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AN INTEGRAL COUNTEREXAMPLE TO GERSTEN'S CONJECTURE
(full proof, readable edition)
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CONTENTS
0. Introduction and proof architecture
1. Coefficient K-theory and the Bott multiplication
2. The arithmetic input: an integral cyclotomic class
3. The specialization criterion
4. The geometric detector: an elliptic quotient of the Fermat quartic
5. Assembling the counterexample
A. Supplementary: on a claimed vanishing theorem
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PLAIN-ENGLISH ORIENTATION (read this first)
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What the theorem says. Take one specific two-dimensional regular local ring A,
built from 5-adic numbers and the equation 5 + x^4 + y^4 = 0. There is a nonzero
element of an abelian group K_5(A) whose image in K_5 of the fraction field of A
is zero. That failure of injectivity contradicts Gersten's conjecture.
Why it is not obvious. Any class pushed forward from a closed divisor inside A
dies automatically over the fraction field. The whole difficulty is proving the
pushed-forward class is itself nonzero.
The two-step idea.
Step 1 (arithmetic, Section 2): build a specific integral class c, and remove
the only obstruction to lifting it (a group K_4(S)[5] that we prove vanishes).
Step 2 (geometry, Sections 3-5): push c forward to a class gamma on A, then
prove gamma != 0 by reducing modulo 5 and testing it with a homomorphism built
from an elliptic curve E: w^2 = s^4 + 1. The test value is 4Q, which is
nonzero.
How to read it. Section 0.5 is the map; Section 5 is the assembled proof.
Sections 1-4 are the machinery each step needs; skim them for the statements on
a first pass.
The mathematics begins in Section 1.
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0. INTRODUCTION AND PROOF ARCHITECTURE
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0.1 The statement
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THEOREM (main).
Let V be the ring of integers in the unramified extension of Q_5 of degree 8,
and put
A = ( V[x,y] / (5 + x^4 + y^4) )_(5,x,y).
Then A is a Noetherian regular local domain of dimension 2, and the map
K_5(A) --> K_5( Frac A )
has nonzero kernel. Consequently Gersten's conjecture, in its unrestricted
integral form, is false.
The maximal ideal of A is m = (x,y), and 5 lies in m^4. The special fiber of A
(the quotient A/5A) is the plane curve x^4 + y^4 = 0, which is one-dimensional
with an ordinary singularity at the origin. So A is regular, but the structure
map V -> A is not smooth: the special fiber is singular. This is exactly the
regime not covered by the known positive results (see 0.4).
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0.2 Two pieces of background, stated once
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What is a regular local ring. A Noetherian local ring R with maximal ideal m
is regular when its Zariski tangent space m/m^2 has dimension equal to the
Krull dimension of R. Such a ring behaves, for the purposes of the residue
complex below, like a germ of a smooth manifold. The ring A in the Theorem is
a regular local ring even though A/5A is a singular curve.
What is K-theory, in one paragraph. K_0 of a ring is the group of finitely
generated projective modules, with the operation of direct sum. K_1 is the
abelianization of GL(R). Higher K-groups K_i(R) are the homotopy groups of a
certain spectrum K(R); the only feature we use is that K-theory is compatible
with localization and with finite coherent pushforward, and that for a field F,
K_1(F) = F^x and K_0(F) = Z. All of this is the standard higher algebraic
K-theory of Quillen. Coefficient K-theory K_i(R; Z/5) means K_i of the mod-5
localization; we state its exact properties in Section 1.
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0.3 Gersten's conjecture
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Let R be a Noetherian regular local domain with fraction field F. Gersten's
conjecture asserts that the canonical residue complex
0 -> K_n(R) -> K_n(F) -> (direct sum over height-one primes p)
K_{n-1}(kappa(p)) -> K_{n-2}(kappa(q)) -> ...
is exact for every n >= 0. Here kappa(p) is the residue field of p, and the
arrows are the boundary maps of localization.
The very first assertion of exactness, the one we disprove, is the injectivity
of K_n(R) -> K_n(F): any K-class that dies over the fraction field must already
have been zero on R itself. This is called generic injectivity. The Theorem
produces a nonzero class gamma in K_5(A) whose image in K_5(Frac A) is zero.
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0.4 What is already known (why the ramified hypothesis matters)
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Quillen proved the conjecture for semilocalizations of finite-type algebras
over a field (regular primes). Panin proved the full equicharacteristic case.
In mixed characteristic, the smooth case is due to Gillet-Levine (relative
version), and finite-coefficient versions are due to Geisser-Levine and, for
unramified regular local rings, to Skalit.
"Unramified" in mixed characteristic (0,p) means p is not in m^2. Our ring has
5 in m^4, so it is maximally ramified. Every positive theorem above assumes
either equicharacteristic or smoothness/unramifiedness, and so none applies.
The point of the example is precisely that it sits in the ramified,
non-smooth corner.
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0.5 The proof in one page
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We must produce an integral class gamma in K_5(A) that is killed by restriction
to Frac A, and we must prove that gamma is nonzero. These are two genuinely
different tasks.
(A) Construct gamma (arithmetic part).
Fix a primitive fifth root of unity zeta, and put
S = Z[zeta, 1/5], u = zeta - 1.
The strategy is to push forward a class from a closed divisor. The divisor
we use is Spec(OO) sitting inside Spec(A), where OO is the valuation ring of
L = H_0(zeta), with H_0 the fraction field of V. The map A -> OO, x->tX_0,
y->tY_0, is constructed in Section 4 (Lemma 4.2).
To get a class on OO that will survive and then witness non-injectivity, we
proceed in steps:
step 1. Build an equivariant coefficient class beta in K_2(S; 5) with
coefficient boundary d_coef(beta) = [zeta]. (Section 2.)
step 2. Prove that the integral obstruction group K_4(S)[5] vanishes.
(This is the hard arithmetic input; Section 2.)
step 3. Deduce that the coefficient class b^2 [u]_5 (two Bott
multiplications by beta, applied to u) lifts to an honest integral
class c in K_5(S). (Section 2.)
step 4. Restrict c to L, and lift the resulting class in K_5(L) to a class
c-tilde in K_5(OO). The lift exists because the boundary
K_5(OO) -> K_5(L) -> K_4(k) hits K_4 of the finite residue field,
which is zero.
step 5. Push c-tilde forward along the closed immersion Spec(OO) into
Spec(A). Every module supported on this closed subscheme is
killed by restriction to Frac A. So gamma is automatically in the
kernel of K_5(A) -> K_5(Frac A). This is done in Section 5.
(B) Show gamma is nonzero (geometric part).
We cannot detect gamma directly; we detect its reduction mod 5. Base change
V -> L splits the divisor Spec(OO) into four rational points p_delta on the
generic Fermat quartic C : X^4 + Y^4 = Z^4. Concretely,
gamma restricted to T = Spec(A (x)_V L) = sum_delta (p_delta)_* res(delta c).
Reducing mod 5 turns each term into chi(delta)^2 b^2 [delta u]_5, where
chi : Gal(Q(zeta)/Q) -> F_5^x is the cyclotomic character. The four points
p_delta specialize to four points g_{chi(delta)} P on the special Fermat
quartic D over F_{5^8}.
To prove this mod-5 class is nonzero, we use a "Picard specialization
criterion" (Section 3). In one sentence: if a sum of pushforwards of
coefficient classes from points were zero, then a certain divisor on the
special fiber would be principal; a suitable homomorphism from the Picard
group of that special fiber can refuse any such relation.
The homomorphism is built in Section 4 from a degree-two map
h : D --> E, E : w^2 = s^4 + 1,
from the Fermat quartic onto a genus-one curve. The four points g_a P map to
Q or -Q on E, with the sign exactly matching the character square. The
weighted sum is then 4Q, which is nonzero modulo 5 (Q is chosen to be
nondivisible by 5). Hence gamma is nonzero. Done.
The rest of this document carries out these steps in full.
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0.6 A word on organization
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Sections 1-2 are the arithmetic input: coefficient theory, the Bott
multiplication, the vanishing of K_4(S)[5], and the integral lift c.
Section 3 is a self-contained geometric lemma, proved once and reused.
