= -4, this map is exactly
multiplication by tau_i in the polynomial ring, dualised. So the
action of the classes on negative degrees is the dualisation of
polynomial multiplication.
There is NO nonzero product crossing from negative degree into
nonnegative degree. The only negative degree that admits a
multiplication by tau_i into degree >= 0 is H^{-1} (multiplying by
tau_i lands in H^2); but H^2 = 0 by the computation above. Therefore no
such cross products exist. We record this because it enters the
Koszul argument in 5.3.
5.2 FINITE BIMODULES FROM PROJECTIVE TAILS
------------------------------------------------------------------------
For a proof of the Hom profile in 5.3 we need the two-cone bimodule to
have a finite representative that is projective on each A-side. The
following tail construction produces one.
DEFINITION 5.1 (TAIL). Let V be a bounded-below complex of finite
projective R-modules that is exact in all sufficiently high degrees.
For N much larger than the top nonzero degree of the homology of V,
define
<V> = (coker(V_{N+1} -> V_N))[N] in the stable category.
This does not depend on N: writing V_N^c for this cokernel, exactness
gives
0 -> V_{N+1}^c -> V_N -> V_N^c -> 0,
so V_{N+1}^c ~ V_N^c[-1] stably. A chain map defined in sufficiently
high degrees induces a map on these cokernels, compatibly with
increasing N and with shifts. A chain homotopy changes the bottom
induced map only by a factorization through a projective, so homotopic
tail maps give the same stable map.
LEMMA 5.2 (SIDE-PROJECTIVE TAIL). Let p : P -> P[-3] be the comparison
map of Section 3.3, viewed as a chain self-map of the bimodule bar
resolution P increasing the homological degree by 3 (equivalently,
lowering the bar index by 3). Form the ordinary cone complex
K = Cone(p : P[-3] -> P) (x)_k Cone(p : P[-3] -> P).
Then:
(a) K has finite projective R-terms and is exact in degrees > 0.
(b) For every N >= 1, C_N = coker(K_{N+1} -> K_N) is finite and
SIDE-PROJECTIVE (projective as a left A-module and as a right
A-module).
(c) Consequently Ccal = <K> has a finite side-projective module
representative in the stable category of R-modules.
The same conclusions hold for the first two filtration levels of K.
REMARK ON SIDE-PROJECTIVITY. A finite stable shift of a finite
side-projective bimodule has a finite side-projective representative:
for a syzygy, take a finite R-projective surjection, whose kernel
sequence splits on each side because the quotient is side-projective;
for a cosyzygy, take a finite R-projective-injective embedding, whose
cokernel sequence splits on each side by symmetry of A. A stable map
between side-projective bimodules has a finite side-projective cone:
represent by f, embed the source into a finite R-projective I, and take
the cone as the cokernel of (f, i).
PROOF OF LEMMA 5.2. Write p for the comparison map of Section 3.3 and
P^tot = P (x)_k P for the external tensor, which resolves the regular
A-bimodule A (tensor of two bar resolutions of T, together, resolves
T (x) T = A). In each total degree there are only finitely many
summands, each an external tensor of projective bimodules, hence
projective over R. So P^tot is a complex of finite projectives.
The cone Cone(p : P[-3] -> P) has two "cells": P at degree 0 and P[-3]
shifted. Taking the external tensor of the two cones gives four cells
of K, labelled by subsets S of {1, 2}. The convention: i in S means the
shifted cell (the P[-3] cell) is chosen in the i-th tensor factor. So
cell S is P^tot[-2|S|], and its differential to cell S \ {i} is the
shifted comparison map p_i (the i-th external copy of p). Filtering by
|S| gives the three associated graded levels
LEVEL 2: P^tot,
LEVEL 1: (P^tot[-2])^{(+) 2},
LEVEL 0: P^tot[-4].
The cell cohomology is supported at degrees 0, -2, -4 respectively.
The filtration long exact sequences therefore show that K is exact in
all positive degrees.
For N >= 1, the cokernel filtration is exact by the preceding
observation. Its graded factors, with multiplicities 1, 2, 1 for
r = 0, 1, 2, are
coker( P^tot_{N + 2r + 1} -> P^tot_{N + 2r} ), r = 0, 1, 2.
The augmented resolution of A (which is P^tot) splits on either A-side,
because the regular module is projective; inductively all its syzygies
are projective on each side. So these filtration factors are
side-projective. Their extensions (the multiple copies in level 1)
also split on each side, proving that C_N is side-projective. Finally
C_N[N] has a finite side-projective representative by the shift
construction above. In particular one may take N = 1 and one finite
cosyzygy of C_1 to keep all dimensions bounded. QED.
EVALUATION ON X. Forming the external tensor of ordinary complexes
gives all four cells of K as projective resolutions of shifts of the
regular A-bimodule. We pass to their stable representatives only after
this construction.
Evaluation on the finite left module X means tensoring over A on the
right. Tensoring with a side-projective bimodule is an exact functor
of left A-modules and preserves projectives. Every R-projective
bimodule evaluates to an A-projective module (a free bimodule
A (x)_k V evaluates to the free module A (x)_k V; sums and summands
preserve projectivity). For evaluation at a fixed module X, the
right-split sequences used above remain exact, so their evaluations
preserve the stable shifts and the long exact Hom sequences used in
the construction. Consequently the filtration of K, after evaluation
on X, has stable cells
X, X[-2]^{(+) 2}, X[-4],
and the adjacent connecting maps are the shifted actions of tau_1 and
tau_2: they are induced by the original bimodule comparison maps p_1,
p_2 tensored with X. This specifies the attaching maps whose Hom
groups we compute next.
Part V continues in 5.3, where the Hom profile of Ccal (x)_A X is
computed.
5.3 THE HOM PROFILE AND ITS TOP PROJECTION
------------------------------------------------------------------------
We now compute the stable Hom groups from X into (shifted) evaluations
of the two-cone bimodule. Recall from 5.2 that Ccal = <K> is the
stable bimodule represented by the tail of K, and that the first two
filtration levels of K, evaluated on X, have the form
X, X[-2]^{(+) 2}, X[-4]
with adjacent connecting maps given by tau_1 and tau_2. Define, for
every integer a,
W^a = stHom_A( X, (Ccal (x)_A X)[a] ).
PROPOSITION 5.3 (HOM PROFILE). For the finite side-projective bimodule
Ccal of Lemma 5.2 the groups W^a are
W^a = k if a = 0 or a = 3,
W^a = 0 for every other integer a.
Moreover the projection to the top cell induces an identification
W^3 -> H^{-1}, (TOP PROJECTION)
which is an isomorphism of one-dimensional k-vector spaces.
PROOF. Let Ccal_{<=1} denote the stable bimodule representing the
first two filtration levels of K. It has the evaluation
Ccal_{<=1} (x)_A X = (top cell X)
+ (two middle cells X[-2])
together with the attaching maps tau_1, tau_2 into the top cell. In
cohomological degree a, the stable Hom groups of the three associated
graded levels are
B^a = H^a, (top cell, shift 0)
D^a = (H^{a-2})^{(+) 2}, (middle cells, shift -2 each)
E^a = H^{a-4}. (bottom cell, shift -4)
The adjacent connecting stable maps are, with the coordinate order
that is visible below,
d_a : D^a -> B^{a+1}, d_a(u, v) = tau_1 u + tau_2 v,
d'_a : E^a -> D^{a+1}, d'_a(z) = (tau_2 z, tau_1 z).
The entry order in d'_a records that deleting the first slot from the
full cell leaves the second singleton, and deleting the second leaves
the first. So the entries are (tau_2, tau_1), not (tau_1, tau_2).
KOSZUL SEQUENCE. For every m >= 0 there is a degreewise exact Koszul
sequence
0 -> P_{m-1} --(tau_2, tau_1)^t--> P_m^{(+) 2}
--(tau_1, tau_2)--> P_{m+1} -> 0, (KOSZUL)
where P_j = 0 for j < 0. Here P_m is the degree-m part of
kk[tau_1, tau_2] (each variable degree one). Exactness:
* the row (tau_1, tau_2) is surjective, because every positive-
degree monomial is divisible by one of the two variables;
* the column (tau_2, tau_1)^t is injective, because kk[tau_1,tau_2]
is a domain;
* a relation tau_1 u + tau_2 v = 0 forces (u, v) = (tau_2 w,
tau_1 w) by coprimality of the two variables.
On the negative-degree groups, dualize (KOSZUL) with D and interchange
its two middle components. By the action calculation of 5.1 (the
action of the classes on negative degrees is the dualisation of
polynomial multiplication), this gives exactly the sequence whose
middle term is D^a, in every negative degree. At the "seam" degree a = 2
the terminal zero space is the actual group H^2 = 0 (not DP_0 = k). So
we obtain, in all degrees:
ker d_a = im d'_{a-1},
coker d_{a-1} = k if a = 0, 0 if a != 0,
ker d'_a = k = H^{-1} if a = 3, 0 if a != 3.
Here "ker d'_a" at a = 3 uses the seam: for m >= 1 the maps of the
column of (KOSZUL) after dualisation are dual to a surjective row and
hence injective, and the remaining source is D P_0 = H^{-1}, whose two
products (by tau_1, tau_2) vanish in H^2 = 0.
FIRST FILTRATION LES. Write K_1^a = stHom_A(X, (Ccal_{<=1} (x)_A X)[a]).
The first filtration long exact sequence, from the sub-bimodule with
just the top and middle cells, gives, for every integer a,
0 -> coker d_{a-1} -> K_1^a --q_a--> ker d_a -> 0. (LES1)
The top-cell connecting map is r_a : E^a -> K_1^{a+1}, and its
projection satisfies q_{a+1} r_a = d'_a (by definition of d'_a).
Thus ker r_a is contained in ker d'_a, which by the Koszul consequences
can be nonzero only when a = 3. At that index K_1^4 = 0 by (LES1),
since D^4 = (H^2)^{(+)2} = 0, so the inclusion ker r_3 ⊆ E^3 = H^{-1}
is an equality: ker r_3 = E^3 = H^{-1}.
Also, im r_{a-1} surjects onto ker d_a by the Koszul consequences. When
a != 0, the left term of (LES1) vanishes, so r_{a-1} is surjective.
When a = 0, (LES1) gives K_1^0 = k (from coker d_{-1} = k), whereas the
only incoming source is E^{-1} = H^{-5} = 0. Therefore
coker r_{a-1} = k if a = 0, 0 if a != 0.
SECOND FILTRATION LES. The second filtration long exact sequence, from
the sub-bimodule of the bottom cell inside Ccal, gives
0 -> coker r_{a-1} -> W^a -> ker r_a -> 0. (LES2)
Combining the two previous displays: coker r_{a-1} = k at a = 0 and 0
otherwise; ker r_a = H^{-1} at a = 3 and 0 otherwise. So W^0 = k,
W^3 = k, and W^a = 0 for every other a.
TOP PROJECTION. At a = 3, the left term of (LES2) vanishes
(coker r_2 = 0 since 3 != 0), and the right term is all of
E^3 = H^{-1}. So the second map in (LES2), which is precisely the
projection to the top cell, is an isomorphism W^3 -> H^{-1}. This is
(TOP PROJECTION). QED.
REMARK 5.4. The exact sequences (LES1) and (LES2) and the Koszul
identities pin down the complete filtration, including attaching maps
not visible from adjacent components alone; K_1^a and W^a are
determined degree by degree.
CONSEQUENCE FOR THE CONSTRUCTION. Proposition 5.3 gives the "two-
scalar profile": after shifting the two-cone bimodule by three, the
only target for a stable map from X into Ccal (x)_A X [3] is the top
cell of Ccal, where it lands in the one-dimensional group H^{-1}. Any
nonzero class in W^3 is detected by the projection to the top cell,
equivalently by the projection to X[-1].
This is exactly the input needed in Part VI: we construct maps
f_H : U_H -> Ccal[3] whose projection to the top cell is a specific
nonzero class in H^{-1}, and use the inverse of (TOP PROJECTION) to
guarantee that f_H itself (as an honest stable bimodule map) is
nonzero.
That finishes Part V.
PART VI. THE TWO TWISTS AND THE FINITE FIBER
============================================================================
This part supplies the input (A, F, X, v) to Proposition 2.4. Section
6.1 constructs two stable bimodule maps f_H : U_H -> Ccal[3] with a
prescribed top-cell projection, one for each twist parameter H. Section
6.2 assembles these two maps into the finite side-projective fiber F
and shows that the fibre carries the diagonal stable map
v : X -> F (x)_A X. Section 6.3 computes the twist eigenvalues and
proves invertibility of the delta maps, i.e. hypotheses (C1) and (C2) of
Proposition 2.4.