Section 4 builds the specific geometry of the Fermat quartic and the elliptic
detector on its special fiber.
Section 5 assembles the main proof.
The reader who wants only the logical thread can read the statements in 1-4 and
then all of Section 5; the proofs inside Sections 2-4 are the technical core.
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0.7 Notation used throughout
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Z, Q, F_p integers, rationals, the field with p elements.
zeta a fixed primitive fifth root of unity.
N = Q(zeta) the fifth cyclotomic field.
S = Z[zeta,1/5] the cyclotomic ring with 5 inverted.
u = zeta - 1 the standard uniformizer at the prime above 5.
Delta = Gal(N/Q) cyclic of order 4.
chi = chi:Delta -> F_5^x, defined by delta(zeta) = zeta^chi(delta).
V the integers of the unramified degree-8 extension of Q_5.
k = F_{5^8} its residue field (also identified with a residue field of L).
k_0 = F_{5^4} the subfield used in the elliptic count.
H_0 = Frac(V) the fraction field of V (unramified degree-8 over Q_5).
L = H_0(zeta) the totally ramified degree-4 extension of H_0.
OO the valuation ring of L (a DVR with residue field k).
A the counterexample ring (defined in the Theorem).
K_i(R), K_i(R;5) integral and mod-5 coefficient K-groups.
G_i(X;5) coherent (mod-5) K-groups of a Noetherian scheme X.
[a]_5 the class of the unit a in K_1(-;5) = (-)^x/(-)^{x5}.
Every module, every scheme, and every class below is taken with these
conventions fixed.
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1. COEFFICIENT K-THEORY AND THE BOTT MULTIPLICATION
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This section assembles the coefficient machinery used in both halves of the
proof. We fix the mod-5 coefficient convention, prove that mod-5 coefficient
K-theory carries a canonical product, define the operation
b = left multiplication by a chosen coefficient class beta,
and prove the precise Bott isomorphisms in low degrees for the fields we use.
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1.1 Coherent K-theory and coefficient groups
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Let G(X) denote the K-theory spectrum of coherent sheaves on a Noetherian
scheme X. For the schemes used below, all of which are regular, of finite
dimension, and affine or quasiprojective over an affine base, the canonical
resolution theorem gives a natural equivalence between K(X) (vector bundles)
and G(X) (coherent sheaves). So we may freely pass between them.
We use Quillen localization and devissage, including localization by a
multiplicative set and the resulting residue maps. The specific
references are Quillen, Higher algebraic K-theory I, sections 4, 5, and 7.
Let M = S/5 be the cofiber of multiplication by 5 on the sphere spectrum S.
The mod-5 coefficient groups are
K_i(X;5) = pi_i( K(X) (cup) M ), G_i(X;5) = pi_i( G(X) (cup) M ).
For i >= 1 there is a natural short exact sequence
0 -> K_i(X)/5 -> K_i(X;5) -> K_{i-1}(X)[5] -> 0 (1.1)
and the identical sequence with G in place of K. Here A/5 = A/5A and
A[5] = ker(5 : A -> A). The boundary map is written d_coef and is called the
coefficient boundary.
For a field H, the identities K_1(H) = H^x and K_0(H) = Z give
K_1(H;5) = H^x / H^{x5}, K_0(H;5) = Z/5. (1.2)
For a unit a we write [a]_5 for its class in K_1(-;5) = (-)^x/(-)^{x5}. We
write K-groups additively throughout, even when the elements are unit classes.
For a scheme X, X^(j) denotes its points of codimension j.
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1.2 The coefficient product
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To handle classes with nonzero coefficient boundary, we make the product
explicit. This is the same product as in Friedlander-Suslin, and it is what
lets localization be compatible with multiplication by coefficient classes.
LEMMA 1.1 (coefficient products).
The spectrum M admits a unique unital pairing mu : M (wedge) M -> M in the
homotopy category. Multiplication by 5 on M is nullhomotopic; consequently
every coefficient group K_i(-;5), G_i(-;5) is an F_5-vector space. The
composition
K(X) (wedge) K(Y) -> K(X x Y) -> K(X x Y) (wedge) M,
using tensor product followed by id (wedge) mu, defines natural bilinear
products in coefficient K-theory, and analogous module actions in coefficient
G-theory.
Proof.
The cofiber sequence for M and the values pi_1(S) = pi_2(S) = Z/2 imply
pi_1(M) = pi_2(M) = 0. Apply the functor [-, M] to the cofiber sequence
S -5-> S -> M. Because pi_1(M) = pi_2(M) = 0, the boundary maps vanish and
restriction along the unit eta : S -> M is injective:
[M, M] -> [S, M] = Z/5.
So 5 * id_M = 0. Now the cofiber sequence
M -5-> M -eta(wedge)1-> M (wedge) M -> Sigma M
admits an extension of id_M along the first two maps to a pairing mu : M
(wedge) M -> M. To see it is unique: any two such extensions differ by an
element of [Sigma M, M], and [Sigma M, M] = 0, because Sigma M again satisfies
pi_1 = pi_2 = 0. Also, restriction of mu along 1 (wedge) eta is the identity,
matching restriction along eta (wedge) 1; so the pairing is unital.
Finally, 5 * id_M = 0, smashed with any spectrum, immediately gives that every
coefficient group is killed by 5. The products are induced by the ordinary
tensor product of K-theory spectra and this fixed pairing on M.
[end proof]
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1.3 Multiplication by a coefficient class
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Fix a base ring R and a class beta in K_2(R;5). For any R-scheme X, define the
operation
b = b_beta : G_i(X;5) -> G_{i+2}(X;5), x |-> beta . x,
left multiplication by the restriction of beta. We write b^2 = b (compose) b;
concretely b^2(x) = beta . (beta . x), with the parentheses as written. We
never assume that coefficient K-theory is an associative ring; only this fixed
two-fold iteration is used.
LEMMA 1.2 (compatibility with localization).
The operation b commutes with every arrow and every boundary homomorphism in the
localization exact sequences of G-theory over R: for a multiplicative set, for
a generic point, and for the devissage residue maps. Under devissage its action
on a residue-field term is multiplication by the restricted class beta. It is
also compatible with finite coherent pushforward and flat pullback over R.
Proof.
Because R is regular, tensoring with a finitely generated projective R-module
is an exact functor on coherent modules, and preserves every support that
appears. So the K(R)-action induced by tensor product is compatible with all
the localization data in sight. Concretely: for each term B of the localization
triangle of G-theory over R, use the composite pairing
(K(R) (wedge) M) (wedge) (B (wedge) M)
-> K(R) (wedge) B (wedge) M (wedge) M
-> B (wedge) M,
whose last arrow is the integral tensor action on K(R) (wedge) B composed with
the fixed pairing mu on M. These pairings commute with every arrow of the
localization triangle by functoriality of tensor product. Precompose with the
representative beta : S^2 -> K(R) (wedge) M. Since beta has even degree 2,
moving it across a localization boundary contributes no sign, and the diagrams
for b commute. The residue-field assertion is the same computation after
devissage: tensoring over the residue field is exactly the restriction of beta
to that field.
The compatibility with finite pushforward and flat pullback is the projection
formula and flat base-change identity; both are identities of tensor functors.
[end proof]
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1.4 The Bott isomorphisms in low degree
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We now state and prove the exact form of field Bott periodicity that we use.
Two features matter:
(i) our statement fixes the multiplication class beta explicitly, rather than
asserting existence of some abstract isomorphism;
(ii) it covers only the small set of field types that appear in Sections 2-5.
LEMMA 1.3 (field Bott maps).
Let H be a perfect field of characteristic different from 5, containing a
primitive fifth root of unity zeta_H. Let beta_H in K_2(H;5) satisfy
d_coef( beta_H ) = [zeta_H],
and let b denote left multiplication by beta_H as in 1.3.
(1) If cd_5(H) <= 3, then for every r >= 1,
b : K_{2r}(H;5) --> K_{2r+2}(H;5)
is an isomorphism.
(2) If cd_5(H) <= 2, then for every r >= 0,
b : K_{2r+1}(H;5) --> K_{2r+3}(H;5)
is an isomorphism.