Throughout, A = T (x)_k T and X = s (x)_k s are as in Part V. Recall
the comparison map p : P -> P[3] of Section 3.3, where P is the
normalized relative bar resolution of T; as a chain map this lowers the
bar degree by three, p_n : P_n -> P_{n-3}.
6.1 TWISTED BIMODULES AND THE LIFT f_H
------------------------------------------------------------------------
For a nonzero scalar H in k, let h = h_H be the automorphism of T that
fixes C and multiplies DC by H. Define the twisted bimodule T_h by
usual left action on T,
right action x . a = x h(a), a in T.
Likewise, the right-twisted bar resolution Q = P_h: its underlying
complex is P, but the right action is twisted by h.
The algebra A twisted by the parameter H is
U_H = A_{h (x) h},
i.e. A with its usual left A-action and right action twisted by
h (x) h (componentwise on the two T-factors). We will take two copies
U_1 = U_{H_1}, U_2 = U_{H_2} at the end of 6.2. Meanwhile Q^tot = Q (x)_k Q
is the external tensor of the right-twisted resolution, and is a
right-twisted bar resolution of U_H by two factors.
EXTENSION FORMULA. A bar-map on the resolution extends to a right-
linear map of bimodules by
psi( a_0 [a_1|...|a_i] a_{i+1} ) = a_0 psi([a_1|...|a_i]) h^{-1}(a_{i+1}),
where the displayed terminal coefficient on the right is its
underlying element of T. This convention makes psi right-linear from
the twisted source to an untwisted target.
PROPOSITION 6.1 (THE LIFT). For each H in k* there is a stable
bimodule map
f_H : U_H -> Ccal[3]
whose composite with the top-cell projection Ccal[3] -> A[-1] is
represented by the bimodule map beta_H : U_H -> Omega A subset P_0^tot
determined by
beta_H(1) = ( SUM_w h(w) (x)_I w* ) (x) ( SUM_w h(w) (x)_I w* ),
where w ranges over the letter basis of T, h(w) is the image under h,
and w* is the trace-dual letter.
Under the canonical identification U_H (x)_A X = X, its projection after
evaluation sends
1 |-> H^2 (f* (x) f*) in Omega X = rad(A) (f (x) f) subset A (f (x) f).
The evaluated map is nonzero modulo stable maps factoring through a
projective A-module. In particular f_H X is nonzero in
stHom_A(X, Ccal (x)_A X [3]).
6.1.1 A twisted trace-dual tensor
------------------------------------------------------------------------
For basis letters w, w* of T, set in P_0 = T (x)_I T
xi_H = SUM_w h(w) (x)_I w*.
The coevaluation identity a SUM_w w (x) w* = SUM_w w (x) w* a (Broue
2009), with h on the first factor and projected over I, gives
h(a) xi_H = xi_H a; hence b(x) = x xi_H is a bimodule map U -> P_0.
We need b to land in ker pi where pi : P_0 -> T is multiplication.
The products w w* and w* w are the duals of the left and right vertex
idempotents of w (evaluate at positive-degree elements by the grading,
and at the idempotents). Since C has 6 elements with left vertex e and
4 with left vertex f (and similarly on the right), both sums are
6e* + 4f* = 0 in characteristic 2, so
pi xi_H = (6e* + 4f*) + H (6e* + 4f*) = 0.
The same calculation on each left vertex r in {e, f} gives
SUM_{w in rad, left(w)=r} h(w) w* = r*,
because the full radical sum vanishes and the omitted idempotent term
is r r* = r*.
Take the external square:
beta_H = b (x)_k b : U_H -> ker( P_0^tot -> A ) = Omega A,
where P^tot = P (x)_k P; the image lands in the kernel because
pi b = 0.
6.1.2 Two homotopies on the bar resolution
------------------------------------------------------------------------
Let D_b : Q -> P be the map that is b epsilon in degree zero (where
epsilon : Q_0 -> U is the augmentation) and zero in every positive
degree. It is a chain map. A homotopy for D_b will supply a comparison
map representing beta_H in the top cell.
Define maps B_i : Q_i -> P_{i+1} for every i >= 0 by
B_i( [a_1|...|a_i] ) = SUM_{w a basis letter of rad} [ a_1|...|a_i | h(w) ] w*,
retaining only composable terms and extending by the extension formula.
LEMMA 6.2 (dB + Bd = D_b). With B_{-1} = 0, one has
d B + B d = D_b
on the nonnegative resolutions.
PROOF. For a in rad with right vertex r, project the first factor of
a xi_H = xi_H h^{-1}(a) to rad along I. Removing the idempotent terms
leaves
SUM_{w in rad} a h(w) (x) w* + SUM_{w in rad} h(w) (x) w* h^{-1}(a)
= a (x) r*,
with the appropriate vertex restrictions in each sum. On the left side
of the original identity the omitted term is a (x) r*; on the right
side the omitted first factors lie in I and disappear under the
projection.
For i > 0, expand d B_i + B_{i-1} d on a bar generator. All
multiplications strictly before the old final bar cancel in pairs. The
two terms adjacent to the old final letter have sum [a_1|...|a_i] r*,
by the projected Casimir identity just proved. The last multiplication
in d B_i is exactly the same expression, by the vertex identity
SUM_{left(w)=r} h(w) w* = r*, so these terms also cancel.
At degree zero, on the generator r[ ]r, the differential gives
d B_0(r[ ]r) = SUM_{left(w)=r} h(w)[ ]w* + r[ ] SUM_{left(w)=r} h(w) w*.
The second term is r[ ] r* by the vertex identity. It restores the
omitted idempotent term of the full tensor xi_H; the result is
r xi_H = b epsilon(r[ ]r). Bimodule linearity proves the identity on
every degree-zero element. QED.
We now use B to represent the prescribed top projection. In K[3], the
cell indexed by {1, 2} is P^tot[-1]. Put
V = B (x) D_b,
which raises the underlying total degree by one, as the shift requires.
Lemma 6.2 gives
d V + V d = (b (x) b) epsilon^tot,
where the right-hand side is supported in degree zero. Thus
d V_0 = beta_H epsilon^tot_0, and at positive indices V satisfies the
comparison equation. The complex with terms P^tot_{i+1} and
augmentation to Omega A resolves Omega A, so V is a comparison map
lifting beta_H.
EXTENSION TO K[3]. To extend V from the top cell to K[3], we must
account for the two attaching maps. Deleting slot 1 contributes
p B (x) D_b to the cell indexed by {2}. The suffix comparison
p : P -> P[3] of Section 3.3 lowers the underlying bar degree by three,
so p D_b = 0 (D_b is concentrated in bar-degree zero). Hence the
contribution from deleting slot 2 is zero. It remains to cancel
p B (x) D_b by a component in cell {2}. Lemma 6.2 shows that p B : Q -> P[2]
is a chain map; the additional boundary identity (Lemma 3.4) supplies
its nullhomotopy. Cocycle closure alone would not suffice.
LEMMA 6.3 (pB = dG + Gd). There are bimodule maps G_i : Q_i -> P_{i-1}
for all i >= 0 such that
p B = d G + G d.
PROOF. Let z' : Q_1 -> T be the cochain given by z'([a]) = z_H(a) from
Lemma 3.4, with the extension formula. Its boundary on [a|b] is
a z'(b) + z'(a b) + z'(a) h^{-1}(b).
By the boundary identity for p (Lemma 3.4), this equals
SUM_{w in rad} p(a, b, h(w)) w*,
which is the augmentation of (p B)_2. Hence
pi (p B)_2 = z' d : Q_2 -> T. (BOUNDARY)
Set G_0 = 0 and choose G_1 : Q_1 -> P_0 with pi G_1 = z'. This is
possible because Q_1 is projective and pi is surjective. By (BOUNDARY),
the map (p B)_2 + G_1 d has image in ker pi = im(d : P_1 -> P_0).
Projectivity of Q_2 therefore supplies G_2 with
d G_2 = (p B)_2 + G_1 d.
Suppose i >= 3 and the homotopy equation has been established below i.
Since p B is a chain map,
d( (p B)_i + G_{i-1} d ) = (p B)_{i-1} d + ( (p B)_{i-1} + G_{i-2} d ) d = 0.
Exactness of P in positive degree and projectivity of Q_i lift this
map through d : P_{i-1} -> P_{i-2}, giving G_i. This proves the equation
for every i >= 2. At i = 0, 1 both sides are zero, since pB has
negative target index and G_0 = 0. QED.
6.1.3 The tail map and its evaluation
------------------------------------------------------------------------
The homotopy G supplies the required correction in the singleton cell
{2}, whose shift in K[3] is 1. Together with the top component V it
defines a graded map Phi : Q^tot -> K[3]:
Phi_{1,2} = B (x) D_b,
Phi_{2} = G (x) D_b,
Phi_{1} = Phi_empty = 0.
The second component lowers the underlying degree by one, as the
singleton shift requires; the first is the comparison V constructed
above.
In the top cell, the chain defect d Phi + Phi d is D_b (x) D_b. In cell
{2}, deletion of slot 1 gives p B (x) D_b, canceled by
(dG + Gd) (x) D_b. Deleting slot 2 from either nonzero component gives
zero because p D_b = 0. Thus the only defect is D_b (x) D_b, supported in
total source degree zero. The map therefore commutes with the
differential in every positive total degree. Passing to high cokernels,
at a common index at least 4 as in Lemma 5.2, gives the stable bimodule
map f_H : U_H -> Ccal[3]. The top component of f_H is still V, so the
resulting stable projection to the top cell is beta_H, as identified
above.
LEMMA 6.4 (EVALUATION). Under U_H (x)_A X = X, the evaluated map
beta_H X sends 1 to H^2 (f* (x) f*) in Omega X subset A (f (x) f). It
does not factor through a projective A-module.
PROOF. In one tensor slot, tensoring xi_H (of the Casimir construction)
with the simple s leaves only the term whose right coefficient is f.
It is the term w = f*, so its value is H f*. Taking the external square
gives the stated value H^2 (f* (x) f*).
The projective cover sequence stays exact on evaluation because it
splits on the right. Its evaluated degree-zero term is A (f (x) f) =
P_0^tot (x)_A X, the projective cover of X, with kernel
rad(A) (f (x) f).
Suppose a map u : X -> rad(A)(f (x) f) factors through a projective
module. Then it factors through a finite free module. Every component
of a map X -> A has image in the left socle, since rad(A) X = 0. In a
symmetric algebra the left socle also annihilates the radical on the
right: if rad(A) u = 0, then for j in rad(A) and a in A,
tr(a u j) = tr(j a u) = 0,
so nondegeneracy of the trace gives u j = 0. A map A -> rad(A)(f (x) f)
is right multiplication by its value at 1, which lies in rad(A), and
hence kills the left socle. Every such factorization is therefore zero.
Our displayed map has the nonzero value H^2 (f* (x) f*) (because H is
invertible and f* is a nonzero basis letter, and the external square is
nonzero), proving the claim. QED.
By Proposition 5.3, the top-cell projection is an isomorphism on the
degree-three group W^3. Lemma 6.4 therefore proves that f_H X is
nonzero, completing Proposition 6.1. We now take the two values
H = H_1, H_2 and combine them into the finite fiber in Section 6.2.
6.2 THE FINITE SIDE-PROJECTIVE FIBER F
------------------------------------------------------------------------
We now combine the two lifts f_{H_1}, f_{H_2} into a finite bimodule F
that will be the side-projective fiber of Proposition 2.4.
FINITE REPRESENTATIVE OF Ccal[3]. Write R = A^e. From Lemma 5.2 we have
the finite bimodule
C_1 = coker( K_2 -> K_1 ), Ccal = C_1[1]
in the stable category of R-modules. To fix actual finite
representatives of its shifts, for a finite left R-module M use the
injection
j_M : M -> I_R(M) := Hom_k(R, M), j_M(m)(t) = t m,
(r phi)(t) = phi(t r).
Evaluation at 1 proves j_M is injective. Since R is symmetric,
I_R(M) = D R (x)_k M = R (x)_k M as a left R-module, so I_R(M) is
finite projective. Put Sigma_R M = coker j_M. When M is projective on
each A-side, the sequence defining Sigma_R M splits on each side: A is
self-injective, so M is injective on either side. Thus Sigma_R M is
again side-projective and represents M[1] in the stable category.
In particular, fix the actual finite bimodule
Y = Sigma_R^4 C_1.
It is projective on both sides and represents Ccal[3]. This fixes one
target for both lifts of Proposition 6.1. Concretely, the tail map
there at index 4 induces Omega_R^4 U_H -> C_1, because
coker( K[3]_5 -> K[3]_4 ) = coker( K_2 -> K_1 ).
Taking four cosyzygies gives the stable map U_H -> Y, using
Sigma_R^4 Omega_R^4 U_H = U_H stably. It therefore has an actual
bimodule-map representative between these finite modules.