(3) If H is finite, then b : K_i(H;5) -> K_{i+2}(H;5) is an isomorphism for
every i >= 0.
Over an algebraically closed field H-bar of characteristic different from 5, the
group K_{2j}(H-bar;5) is generated by b^j(1) (where 1 in K_0(H-bar;5) is the
unit class and b^0(1) = 1), and K_{2j+1}(H-bar;5) = 0.
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1.4.1 The instances actually used
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In the construction we need precisely these:
number fields and finite extensions of Q_5: cd_5 = 2. Hence
b^2 : K_1(H;5) -> K_5(H;5) is an isomorphism; (1.3a)
the function field J of a one-variable curve over a finite extension of Q_5:
cd_5(J) <= 2 + 1 = 3. Hence
b^2 : K_2(J;5) -> K_6(J;5) is an isomorphism; (1.3b)
finite fields: cd_5 = 1. Hence
b^2 : K_0(k_v;5) -> K_4(k_v;5) is an isomorphism. (1.3c)
The third instance raises the target of a valuation homomorphism from degree
zero to degree four in the arithmetic step of Section 2.
The primitive roots and the classes beta_H needed in each case are obtained by
restriction from the fixed ground field, so the maps in (1.3) are instances of
Lemma 1.3 rather than of some abstract isomorphism between groups.
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1.4.2 Proof of Lemma 1.3
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We work in the corrected, strongly convergent motivic spectral sequence with its
mod-5 multiplicative structure,
E_2^{p,q} = H^{p-q}( H, Z/5(-q) ) ==> K_{-p-q}( H; 5 ). (1.4)
This form of the spectral sequence, and the finite-coefficient multiplicative
structure on it, are recorded in Weibel, K-book, VI.4.2, Addendum VI.4.2.1, and
VI.4.4; the corrected construction is Suslin's (On the Grayson spectral
sequence, Introduction and Theorem 6.1). The product on the abutment is the
usual K-theory product; the product on E_2 is the motivic cup product
(Friedlander-Suslin, Proposition 16.1 and Theorem 16.2). The
Moore-space comultiplication used there corresponds under stable duality to a
unital pairing on M; by Lemma 1.1 that pairing is our mu. So our products
coincide with the ones in the spectral sequence.
Reparametrize an E_2-term as H^s(H, Z/5(j)) with total K-degree 2j - s:
namely s = p - q and j = -q. The nonzero range of the spectral sequence is
0 <= s <= j. (1.5)
Indeed: for j < 0 the weight vanishes; and for j >= 0 the defining motivic
complex C_* Z_tr(G_m^{wedge j})[-j] is concentrated in cohomological degrees
at most j, so H^s(H, Z/5(j)) = 0 for s > j. For 0 <= s <= j the norm-residue
and Beilinson-Lichtenbaum comparison theorems (Voevodsky, Theorems 6.16-6.17,
together with the completion of the norm-residue argument) identify
H^s(H, Z/5(j)) = H^s_et( H, mu_5^{tensor j} ). (1.6)
Under the dimension bounds (1.3a-c) only finitely many terms lie in each total
degree, so the resulting filtration is finite and the abutment is computed.
A differential d_rho (rho >= 2) acts by
(s, j) |-> ( s + 2*rho - 1, j + rho - 1 ).
Step 1: the algebraically closed case. If H-bar is an algebraic closure of H,
then (1.6) leaves only s = 0 terms. In total degree 2j >= 0 there is exactly
one E_2-term, H^0(H-bar, mu_5^{tensor j}) = F_5; in odd degree there is none.
There is no possible differential. Hence
K_{2j}(H-bar;5) = F_5, K_{2j+1}(H-bar;5) = 0. (1.7)
Step 2: restriction of beta_H is nonzero. The coefficient sequence (1.1)
surjects K_2(H-bar;5) onto H-bar^x [5] = mu_5(H-bar). By (1.7) both sides
have dimension one, so the coefficient boundary is an isomorphism in this
degree. Therefore the restriction of beta_H to H-bar is nonzero, and so its
edge image is nonzero. Multiplicativity of the spectral sequence (Lemma 1.1)
now shows that each b^j(1) has nonzero edge image, hence generates the
corresponding even group. This proves the last assertion of the lemma,
without any separate rigidity input.
Step 3: the edge image. By naturality the edge image alpha in H^0(H, mu_5) of
beta_H is nonzero: the geometric restriction of alpha is the nonzero class
just computed, and restriction on H^0 is injective. The edge image of b^r(1)
is then alpha^{tensor r}, which generates the one-dimensional group
H^0(H, mu_5^{tensor r}) because H contains mu_5. Therefore the whole line
F_5 * b^r(1) consists of permanent cycles, for every r >= 1. (Only
alpha != 0 is needed; we never choose an identification of alpha with a
particular primitive root.)
Step 4: cd_5(H) <= 3, even degrees. In total degree 2r the only possible E_2
terms are
H^0(H, mu_5^{tensor r}) and H^2(H, mu_5^{tensor (r+1)}). (1.8)
Neither has an incoming differential. From the first, the only possible
outgoing differential is d_2, into an H^3-term; it vanishes by permanence of
F_5 * b^r(1). From the second, every potential outgoing differential lands in
cohomological degree at least 5, hence is zero. So we get the two-step
filtration
0 -> H^2(H, mu_5^{tensor (r+1)}) -> K_{2r}(H;5)
-> H^0(H, mu_5^{tensor r}) -> 0. (1.9)
Multiplication by beta_H preserves (1.9) and acts on both graded pieces by
tensoring with alpha. Each such act is an isomorphism. Hence (1) of the
lemma follows.
Step 5: cd_5(H) <= 2, odd degrees. In total degree 2r+1 the sole E_2 term is
H^1(H, mu_5^{tensor (r+1)}), with no possible nonzero differential.
Multiplication by beta_H acts by tensoring with alpha, an isomorphism. This
proves (2).
Step 6: finite fields. For a finite field, cd_5 = 1, so the calculation of
Step 5 applies in every positive degree. In degree zero, 1 generates
K_0(H;5) = F_5, and b(1) = beta_H generates K_2(H;5) = H^0(H, mu_5). This
proves (3).
[end proof]
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1.5 A first consequence for the arithmetic step
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Combine Lemma 1.2 and Lemma 1.3 in the following concrete form, which is the
template for the entire Section 2. Suppose H is a number field or a finite
extension of Q_5, equipped with a primitive root zeta_H and a class beta_H
with d_coef(beta_H) = [zeta_H]. Suppose the normalization map
H^x -> (direct sum over closed points v of a fixed Dedekind ring inside H)
Z, a |-> (v(a))_v
is surjective modulo 5. Then, since b^2 : K_1(H;5) -> K_5(H;5) and
b^2 : K_0(k_v;5) -> K_4(k_v;5) are isomorphisms, and since b commutes with the
localization boundary (Lemma 1.2), the induced map
K_5(H;5) -> (direct sum over v) K_4(k_v;5)
is surjective onto the direct sum of residue fields. This is exactly the input
we need in Section 2.3.
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2. THE ARITHMETIC INPUT: AN INTEGRAL CYCLOTOMIC CLASS
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The coefficient exact sequence (1.1) identifies the obstruction to lifting a
degree-5 coefficient class to an integral class: it is the subgroup of integral
K_4 killed by 5. This section proves that the obstruction vanishes for one
cyclotomic ring, then extracts the lift we need. The plan:
(2.1) record the arithmetic of the cyclotomic ring S = Z[zeta, 1/5];
(2.2) choose an equivariant coefficient class beta in K_2(S; 5);
(2.3) prove that unramified degree-2 classes vanish (Lemma 2.1);
(2.4) prove that the integral K_4 obstruction K_4(S)[5] vanishes
(Lemma 2.2), using Lemma 1.3, Lemma 2.1, and finite generation;
(2.5) take the integral lift c of b^2[u]_5 and record its behavior under
the four cyclotomic automorphisms (Proposition 2.3).
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2.1 The cyclotomic ring
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Fix once and for all a primitive fifth root of unity zeta. Put
N = Q(zeta), S = Z[zeta, 1/5], u = zeta - 1, Delta = Gal(N/Q).