TWO TWISTS. For i = 1, 2, put
sigma_i = h_{H_i} (x) h_{H_i}, U_i = A_{sigma_i}.
So U_i has its usual left action and right action u . a = u sigma_i(a).
Each U_i is free of rank one on both A-sides. The canonical
identification U_i (x)_A X = X uses the fact that sigma_i fixes the
simple character of X (h_{H_i} fixes the C-part, and s is the f-simple
of C, on which the DC part acts by zero).
RESCALING. By Lemma 6.4 the top projections of the evaluated lifts are
H_i^2 beta_0 for one fixed nonzero class beta_0 in
stHom_A(X, X[-1]) = H^{-1}. By Proposition 5.3 the top projection
W^3 = stHom_A(X, Ccal (x)_A X [3]) -> stHom_A(X, X[-1]) = H^{-1}
is an isomorphism. Rescale the two lifts by H_i^{-2}, and choose actual
representatives g_i : U_i -> Y of these rescaled maps. Their
evaluations are the same nonzero element w in W^3.
THE FIBER. Take the finite free R-module Q = R (x)_k Y and its
surjection pi : Q -> Y, pi(r (x) y) = r y. Define F by the exact
sequence
0 -> F -> U_1 (+) U_2 (+) Q --(g_1, g_2, pi)--> Y -> 0,
and let rho_i : F -> U_i be the two projections (i = 1, 2). This is a
construction of finite modules: C_1 is a cokernel of two finite terms,
four finite cokernels give Y, and one finite kernel gives F. Since Y
is projective on each side, the sequence splits on each side.
Consequently F is projective as both a left and a right A-module:
F is a direct summand of (U_1 (+) U_2) (+) Q on each side, and each of
U_1, U_2, Q (= R (x)_k Y = A (x)_k Y as an A-bimodule) is projective on
both sides.
EVALUATED TRIANGLE. Tensoring the exact sequence above with X preserves
exactness (F is right projective). Furthermore Q (x)_A X is projective
over A: evaluation of a free A^e-module R = A (x)_k A against A gives
A (x)_k (A (x)_A X) = A (x)_k X, a free A-module, and evaluation
preserves direct summands. Thus the evaluated stable triangle is
FX -> X (+) X --(w, w)--> Ccal (x)_A X [3] -> FX[1]. (TRIANGLE)
Here FM = F (x)_A M. Tensoring with F is exact because F is right
projective; it preserves projectives because F is left projective. It
therefore acts on stable maps compatibly with shifts. The actual maps
rho_i give natural transformations from this functor to tensoring with
U_i.
LEMMA 6.5 (THE DIAGONAL v AND THE POSITIVE PROJECTION). Put
V^a = stHom_A(X, F (x)_A X [a]) for each integer a. Then there is a
unique v in V^0 whose two projections rho_1 v, rho_2 v are both id_X,
and it spans V^0. For every integer a > 0, projection induces an
isomorphism
(rho_1[a], rho_2[a]) : V^a -> (H^a)^{(+) 2}, (POSPROJ)
where H^a = stHom_A(X, X[a]).
PROOF. Apply stable Hom from X to (TRIANGLE). The result is the exact
segment
(H^{a-1})^{2} -> W^{a+2} -> V^a -> (H^a)^{2} -> W^{a+3}. (SEG)
At a = 0: the first relevant group is W^2 = 0, and the last map is
k^2 -> W^3, (c, d) |-> (c + d) w. This is surjective (send 1 to w with
c + d = 1 in F_2, e.g. (1, 0) or (0, 1)). So V^0 identifies with the
diagonal line {(c, d) : c + d = 0} = {(c, c)} in k^2. This line is
spanned by v with rho_1 v = rho_2 v = id_X.
At a = 1: the preceding map (H^0)^2 -> W^3 is the same surjection (with
H^0 = k). It kills the potential connecting term from W^3 to V^1, and
W^4 = 0. So (POSPROJ) is an isomorphism in degree 1 as well.
For a >= 2: both W^{a+2} and W^{a+3} vanish by Proposition 5.3, giving
(POSPROJ) directly. QED.
We have obtained an actual finite bimodule F and a nonzero stable map
v : X -> F X with rho_i v = id_X for i = 1, 2. It remains to compare
tensoring an extension with F against composing that extension with v.
6.3 THE TWIST EIGENVALUES AND THE COMPARISON
------------------------------------------------------------------------
LEMMA 6.6 (TWIST EIGENVALUE). For H in {H_1, H_2}, tensoring with
U_H = A_{h (x) h}, under its canonical identification on X, acts on the
whole space H^{3m} as multiplication by H^{-m}, for every integer
m >= 0.
PROOF. The inverse automorphism in the exponent comes from the
right-twist convention. For an automorphism h of T and a left
T-module M, the balanced map
T_h (x)_T M -> {}_{h^{-1}} M, a (x) m |-> h^{-1}(a) m,
where the target action is b . m = h^{-1}(b) m, is an isomorphism. For
M = s it reduces to the canonical simple identification (h^{-1} fixes
the character of s).
In the normalized bar resolution of s, a comparison into its T_h-tensor
image applies h^{-1} to every bar letter. It commutes with the
differential because h^{-1} preserves products; the first multiplication
agrees by the balancing relation for the right-twisted factor, and the
last action agrees because h fixes the character of s. The
simple-valued degree-three cocycle obtained from p has exactly one
dual-ideal input in each nonzero component. Indeed p lowers the
dual-letter weight by one (Section 3.3), and only weight-zero outputs
can act nontrivially on s. The bar comparison therefore multiplies this
cochain, and hence its class tau, by H^{-1}.
The tensor functor preserves Yoneda products. In the two-factor algebra
A, both factors of one branch use the same H, so both tau_1 and tau_2
acquire the factor H^{-1}. Every basis monomial tau_1^r tau_2^{m-r} of
H^{3m} therefore acquires the factor H^{-m}. Linearity gives the
assertion on the whole homogeneous piece. QED.
PROPOSITION 6.7 (THE CONVERSION HYPOTHESES). The finite bimodule F is
projective on both A-sides, and the map v of Lemma 6.5 satisfies the
following conditions. For every integer a >= 0 define
delta^a : H^a (+) H^a -> V^a, (g, j) |-> F(g) v + v[a] j.
Then delta^0 is surjective with a one-dimensional kernel (the diagonal
line in k^2), and delta^a is an isomorphism for every integer a > 0.
PROOF. Finiteness and both projectivity assertions were proved in 6.2.