Delta is cyclic of order 4. We use the character
chi : Delta -> F_5^x, delta(zeta) = zeta^{chi(delta)}.
The polynomial of u is obtained by substituting zeta = 1 + u in
Phi_5(X) = X^4 + X^3 + X^2 + X + 1:
Phi_5(1+U) = U^4 + 5 U^3 + 10 U^2 + 10 U + 5. (2.1)
This polynomial is Eisenstein at 5. Consequence: it is irreducible over Q_5,
so N has exactly one place above 5.
Consider the ring Z_5[u] inside N (tensored with Z_5). It is finite free over
Z_5 by (2.1). Since (2.1) is irreducible over Q_5, it is a domain; being
integral over Z_5 it has dimension one. Reduce (2.1) modulo 5: the reduction is
U^4, so the unique maximal ideal of Z_5[u] is (5, u) and the residue field is
F_5. From (2.1),
u^4 = -5 (1 + 2u + 2u^2 + u^3). (2.2)
The factor in parentheses on the right is a unit of Z_5[u], since its image
modulo (5, u) is 1. Therefore (5, u) = (u) in Z_5[u]: the maximal ideal is
principal. A one-dimensional Noetherian local domain whose maximal ideal is
principal is a discrete valuation ring. Being integrally closed in its fraction
field Q_5(u), it is exactly the integer ring of that field, with residue field
F_5 and ramification index 4.
Ring of integers. The norm of u is N_{N/Q}(u) = Phi_5(1) = 5. Since
Phi_5'(zeta) = 5 zeta^4 / (zeta - 1) = 5 zeta^4 / u,
the discriminant of the power basis 1, zeta, zeta^2, zeta^3 is
disc = N_{N/Q}( Phi_5'(zeta) )
= 5^4 / N_{N/Q}(u)
= 5^4 / 5
= 125. (2.3)
Let O_N be the full ring of integers of N. By the discriminant-index formula,
the index [O_N : Z[zeta]] has no prime divisor other than 5. The local
computation at the prime above 5, made above, already shows that Z[zeta] is
maximally integral at 5. Hence the index is 1:
O_N = Z[zeta], disc(N) = 125. (2.4)
Class number. N has two complex places and no real places (it is a totally
imaginary degree-4 field, since the four conjugates of zeta all have absolute
value 1). Minkowski's ideal-class bound reads
bound = (4! / 4^4) (4/pi)^2 sqrt(125) < 2.
Every ideal class has an integral ideal representative of norm at most this
bound, i.e. of norm 1, i.e. the class is trivial. Therefore
Pic(O_N) = 0. (2.5)
Since S = O_N[1/5] is a localization of O_N, we have Pic(S) = 0 as well.
Units. The norm of u is 5, so u is invertible in S. Every conjugate delta(u)
is also invertible in S. Consequently every closed point v of Spec(S) has
finite residue field k_v of characteristic different from 5, and the class of
zeta in k_v has order exactly 5: its order divides 5 (because zeta^5 = 1 in any
ring), and cannot equal 1 (because if zeta = 1 in k_v, then u = zeta - 1 maps
to 0 in k_v, contradicting that u is a unit at v). This is the precise residue
field property used when we apply Lemma 1.3(3) below.
--------------------------------------------------------------------------------
2.2 The equivariant coefficient class
--------------------------------------------------------------------------------
In K_1(S) = S^x, the class [zeta] has order five. The determinant
S^x -> K_1(S) splits off, so this class is genuinely of order five (not
accidentally trivial). The coefficient exact sequence (1.1) in degree 2,
0 -> K_2(S)/5 -> K_2(S;5) -> K_1(S)[5] -> 0,
supplies a class beta' in K_2(S;5) with
d_coef( beta' ) = [zeta]. (2.6)
The coefficient group K_2(S;5) is an F_5-vector space by Lemma 1.1, and Delta
acts on it by functoriality, because each delta in Delta is an automorphism of S.
Define the idempotent
e_chi = (1/4) sum_{delta in Delta} chi(delta)^{-1} * delta in F_5[Delta],
where 1/4 is the inverse of 4 in F_5. Let beta = e_chi * beta'. Since the
coefficient boundary d_coef is Delta-equivariant, and delta([zeta]) =
chi(delta)[zeta],
d_coef( beta )
= e_chi ( [zeta] )
= (1/4) sum_delta chi(delta)^{-1} chi(delta) [zeta]
= [zeta]. (2.7)
Relabeling the sum gives tau * e_chi = chi(tau) e_chi for each tau in Delta.
Therefore:
d_coef( beta ) = [zeta], delta( beta ) = chi(delta) * beta (2.8)
for all delta in Delta.
This is the one algebraic input that later produces the "square of the
cyclotomic character" in the detection step.
The operation b, and b^2, are as in Section 1.3. For any S-scheme Y on which
we use the operation, b denotes left multiplication by the restriction of beta,
and b^2 = b (compose) b. In particular b^2(v) = beta . (beta . v), with the
parentheses as written. Whenever we discuss localization boundaries
distinct from d_coef, Lemma 1.2 states that b commutes with them.
--------------------------------------------------------------------------------
2.3 Unramified classes of degree two
--------------------------------------------------------------------------------
The arithmetic reason that localization leaves only a single coefficient line is
the following. Note carefully that it concerns reductions of integral K_2
classes, not the whole of K_2(N; 5).
LEMMA 2.1 (unramified degree-two classes).
Let N = Q(zeta) and S = Z[zeta, 1/5] as above. If a class in K_2(N)/5 has
zero tame residue in k_v^x / k_v^{x5} at every closed point v of Spec(S), then
the class is zero.
Proof.
By Matsumoto's theorem, K_2 of a field has the presentation by Steinberg symbols
{-, -}; see Weibel's K-book III.6.1 for the modern formulation. The
classical-to-Quillen comparison (Weibel IV.1.10.1 and IV.7.2) identifies this
group with K_2(N). The degree-two norm-residue theorem (Voevodsky, Theorem
6.16) then gives
K_2(N)/5 = H^2(N, mu_5^{tensor 2}). (2.9)
Use zeta to identify one copy of mu_5 with the constant module Z/5. Under the
Kummer exact sequence, the resulting group H^2(N, mu_5) is identified with
Br(N)[5]. Write a for the Brauer class corresponding to the class in the
lemma's statement.
Tame residue vs Brauer residue. For a closed point v of Spec(S), the residue
characteristic is not five. The Quillen boundary on K_2(N) is the tame symbol.
Under norm residue and the twist identification above, its reduction modulo
five and the Brauer residue of a agree up to a single overall sign depending on
boundary conventions; in particular, their vanishing is equivalent. A
check: write each field element as (power of a uniformizer) times (unit). Both
residues kill unit-unit symbols. On a symbol consisting of a unit epsilon and a
uniformizer pi, both residues give the class of epsilon-bar (modulo 5) in
k_v^x/k_v^{x5}, up to sign. Bilinearity and the Steinberg relations cover the
general case; references: Weibel V.6.1.2 and V.(6.6); Harari, Brauer, Prop.
7.10. Therefore a has zero Brauer residue at every finite place v with v not
dividing 5.
Localization at other places. The completion N_v has finite residue field k_v.
For the henselian valuation ring of N_v, the kernel of the Brauer residue
map is Br(k_v), which is zero because k_v is finite; see Harari, Prop. 6.7 and
Example 6.8(a). Consequently a|_{N_v} = 0 at every place v not dividing 5.
The infinite completions of N are complex, and Br(C) = 0. So only the single
place above 5 remains.
The single remaining place. In the global Brauer invariant sequence for N,
the sum over all places of the local invariants is zero. Since we already
showed that all invariants away from 5 (and at infinity) are zero, the
invariant at the remaining place is zero as well. The local invariant is
injective at that place, and the global restriction map on Br(N) is
injective; both are in Milne's CFT, IV Example 2.14(c)-(e) and Proposition 4.3
(see also Weibel III.6, VI.8.1.1). These implications combine to give a = 0.
Thus the original class is zero. We never needed a residue calculation at
the place above 5.