The actual bimodule projections rho_i are natural with respect to
tensoring module maps and their stable shifts. Since rho_i v = id_X,
they give
rho_i[a] F(g) v = U_i(g), rho_i[a] v[a] j = j.
For a = 0, k-linearity gives delta^0(c id_X, d id_X) = (c + d) v. This
is onto the line k v = V^0 and has the diagonal line
{(c, d) : c = d} as its kernel.
Let a = 3m > 0. Under (POSPROJ) and Lemma 6.6 the map delta^a is
(g, j) |-> ( H_1^{-m} g + j, H_2^{-m} g + j ).
This is a 2 x 2 scalar matrix
MAT(m) = ( H_1^{-m} 1 )
( H_2^{-m} 1 )
tensored with the identity of H^{3m} (which has dimension m + 1 over k).
The scalar matrix has determinant H_1^{-m} + H_2^{-m}, which is nonzero:
if H_1^{-m} + H_2^{-m} = 0, then H_1^{-m} = H_2^{-m}, so
H_1^m = H_2^m after multiplying by H_1^m H_2^m, contradicting algebraic
independence of H_1 and H_2 over F_2 (the two distinct monomials remain
distinct in the fraction field of F_2[q, H_1, H_2]). This argument holds
for every positive integer m, including those divisible by the
characteristic. Hence MAT(m) is invertible and delta^{3m} is an
isomorphism.
If a > 0 is not divisible by 3, then H^a = 0, and Lemma 6.5 gives
V^a = 0. Thus delta^a is an isomorphism (between zero spaces) in these
degrees too. QED.
The hypotheses of Proposition 2.4 are now verified, for the finite
symmetric algebra A, its finite module X, and the actual finite bimodule
F with the stable map v. Applying that proposition to
Lambda = Tri(A; F) = ( A 0 )
( F A )
produces the required finite nonprojective Gorenstein-projective left
module Z with Ext^a_Lambda(Z, Z) = Ext^a_Lambda(Z, Lambda) = 0 for
every a >= 1. In particular the degree-one vanishing uses the
surjectivity at degree zero (the kernel being the diagonal line) in
addition to the positive-degree invertibility.
PART VII now verifies the structural assertions about Lambda and
completes the proof of Theorem 1.1.
PART VII. THE RADICAL, NONCOMMUTATIVITY, AND BASE CHANGE
============================================================================
Proposition 2.4 has produced the module Z over the triangular algebra
Lambda = Tri(A; F) with the two Ext vanishings and nonprojectivity. This
part completes the proof of Theorem 1.1 by verifying the structural
assertions about Lambda (Proposition 7.1) and the persistence under
field extension (Proposition 7.2).
7.1 THE RADICAL, NONCOMMUTATIVITY, AND THE EIGHT SIMPLES
------------------------------------------------------------------------
PROPOSITION 7.1. The algebra Lambda is noncommutative, split, and has
Lambda / rad(Lambda) = k^8,
so it has eight isomorphism classes of simple modules. Moreover
rad(Lambda)^4 != 0.
PROOF. Put J_T = rad(T). Since T = C (x) DC and C / rad(C) = k^2 (the
two idempotents e, f), we have, using the positive grading in which the
span of all non-idempotent basis letters is the radical,
J_T = rad(C) (+) DC, T / J_T = k^2.
The ideal J_A = J_T (x) T + T (x) J_T in A = T (x) T is nilpotent: the
two summand ideals commute, and each is nilpotent (being generated by
the nilpotent J_T on one tensor factor). Its quotient is
A / J_A = (T / J_T) (x) (T / J_T) = k^2 (x) k^2 = k^4, which is
semisimple. A nilpotent ideal with semisimple quotient equals the
radical, so J_A = rad(A).
In the triangular algebra, the two-sided ideal
J_Lambda = ( J_A 0 )
( F J_A )
is nilpotent as well. Indeed, a product of elements of J_Lambda has at
most one off-diagonal F factor, with powers of J_A on its two sides, and
sufficiently long products vanish. Its quotient in Lambda is
Lambda / J_Lambda = (k^4) (+) (k^4) = k^8, which is semisimple. Hence
J_Lambda = rad(Lambda), and Lambda / rad(Lambda) = k^8 is split with
exactly eight simple modules.
Let chi : T -> k be the character of s (chi(f) = 1, chi(e) = chi(rad T) = 0).
The composite Lambda -> A -> (id_T (x) chi) T -> C is a surjective
algebra map. For a surjection of finite-dimensional algebras, rad maps
onto rad (nilpotent image, semisimple quotient). So
rad(C) = image of rad(Lambda).
In C, the multiplication table gives
(u t)^2 = (y + q x)^2 = q (1 + q) z != 0
(Lemma 3.1). Since u, t are in rad(C), the element (ut)^2 lies in
rad(C)^4 and is nonzero. As rad(C) is the image of rad(Lambda), it
follows that rad(Lambda)^4 != 0. (Concretely, if rad(Lambda)^4 were
zero, its image rad(C)^4 would be zero too, contradiction.)
Finally Lambda is noncommutative: e u = u != 0 = u e in C (corners),
and the same relation holds in the upper-left A-block of Lambda. QED.
7.2 PERSISTENCE UNDER FIELD EXTENSIONS
------------------------------------------------------------------------
PROPOSITION 7.2. For every field extension K / k, the algebra
Lambda_K = K (x)_k Lambda and module Z_K = K (x)_k Z satisfy all
conclusions of Theorem 1.1 over K.
PROOF. BASE CHANGE OF EXT. For finite Lambda-modules M, N, take a
projective resolution P of M with finite projective terms (finite over
Lambda, hence finite over Lambda_K after K (x)_k -). In each degree,
finite projectivity (over k, and over K after base change) gives
K (x)_k Hom_Lambda(P_i, N) = Hom_{Lambda_K}(K (x)_k P_i, K (x)_k N).
Flatness of K over k commutes with kernels and images, hence
K (x)_k Ext_Lambda^i(M, N) = Ext_{Lambda_K}^i(K (x)_k M, K (x)_k N) (BASE)
for every i >= 0.
BOTH EXT VANISHINGS. Applying (BASE) with M = N = Z and with N = Lambda
gives, for every i >= 1, that K (x)_k Ext_Lambda^i(Z, Z) = 0 and
K (x)_k Ext_Lambda^i(Z, Lambda) = 0. So the same vanishings hold over
Lambda_K.