[end proof]
--------------------------------------------------------------------------------
2.4 The integral obstruction vanishes
--------------------------------------------------------------------------------
We are ready to eliminate the obstruction to lifting a degree-five coefficient
class. The first part of the proof detects coefficient classes after passage to
an algebraic closure; the last part converts detection at the coefficient level
into an assertion about integral five-primary torsion using finite generation.
LEMMA 2.2 (integral K_4 obstruction vanishes).
For S = Z[zeta, 1/5], the group K_4(S)[5] is zero.
Proof.
Because S is a Dedekind ring with Pic(S) = 0, every divisor on Spec(S) is
principal. Equivalently, the normalized valuation map
N^x -> (direct sum over closed points v of Spec S) Z,
f |-> ( v(f) )_v
is surjective. Quillen localization and devissage for S (Quillen, Sec. 7,
Prop. 3.2 and paragraph 3.4) identify the boundary map
K_1(N;5) -> (direct sum over v) K_0(k_v;5)
with this valuation map modulo 5. Since the valuation map is surjective and
K_0(k_v;5) = Z/5 by (1.2), this boundary is surjective.
Convert to a statement in degree 5. N has 5-cohomological dimension 2
(Weibel VI.7 and VI.8.2). By Lemma 1.3(2) (odd case, r = 0) the operation
b^2 : K_1(N;5) -> K_5(N;5)
is an isomorphism. For every v, the finite residue field k_v contains the
primitive image of zeta (see 2.1), so Lemma 1.3(3) gives
b^2 : K_0(k_v;5) -> K_4(k_v;5)
as an isomorphism. The same restricted coefficient class beta acts on every
term, and by Lemma 1.2 the operation b commutes with the localization boundary.
Composing the isomorphisms with the surjection above, we obtain a surjection
K_5(N;5) -> (direct sum over v) K_4(k_v;5). (2.10)
Now take the exact localization segment
K_5(N;5) -> (direct sum over v) K_4(k_v;5)
-> K_4(S;5) -> K_4(N;5). (2.11)
By (2.10) the middle arrow (the first displayed) is zero on the right; so the
map K_4(S;5) -> K_4(N;5) is injective:
K_4(S;5) -> K_4(N;5). (2.12)
Bott detection inside K_4(N;5). Take x in K_4(S;5); write x_N for its image in
K_4(N;5). By Lemma 1.3(1) (even case r = 1, cd_5 N = 2 <= 3),
b : K_2(N;5) -> K_4(N;5)
is an isomorphism; so x_N = b(z) for a class z in K_2(N;5). The localization
residues of x_N in K_3(k_v;5) are zero, because x extends over S. Write
d_v for the coefficient localization boundary at v. By Lemma 1.2,
0 = d_v( x_N ) = b( d_v( z ) ).
But b : K_1(k_v;5) -> K_3(k_v;5) is injective by Lemma 1.3(3). So d_v(z) = 0
for every v.
Correct z by a multiple of beta. The coefficient boundary d_coef(z) lies in
K_1(N)[5] = N^x[5] = mu_5(N) = <zeta>. Choose m in F_5 such that
d_coef( z ) = m [zeta].
Consider z - m * beta_N, where beta_N is the restriction of beta to N.
Equation (2.8) makes the coefficient boundary of z - m beta_N equal to zero.
Its localization residues at every v are still zero, because beta extends over
S, so beta_N has zero residues at every v. The coefficient exact sequence now
identifies z - m beta_N with an element of K_2(N)/5, and naturality of coefficient
reduction with localization identifies its coefficient residues with the tame
residues of that element. Lemma 2.1 therefore makes z - m beta_N = 0. Hence
z = m beta_N, x_N = m * b( beta_N ). (2.13)
Geometric detection. Let N-bar be an algebraic closure of N. By the last
clause of Lemma 1.3, the restriction of b( beta_N ) to N-bar is the nonzero
square of a geometric Bott generator in K_4(N-bar;5). Restriction is
consequently injective on the F_5-line F_5 * b( beta_N ). Together with (2.12)
this proves
K_4(S;5) -> K_4(N-bar;5) is injective. (2.14)
Passage to integral torsion. The last clause of Lemma 1.3 also gives
K_5(N-bar;5) = 0. The coefficient exact sequence (1.1) with i = 5 and X =
Spec(N-bar) then gives:
0 -> K_5(N-bar)/5 -> K_5(N-bar;5) -> K_4(N-bar)[5] -> 0,
so K_4(N-bar)[5] = 0. In fact K_4(N-bar) has no nonzero element of any finite
five-power order: if x has order 5^k, then 5^{k-1} x has order 5, hence is
zero, hence x = 0.
Set G = K_4(S), and let rho denote coefficient reduction at each ring. By
naturality there is a commutative square
G --rho--> K_4(S;5)
| |
| |
K_4(N-bar) --rho--> K_4(N-bar;5).
The right vertical arrow is injective by (2.14). Suppose g in G has finite
five-power order. Its integral image in K_4(N-bar) is zero (as we just saw).
The square therefore forces rho(g) = 0. But rho(g) = 0 means g lies in the
kernel of the reduction map G -> G/5, i.e. g lies in 5G. So:
every element of G of finite five-power order lies in 5G. (2.15)
This holds for every element of five-power order, not only for elements killed
by five.
Finite generation eliminates a nonzero divisible subgroup. Quillen's theorem
makes K_4(O_N) finitely generated (Quillen, Finite generation, Sec. 1, Thm
1 and Remark (2); see also Weibel IV.6.9). Integral localization at the single
place above 5 gives the segment
0 = K_4(F_5) -> K_4(O_N) -> G -> K_3(F_5) = Z/24.
The two finite field values are Quillen's integral computation (Quillen, Finite
fields) and Z/24 is generated by the class of the cyclotomic field; regardless
of the exact value, they are finite and known. So G is finitely generated.
Write G = Z^r (direct sum) P (direct sum) T, where P is the finite five-primary
torsion subgroup of G and T is finite of order prime to 5.
By (2.15), P is contained in 5G. Take p in P, p = 5g for some g in G. The
free component of g is zero: 5g = p is torsion. The T-component of g is zero:
multiplication by 5 is an automorphism of T (its order is prime to 5,
so 5 is invertible in End(T)). So g itself lies in P. Thus P is contained
in 5P. The reverse inclusion 5P contained in P is trivial. Iterating
P = 5^n P to the exponent of the finite group P gives P = 0. In particular
K_4(S)[5] = 0.
[end proof]
--------------------------------------------------------------------------------
2.5 The integral lift and its conjugates
--------------------------------------------------------------------------------
The obstruction is now gone, so the coefficient class b^2[u]_5 lifts to an
honest integral class. We also record how the four cyclotomic automorphisms
act on the reduction of that class.
PROPOSITION 2.3 (integral lift and conjugates).
Let u = zeta - 1 in S^x, and let b be multiplication by the class beta of (2.8).
There exists c in K_5(S) with
c mod 5 = b^2 [u]_5 in K_5(S;5). (2.16)
For every delta in Delta, the reduced conjugate satisfies
( delta c ) mod 5 = chi(delta)^2 * b^2 [ delta u ]_5 in K_5(S;5). (2.17)
Restriction along any fixed map iota : S -> H to a field gives the corresponding
identities in K_5(H;5); in the second one, chi(delta)^2 * b_iota^2[ iota(delta u) ]_5
where b_iota is multiplication by the restriction of beta. No action of delta
on H is required.
Proof.
Lemma 2.2 and the coefficient exact sequence (1.1) with i = 5 give
0 -> K_5(S)/5 -> K_5(S;5) -> K_4(S)[5] = 0.
The boundary is zero because K_4(S)[5] = 0, so K_5(S) -> K_5(S;5) is
surjective. Therefore b^2[u]_5, which lies in K_5(S;5), has an integral
preimage c in K_5(S). This proves (2.16).
For (2.17), apply delta after reduction and use naturality and bilinearity of
the parenthesized product:
( delta c ) mod 5
= delta( beta . ( beta . [u]_5 ) )
= ( delta beta ) . ( ( delta beta ) . [ delta u ]_5 )
= chi(delta) beta . ( chi(delta) beta . [ delta u ]_5 )
= chi(delta)^2 * beta . ( beta . [ delta u ]_5 ).
The last expression is chi(delta)^2 b^2[ delta u ]_5, as claimed. The
computation made no equivariant choice of c: any integral lift of b^2[u]_5
satisfies (2.16) and hence (2.17). Restriction along iota is by functoriality of
the whole construction, and it introduces no action of delta on H.
[end proof]
--------------------------------------------------------------------------------
3. THE SPECIALIZATION CRITERION
--------------------------------------------------------------------------------
In the detection step of the main proof (Section 5) we must show that a sum of
pushforwards
sum_i (p_i)_* ( e_i * b^2 [a_i]_5 ) in G_5(U;5) (3.1)
of degree-five coefficient classes from four rational points p_i on a curve is
nonzero. The tool is a general divisor-theoretic criterion: a vanishing of such
a sum forces a certain divisor on the special fiber to be principal, and a
homomorphism from the Picard group can refuse any relation.
This is a self-contained geometric statement, proved once here and applied in
Section 5. It uses only the Bott isomorphisms of Lemma 1.3 and the
compatibility Lemma 1.2, plus the codimension-two vanishing d_1^2 = 0 in the
coniveau spectral sequence for coherent K-theory of a smooth surface.
--------------------------------------------------------------------------------
3.1 The statement
--------------------------------------------------------------------------------
PROPOSITION 3.1 (Picard specialization criterion).
Let L be a finite extension of Q_5 containing mu_5, with valuation ring R and
residue field k. Let X -> Spec R be a smooth projective morphism of relative
dimension 1 with geometrically integral fibers
C = X_L, D = X_k. (3.2)
For a closed point q of C, let r(q) be its specialization on D: it is the image
of the closed point under the unique extension of Spec(kappa(q)) -> X to the
valuation ring of kappa(q). For a closed point r of D, write [r] in Pic(D) for
the class of the divisor (r).
Let m >= 0 be an integer, and suppose that:
(H1) U is the spectrum of a localization of the coordinate ring of a
nonempty affine open of C, and p_1, ..., p_m are distinct L-rational
closed points of U;
(H2) Z is a finite set of closed points of D, and every closed point q of C
with r(q) not in Z belongs to U;
(H3) B is an abelian group killed by 5, and psi : Pic(D) -> B is a
homomorphism with psi([r]) = 0 for every r in Z.
Put r_i = r(p_i) in D(k). Fix beta_L in K_2(L;5) with d_coef(beta_L) = [zeta]
for a primitive fifth root of unity zeta; let b be left multiplication by the
restrictions of beta_L, and b^2 = b (compose) b. Let v_L be the normalized
valuation of L. For a_i in L^x and e_i in F_5, the condition
sum_{i=1}^m e_i * v_L(a_i) * psi( [r_i] ) != 0 in B (3.3)
implies
sum_{i=1}^m (p_i)_* ( e_i * b^2 [a_i]_5 ) != 0 in G_5(U;5). (3.4)
Here [a_i]_5 is the class of a_i in K_1(L;5) = L^x/L^{x5}, and the F_5-scalars
in (3.3) act through the natural F_5-module structure on B.
--------------------------------------------------------------------------------
3.2 Proof, stage 1: localization and Bott untwisting
--------------------------------------------------------------------------------
The sum in (3.3) is zero when m = 0, so (3.3) implies m > 0. Suppose, for
contradiction, that the sum in (3.4) is zero. We first convert this assumed
relation into prescribed ordinary tame symbols on the generic curve.
Let J = L(C) be the function field of the generic fiber, and A_U = Gamma(U, O_U)
the coordinate ring of U. A_U is a localization of the coordinate ring of a
smooth affine curve, so it is a Noetherian regular domain of dimension at most
one with fraction field J. Because at least one of the p_i is retained, its
dimension is exactly one. Every finitely generated torsion A_U-module is
killed by some nonzero element; the quotient is then a finite-dimensional
Noetherian module, hence Artinian, hence of finite length with simple factors
the residue fields of the retained closed points. We do not need U to be
semilocal or the inverted set to be finite.
Quillen localization for the torsion category, combined with devissage and the
coefficient convention of Section 1, gives the exact sequence
K_6(J;5) -> (direct sum over q in U^(1)) K_5(kappa(q);5)
-> G_5(U;5). (3.5)
The index set of the direct sum may be infinite, but each class has finite
support. Every q in U^(1) is a closed point of C, so kappa(q) is a finite
extension of L.
By (3.4) being assumed zero, exactness supplies xi in K_6(J;5) whose residue at
each p_i is e_i b^2[a_i]_5, and whose other residues in U are zero. Now
cd_5(J) <= 3 and cd_5(kappa(q)) = 2 (finite extensions of Q_5). All these
fields are perfect and contain the same primitive root, by restriction from L.
So Lemma 1.3 applies with the actual multiplication operation used here:
b^2 : K_2(J;5) -> K_6(J;5) is surjective; (3.6a)
b^2 : K_1(kappa(q);5) -> K_5(kappa(q);5) is injective. (3.6b)
Choose xi_0 in K_2(J;5) with b^2 xi_0 = xi. Write d_q for the coefficient
localization boundary at q. By Lemma 1.2,
d_q( b^2 xi_0 ) = b^2 ( d_q xi_0 ),
with no sign, because b has even degree. Injectivity of b^2 on residue fields
then forces
d_q( xi_0 ) = e_i [a_i]_5 if q = p_i,
d_q( xi_0 ) = 0 for all other q in U^(1). (3.7)
There is still one coefficient obstruction to representing xi_0 integrally:
its coefficient boundary lives in K_1(J)[5] = mu_5(J) = <zeta> (the last
equality holds because L already contains mu_5). So there is lambda in F_5
with
d_coef( xi_0 ) = lambda [zeta].
Let beta_U and beta_J be the restrictions of beta_L. Naturality gives
d_coef( beta_J ) = [zeta], while every localization residue of beta_J on U is
zero, because beta_J extends to the coefficient class beta_U over U. (This uses
an extension over the L-scheme U, not over R.) The class
eta = xi_0 - lambda * beta_J in K_2(J;5)
therefore has zero coefficient boundary and the same residues as xi_0.
The coefficient exact sequence now supplies an integral representative z in
K_2(J) of eta. Write d_q^int z in K_1(kappa(q)) = kappa(q)^x for the ordinary
tame symbol. Reduction of localization residues commutes with coefficient
reduction, so (3.7) turns into
[ d_q^int z ]_5 = e_i [a_i]_5 if q = p_i,
[ d_q^int z ]_5 = 0 otherwise, q in U^(1). (3.8)
In particular, at every retained point q of U different from the p_i, the unit
d_q^int z is a fifth power in kappa(q)^x. We have acquired an ordinary
K_2-class whose tame symbols are completely controlled.
The remaining task is to derive the relation that these tame symbols must
satisfy on the special fiber D. That is done in stage 2.
--------------------------------------------------------------------------------
3.3 Proof, stage 2: the horizontal divisor
--------------------------------------------------------------------------------
For each closed point q of C, let R_q be the valuation ring of the unique
extension of the valuation of L to kappa(q); let k_q be its residue field and
v_q its normalized valuation. The extension kappa(q)/L is finite, so R_q is a
DVR finite over R. Properness supplies an extension
j_q : Spec R_q -> X,
finite over X: indeed, its graph inside Spec(R_q) x_R X is a closed immersion
because X/R is separated, and the projection to X is the base change of the
finite map Spec R_q -> Spec R. Define the residue degree
f_q = [ k_q : kappa(r(q)) ].
Smoothness over the DVR R makes X a regular surface with function field J. The
codimension filtration for coherent K-theory yields the following row of the
coniveau complex:
K_2(J) -> (direct sum over s in X^(1)) kappa(s)^x
-> (direct sum over r in X^(2)) Z. (3.9)
We need only that the composite of the two arrows is zero, which is d_1^2 = 0
in the coniveau spectral sequence (Quillen, Section 7, Theorem 5.4 and eq.