GORENSTEIN-PROJECTIVE. The totally acyclic complex P_Z of Proposition 2.4
defines Z. After tensoring with K, its terms are finite projective over
Lambda_K, and the complex remains exact (K is exact over k). The same
termwise Hom isomorphism, now with target Lambda_K, shows that the dual
complex over Lambda_K is also exact. Hence Z_K is Gorenstein-projective.
NONPROJECTIVITY. Choose a finite projective surjection P ->> Z over
Lambda, with kernel N. It does not split, because Z is not projective;
its class in Ext_Lambda^1(Z, N) is nonzero. Base change is faithful
(K (x)_k - is exact and reflects whether an element is zero), so (BASE)
shows that the extended class in Ext_{Lambda_K}^1(Z_K, N_K) is nonzero.
Therefore the extended sequence does not split, and Z_K is nonprojective.
RADICAL AND SIMPLE COUNT. The ideal J_Lambda of Proposition 7.1 stays
nilpotent after tensoring with K, and its quotient becomes
K (x)_k k^8 = K^8. A nilpotent ideal with semisimple quotient K^8 equals
the radical of Lambda_K. Its fourth power is K (x)_k rad(Lambda)^4,
which is nonzero because rad(Lambda)^4 != 0 and K (x)_k - is faithful on
k-vector spaces. So Lambda_K is split with eight simples, noncommutative,
and has rad(Lambda_K)^4 != 0. QED.
This completes the proof of Theorem 1.1. In particular, taking the
field extension K to be an algebraic closure of k gives a counterexample
over any algebraically closed field of characteristic two.
REMARK 7.3 (THE COMPANION CONSTRUCTION). The companion paper (OpenAI,
"A counterexample to Tachikawa's second conjecture," 2026) constructs a
symmetric counterexample over the same k by a different construction.
The present Lambda is noncommutative and triangular; the two
constructions are independent. Both disprove the module form and
persist under base change; neither addresses the commutative Noetherian
case (Huneke-Leuschke 2004; Christensen-Holm 2010, App. A).
Appendix A gives the finite cochain table; Appendix B is a glossary.
APPENDIX A. THE FINITE COCHAIN TABLE
============================================================================
This appendix gives the complete coefficient table for the cochain p of
Section 3.3, and the finite recurrences that verify its Hochschild
cocycle property and boundary identity. Lower-case letters are the
basis of C; upper-case letters are the dual basis of DC.
READING THE TABLE. A four-letter entry "abcv" means that the indicated
coefficient is the coefficient of v in p(a, b, c). So the entry
"tyJf" in the q^3 row means p(t, y, J) = q^3 f. All unlisted
coefficients are zero. Each entry has composable radical inputs and an
output with the same two endpoints. For example, the entry "EuNT" means
p(E, u, N) has T as coefficient, etc.
The table has 179 nonzero entries, with distinct three-letter inputs. In
each entry, the number of capital (upper-case) input letters exceeds
the number of capital output letters by exactly one. This is the weight
-1 property of p.
COEFFICIENT q^2
EuNT EyZX FNtU JtXZ NtEU NtXV XzJT ZxEY tEun tXuf
tXvn tXzj tuFn tuNf tvNn uFNT vNNT vtJu xEyz zJtx
COEFFICIENT q^3
EyJT JtEY tEyj tyJf
COEFFICIENT q
EuUE EuVX EvVE ExXE EyYE EzZE FNnF FVvF FVxU FtTF
FtYU Ftun Ftxj JFtX JNtZ JUyX JVyZ JjEE JjTT JjXX
NFtU NNtV NUyU NVyV NjJN NjZV NnNN NnVV Ntuf Ntvn
Ntzj TtEE TtXX UuFF UuUU UvNF UvVU UxXU Uyun Uyxj
UzZU VvNN VvVV VxEU VxXV Vyuf Vyvn Vyzj VzZV XxEE
XxXX YuFT YuUY YvNT YvVY YxXY YyEE YyJJ YyTT YyXX
YyYY YyZZ YzZY ZxXZ ZzEE ZzJJ ZzTT ZzXX jJFF jJUU
jJjj jJnn jYyj jZvn jZzj nNFF nNUU ntTn tJjt tTNN
tTVV tTtt tYyt tZzt tzJn uFVX uFty uNFT uNUY uNte
uUyy uVye ujJu vNUE vNVX vNty vVyy vtTv xEuv xJFT
xJUY xJjx xJte xTtx xXuu xXvv xXyy xXzz xYyx xZye
xZzx xuFv xuNu xvNv yxJu yzJv zJNT zJVY zYyz zZuu
zZvv zZyy zZzz
COEFFICIENT 1
JNjX JVzX NNjU NVzU NjYU Njun Njxj Ntxt TNtX TVyX
Vyxt VzYU Vzun Vzxj YuNJ YuVZ jYun juNn jxJn txTn
uNNJ uNVZ uNjy uVzy xJNJ xJVZ xTNT xTVY yxTv zYuv
zuNv zxJv
FINITE RECURRENCES. These verify the identities in 3.4 without
recomputing the whole table by hand. Let B be the twenty-letter basis
and B_r = B \ {e, f} the radical basis. Write
ab = SUM_v mu_{ab}^v v,
p(a,b,c) = SUM_v p_{abc}^v v.
The lower-case coefficients mu are exactly the multiplication table of
Section 3.1, plus the idempotent actions. The remaining coefficients are
determined, for lower-case a, b, c, by
mu_{a, b*}^{c*} = mu_{c a}^{b},
mu_{b*, a}^{c*} = mu_{a c}^{b},
mu_{a*, b*}^{v} = 0.
Unspecified coefficients are zero. In particular every coefficient
belongs to F_2[q].
CLOSURE RECURRENCE (Hochschild cocycle). For each a, b, c, d in B_r and
v in B, substitution of the table and the recurrences gives
SUM_{x in B} ( mu_{a x}^{v} p_{bcd}^{x} + mu_{a b}^{x} p_{xcd}^{v}
+ mu_{b c}^{x} p_{axd}^{v} + mu_{c d}^{x} p_{abx}^{v}
+ p_{abc}^{x} mu_{x d}^{v} ) = 0.
This is the coefficient of v in the normalized Hochschild differential
of p. One may check it on all 18^4 = 104976 radical four-words;
precisely 15250 have composable endpoints, and the other equations
vanish by the corners. Thus the finite calculation covers the whole
relative bar term, not a selected collection of words.
BOUNDARY RECURRENCE (identity of Lemma 3.4). Put epsilon_a = 0 on
lower-case letters and epsilon_a = 1 on capital letters. For every basis
letter define
gamma_a = q if a in {u, v, t, E, X, Y, Z, U, V},
gamma_a = 0 otherwise.