(5.5)).
The codimension-one points of X are of two types:
* the generic point of the special fiber D, with residue field k(D). Its
residue d_D z in k(D)^x contributes the principal divisor
div_D( d_D z ) to the codimension-two sum;
* the closed points q of C. Their closures meet D in the closed points
r(q). Each contributes to the sum at r(q).
We compute the contribution at r(q) without any normal-flatness assumption. On
Spec(R_q) the localization boundary
K_1(kappa(q)) -> K_0(k_q) = Z
is exactly the normalized valuation v_q: a unit has boundary zero, and a
uniformizer pi_q has boundary [ R_q / pi_q R_q ] = [k_q] (Weibel, V.6.1.2 and
V.(6.6)). By functoriality of coherent localization for the finite map j_q,
the horizontal boundary on X equals the pushforward of this DVR boundary
(Quillen, Section 7, paragraphs 2.7-2.8, Prop. 3.2, paragraph 3.4). At the
generic point this pushforward is the identity on kappa(q); at the closed point
it sends [k_q] to [k_q : kappa(r(q))] [kappa(r(q))]. Hence the contribution at
r(q) is exactly
f_q * v_q( d_q^int z ).
There is no extra ramification factor: the boundary already uses the normalized
valuation on kappa(q), and the closed-point pushforward contributes only the
residue degree f_q.
Define the horizontal divisor
H(z) = sum_{ q in C^(1) } f_q * v_q( d_q^int z ) * (r(q)). (3.10)
The sum is finite because the first arrow of (3.9) lands in a direct sum. All
codimension-two points of the proper surface X lie on D. Applying d_1^2 = 0 to
z gives, as an equality of divisors on D,
H(z) + div_D( d_D z ) = 0.
Every principal divisor is linearly trivial, so
[ H(z) ] = 0 in Pic(D). (3.11)
Stage 2 is complete: the tame symbols of z, aggregated on the special fiber,
form a specific principal divisor class.
--------------------------------------------------------------------------------
3.4 Proof, stage 3: the Picard test
--------------------------------------------------------------------------------
Apply psi to (3.11). If r(q) is in Z, the q-th summand vanishes by
psi([r(q)]) = 0. If r(q) is not in Z, hypothesis (H2) says q belongs to U.
For such a q different from the p_i, (3.8) makes v_q(d_q^int z) divisible by 5,
and the term dies in B because B is killed by 5. For q = p_i we have
kappa(p_i) = L and k_{p_i} = k = kappa(r_i), so
f_{p_i} = 1, v_{p_i} = v_L.
The remaining congruence in (3.8) gives
v_{p_i}( d_{p_i}^int z ) = e_i * v_L(a_i) mod 5.
So psi( [H(z)] ) = 0 becomes
sum_{i=1}^m e_i * v_L(a_i) * psi( [r_i] ) = 0 in B,
contradicting (3.3). This proves (3.4).
[end proof]
--------------------------------------------------------------------------------
3.5 Remark: no Gersten exactness is used
--------------------------------------------------------------------------------
The exactness in (3.5) comes from localization with respect to a torsion
category, valid for any multiplicative localization U of a smooth affine curve.
On the mixed-characteristic surface X, we use only d_1^2 = 0 in (3.9), which is
a formal property of the coniveau spectral sequence for coherent K-theory.
Nowhere do we assume Gersten injectivity or exactness. This is important: the
whole point of the paper is to disprove generic injectivity, so the criterion
must be independent of it.
--------------------------------------------------------------------------------
3.6 Where the hypothesis (H2) comes from
--------------------------------------------------------------------------------
The criterion is stated for an "arbitrary localization U". In the main proof
the retained-point hypothesis (H2) is supplied by the specific ring A: every
closed point q of C whose specialization falls outside Z_infinity lies in T.
Section 5 checks this directly from the definition of A. For this document it
is enough that hypothesis (H2) is checkable point by point; the abstract
criterion is what we proved above.
--------------------------------------------------------------------------------
4. THE GEOMETRIC DETECTOR: AN ELLIPTIC QUOTIENT OF THE FERMAT QUARTIC
--------------------------------------------------------------------------------
The Picard specialization criterion (Proposition 3.1) asks for a homomorphism
psi from the Picard group of the special fiber D that kills the "omitted"
points Z but detects a weighted sum of the four chosen specializations. We
construct psi from a degree-two quotient of the Fermat quartic onto a genus-one
curve. The square of the cyclotomic character takes values +1 or -1 in F_5;
the quotient turns precisely these signs into elliptic negation, so the
weighted sum becomes 4Q != 0.
We also construct the local ring A and the closed immersion Spec(OO) -> Spec(A)
that carries the integral class, together with the splitting of this divisor
after flat base change V -> L.
The section is organized as:
4.1 the curves D (Fermat quartic) and E (genus one), the degree-two map h;
4.2 the Picard detector psi, and the killing of infinity;
4.3 the weighted orbit: the point P, the automorphisms g_a, the value 4Q;
4.4 the local ring A and the surjection A -> OO (Lemma 4.2);
4.5 the model and the splitting into four points (Lemma 4.3).
--------------------------------------------------------------------------------
4.1 The curves and the map h
--------------------------------------------------------------------------------
Set
k = F_{5^8}, k_0 = F_{5^4}.
Over F_5, let D be the projective plane curve
D : X^4 + Y^4 = Z^4,
and let E be the smooth projective model of
E : w^2 = s^4 + 1.
D is smooth. Check: the three partial derivatives 4X^3, 4Y^3, -4Z^3 do not
vanish simultaneously at any point of the projective plane over F_5. This
remains true after any extension of the ground field (4 is invertible and
(x,y,z) = 0 is not a projective point). D is geometrically integral: distinct
projective plane components would have nonzero intersection (Bezout), producing
a singular point, contradiction.
E is smooth of genus one. The polynomial s^4 + 1 has four simple roots. At
infinity, use coordinates q = 1/s and r = w/s^2, so that r^2 = 1 + q^4. Above
q = 0 there are two unramified points, r = +1 and r = -1. So the double cover
E -> P^1_s has exactly four branch points; the degree-two Hurwitz formula gives
genus ((4 / 2) - 1) = 1.
There is a finite morphism of degree two
h : D -> E, s = X / Y, w = Z^2 / Y^2. (4.1)
Geometrically, this is the standard passage from a diagonal quartic to its
genus-one quotient by the sign involution v |-> -v, where v = Z/Y. We verify the
particular map on the affine chart Y != 0. Put s = X/Y and v = Z/Y; dividing
X^4 + Y^4 = Z^4 by Y^4 gives
s^4 + 1 = v^4. (4.1a)
So the function field of D over k(s) is obtained by adjoining v with v^4 =
s^4 + 1, while the function field of E over k(s) is obtained by adjoining w
with w^2 = s^4 + 1. The map D -> E sends v |-> v^2 = w, i.e. w = Z^2/Y^2,
which is (4.1). Since v |-> v^2 has kernel +/1 and the involution v |-> -v is
nontrivial, the extension of function fields has degree two. The rational map
extends to the smooth projective curves and is finite. It sends the four
points of D with Z = 0 (v = 0 in the affine chart, so w = 0) to the four branch
points of E -> P^1, i.e. the points with w = 0 and s^4 = -1.
--------------------------------------------------------------------------------
4.2 The Picard detector psi
--------------------------------------------------------------------------------
The eighth roots of unity lie in k_0, because 8 divides 5^4 - 1 = 624. Thus
all four branch points (with s^4 = -1 and w = 0) are k_0-rational. Choose one,
call it o, as the origin of E_{k_0}. Every fiber of E -> P^1 is linearly
equivalent to 2(o); accordingly the involution w |-> -w is negation for this
group law, using the standard identification of degree-zero divisor classes with
points on a genus-one curve with chosen origin (Silverman, Proposition III.3.4).
In particular, every branch point represents a point of E[2](k_0).
Base change D_k and E_k. Continue to write h : D_k -> E_k. The origin o in
E(k_0) defines the group law on E_k. Put
Z_inf = D_k intersect { Z = 0 }, (the four points over Z = 0).