For all a, b in B_r and v in B, the same table gives the pair of
coefficient identities
SUM_{w in B_r, eps_w = 0} SUM_x p_{abw}^{x} mu_{x, w*}^{v}
= eps_b gamma_a mu_{a b}^{v}, (CONST)
SUM_{w in B_r, eps_w = 1} SUM_x p_{abw}^{x} mu_{x, w*}^{v}
= ( gamma_b + gamma_v + (1 - eps_b) gamma_a ) mu_{a b}^{v}. (LIN)
There are 18^2 = 324 input pairs. These two equations are the constant
and linear coefficients in H of the boundary identity of Lemma 3.4: its
left side gives H^{eps_w}, while the three terms on the right give
respectively H gamma_b mu_{a b}^{v}, H gamma_v mu_{a b}^{v}, and
H^{1 - eps_b} gamma_a mu_{a b}^{v}. This verifies the identity for an
indeterminate H, hence for either H_1 or H_2.
NONZERO EVALUATION. Finally the entries p(t, x, J) = 0 and
p(t, y, J) = q^3 f give
q^2 p(t, x, J) + p(t, y, J) = q^3 f,
which is the nonzero value on the explicit cycle zeta of Section 3.3.
All these substitutions are polynomial identities in characteristic two.
They require no specialization of q or H. The appendix therefore both
supplies the data and verifies the two properties (Hochschild cocycle,
boundary identity) used in Parts III-VI.
That completes Appendix A.
APPENDIX B. GLOSSARY OF SYMBOLS
============================================================================
Field and parameters
k F_2(q, H_1, H_2), characteristic 2.
q, H_1, H_2 algebraically independent transcendentals over F_2.
H a nonzero element of k used to twist; H in {H_1, H_2}
for the two twists of the construction.
h_H automorphism of T fixing C and multiplying DC by H.
sigma_i h_{H_i} (x) h_{H_i}, acting on A = T (x) T.
The algebras
C ten-dimensional k-algebra of Section 3.1.
e, f orthogonal idempotents with e + f = 1 in C.
x,y,z,u,v, non-idempotent basis letters of C; eCe = e,x,y,z;
t,j,n eCf = u,v; fCe = t,j; fCf = f,n.
ell_i x + q^i y, right multipliers in the resolution of s.
DC the k-dual of C, a C-bimodule.
T C (x) DC, the trivial extension, dim 20, symmetric.
I k e (+) k f, the semisimple subalgebra of T.
r, J_T radical of T; r^6 = 0, T/r = k x k.
A T (x)_k T, dim 400, symmetric.
J_A, J_Lambda radicals of A and of Lambda.
Lambda Tri(A; F) = (A 0; F A), the finite-dimensional algebra
over which Z lives.
R A^e = A (x)_k A^op, the enveloping algebra.
The dual basis
a* trace-dual of the lower-case basis letter a; e.g.
F = f*, J = j*; star is an involution, (a*)* = a;
upper-case W = w*.
z_H the cochain z_H of Lemma 3.4, valued in T.
The modules
s the one-dimensional left f-simple of T (f acts by 1,
all other letters by 0).
s^r the same as a right simple.
X s (x)_k s, the one-dimensional left A-module.
M, N generic finite left modules.
FM F (x)_A M for a side-projective bimodule F.
The resolutions and classes
P the normalized relative bar resolution of T; P_n = 0
for n < 0.
p the explicit Hochschild 3-cocycle of Section 3.3; also
the comparison map P -> P[3], lowering bar degree by 3.
tau p(s), the degree-three generator of Ext_T(s,s) = kk[tau].
tau_1, tau_2 the two external copies p_1(s (x) s), p_2(s (x) s);
Ext_A(X,X) = kk[tau_1, tau_2].
R_i terms of the projective resolution of s over C.
The two-cone machinery
Ccal the stable A-bimodule built from two cones of tau_1, tau_2.
K the ordinary cone complex Cone(p)(x)_k Cone(p).
C_1 coker(K_2 -> K_1).
Sigma_R M coker(M -> Hom_k(R,M)), a finite-projective replacement.
Y Sigma_R^4 C_1, a finite representative of Ccal[3].
W^a stHom_A(X, (Ccal (x)_A X)[a]); k for a in {0,3}, else 0.
H^a stHom_A(X, X[a]); kk[tau_1, tau_2] in degrees 3m.
P_m degree-m part of kk[tau_1, tau_2] (each variable deg 1).
The twists and the fiber
T_h, P_h the right-twisted bimodule and bar resolution.
U_H A_{h (x) h} = A with right action twisted by h (x) h.
U_1, U_2 U_{H_1}, U_{H_2}.
f_H the lift U_H -> Ccal[3] of Proposition 6.1.
beta_H the bimodule representative U_H -> Omega A of its top
projection.
xi_H the Casimir element SUM_w h(w) (x)_I w* in P_0.
B_i, G_i the homotopy components in Lemmas 6.2 and 6.3.
Phi the tail map Q^tot -> K[3].
Q R (x)_k Y, a finite free A-bimodule.
F the fiber 0 -> F -> U_1 (+) U_2 (+) Q -> Y -> 0;
side-projective, the bimodule input to conversion.
rho_i the projection F -> U_i.
The stable map and the delta maps
v the diagonal stable map X -> F (x)_A X; rho_i v = id_X.
V^a stHom_A(X, F (x)_A X [a]).
delta^a H^a (+) H^a -> V^a, (g,j) |-> F(g) v + v[a] j.
delta^0 is onto with kernel the diagonal line; delta^a is an
isomorphism for a > 0.
The final module
Z (X, Y, iota), the finite nonprojective
Gorenstein-projective left Lambda-module of Theorem 1.1;
Ext^a_Lambda(Z,Z) = Ext^a_Lambda(Z,Lambda) = 0 for a >= 1.
Homological conventions
stHom stable Hom, quotiented by maps through finite projectives.
Ext^a the a-th extension group; Ext^a_R(M,N) = stHom_R(M,N[a])
for a > 0 over a symmetric algebra.
Omega M = M[-1], M[1] the stable shift (syzygy, cosyzygy).
D Hom_k(-, k), the k-dual.
(x) tensor product over k unless a subscript says otherwise.
(+) direct sum (also written "x" of modules in some texts;
here always "x" is tensor and "(+)" is direct sum).
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END OF DOCUMENT