Define
psi : Pic(D_k) -> E(k) / 5 E(k),
[ D' ] |-> [ h_* D' - deg(D') * (o) ] mod 5 E(k). (4.2)
Well-definedness. Every line bundle on the smooth curve D_k has a nonzero
rational section and hence a representing divisor D'. On a closed point r,
finite pushforward of divisors is
h_*(r) = [ kappa(r) : kappa( h(r) ) ] * ( h(r) ).
Its degree is [kappa(r) : kappa(h(r))] * [kappa(h(r)) : k] = [kappa(r) : k],
so finite pushforward preserves degree over k. By the norm formula (Stacks,
Tag 02RT), it takes principal divisors to principal divisors; so psi descends to
a homomorphism. The identification Pic^0(E_k) = E(k) sends [ (R) - (o) ] to
the point R (Silverman, Proposition III.3.4 and Remark III.3.5.1). Hence psi
is a well-defined homomorphism on the entire Picard group, including classes
represented by non-rational closed points.
For a k-rational point r of D_k, psi([r]) = h(r) in E(k)/5E(k).
Infinity is killed. Every point of Z_inf is k-rational (its coordinates satisfy
X^4 + Y^4 = 0 with Z = 0, and Z_inf lies in D_k). Under h it maps to a branch
point of E_k -> P^1_k, hence to an element of E[2]. A two-torsion class is
zero modulo 5 (because 5 is odd and invertible on E[2]). Hence
psi( [r] ) = 0 for every r in Z_inf. (4.3)
This is the point-killing property (H3) of Proposition 3.1, with Z = Z_inf.
--------------------------------------------------------------------------------
4.3 The weighted orbit: 4Q != 0
--------------------------------------------------------------------------------
For a in F_5^x, define an automorphism of D by
g_a (X : Y : Z) = ( aX : aY : Z ). (4.4)
The commutative diagram
D --g_a--> D
| |
h h
v v
E --"a^{-2}"->E
holds on the dense chart Y != 0: g_a fixes s = X/Y and multiplies w = Z^2/Y^2
by a^{-2}. On the branch-point fiber (w = 0), a^{-2} w = 0 = w, so the
statement extends. Since D is integral and E is separated, the identity of
morphisms extends from the dense chart. Hence
h ( g_a P ) = ( s(P), a^{-2} w(P) ) for all a in F_5^x. (4.5)
In Fourier coordinates on w (i.e. depending on whether a^2 = +1 or -1):
h ( g_a P ) = Q if a^2 = 1,
h ( g_a P ) = -Q if a^2 = -1. (4.6)
We now construct the pair Q, P.
LEMMA 4.1 (the weighted orbit).
There are points Q in E(k_0) and P in D(k) such that
Q not in 5 E(k), h(P) = Q, Z(P) != 0, (4.7)
and
sum_{ a in F_5^x } a^2 * psi( [ g_a P ] )
= sum_{ a in F_5^x } a^2 * h( g_a P )
= 4Q != 0 in E(k) / 5 E(k). (4.8)
Proof.
Curve counts. The curve E has four points over F_5. At s = 0: w^2 = 1, so
w = +/-1, two affine points. At each nonzero s, s^4 + 1 = 2 which is a
nonsquare in F_5 (the squares are 0, 1, 4), so no affine points. Two rational
points at infinity (r = +/-1 above q = 0). Total: 4.
Let a_1 = 4 + 1 - 4 = 2 be the trace of Frobenius. The Frobenius polynomial
is T^2 - a_1 T + q = T^2 - 2T + 5, whose roots count points of E over F_{q^m}
via the sequence a_m of power sums of the roots (Silverman, Theorem V.2.3.1):
a_0 = a_1 = 2, a_m = 2 a_{m-1} - 5 a_{m-2} for m >= 2.
Compute
a_2 = 4 - 10 = -6,
a_3 = 2(-6) - 5(2) = -22,
a_4 = 2(-22) - 5(-6) = -44 + 30 = -14.
Hence, with q^4 = 5^4 = 625 and #E(F_{5^4}) = q^4 + 1 - a_4:
#E(k_0) = 625 + 1 - (-14) = 640 = 2^7 * 5. (4.9)
Since 5 divides 640, E(k_0) has a point whose class in E(k_0)/5E(k_0) is
nonzero; choose such a Q.
Q remains nondivisible by 5 over k. Suppose Q = 5R with R in E(k). Apply
the trace for the quadratic extension k / k_0, using the group law over k_0.
The Frobenius of F_{5^4}, call it sigma, generates Gal(k/k_0). Then
Tr( Q ) = Q + sigma(Q) = 2Q (since Q in E(k_0)),
Tr( 5R ) = 5 ( R + sigma(R) ) in 5 E(k_0).
If Q = 5R then 2Q = 5( R + sigma(R) ) lies in 5E(k_0). Since multiplication by
2 is invertible on E(k_0)/5E(k_0), this forces Q in 5E(k_0), a contradiction.
Thus Q is not in 5E(k).
Q is not a branch point. Every branch point is two-torsion, and every
two-torsion point is divisible by 5 (5 is invertible on E[2]); so branch points
lie in 5E(k). Since Q is not in 5E(k), Q is not a branch point.
Lifting to P. A fiber of the degree-two morphism h : D_k -> E_k has a point
over an extension of degree at most two. Since k is the quadratic extension of
k_0 and Q is k_0-rational, we may choose P in D(k) with h(P) = Q. Because Q is
not a branch point, P is not over Z = 0, i.e. Z(P) != 0. This proves (4.7).
Weighted sum. Combining (4.6) and (4.7):
h( g_a P ) = Q if a^2 = 1, (two values of a: a = +/-1),
h( g_a P ) = -Q if a^2 = -1, (two values of a: a = +/-2 in F_5).
Each summand in (4.8) is a^2 * h(g_a P):
- for a^2 = 1: (+1) * Q = Q, sum over two a's = 2Q;
- for a^2 = -1: (-1) * (-Q) = Q, sum over two a's = 2Q.
Total: 4Q. Because 4 is invertible in F_5 and Q is nonzero in E(k)/5E(k), the
product 4Q is nonzero. This proves (4.8).
[end proof]
--------------------------------------------------------------------------------
4.4 The regular local ring and its closed divisor
--------------------------------------------------------------------------------
We now realize the four-point orbit of Lemma 4.1 as the specialization of a
single horizontal divisor. The construction has three parts:
(i) choose a cyclotomic uniformizer t in the local ring OO with controlled
conjugates;
(ii) realize the surjective map from the regular local ring A onto the
cyclotomic valuation ring OO;
(iii) split the divisor by the flat scalar extension V -> L.
Setup. Let H_0 / Q_5 be the unramified extension of degree 8, and let V be its
ring of integers, with residue field identified with k = F_{5^8}. Retain the
cyclotomic notation zeta, u = zeta - 1, Delta = Gal(Q(zeta)/Q), and
chi : Delta -> F_5^x. Choose an embedding of Q(zeta) into an algebraic closure
of H_0, and put
L = H_0 ( zeta ), OO = the ring of integers of L. (4.10)
Cyclotomic uniformizer in OO. The translated cyclotomic equation
(2.1),
u^4 + 5 u^3 + 10 u^2 + 10 u + 5 = 0,
is Eisenstein over V (the constant term is 5 times a unit, all other
coefficients are divisible by 5, and the leading coefficient u^4 has constant
term u^4 not divisible by 5). Consequently L / H_0 is totally ramified of
degree 4, u is a uniformizer of OO, and the residue field is k. Restriction of
automorphisms identifies Gal(L / H_0) with Delta.
We need a uniformizer t that behaves well under the four automorphisms.
Existence: from the cyclotomic equation,
u^4 / (-5) = 1 + 2u + 2u^2 + u^3, (4.11)
and the right-hand side is a unit congruent to 1 modulo u. By Hensel's lemma
(applied in the complete DVR V, since 4 is invertible in V), the inverse of
this unit has a fourth root congruent to 1 modulo u. Multiply u by that fourth
root to get t, an element still a uniformizer, congruent to u modulo u^2. Then
t^4 = -5. (4.12)
The quotient delta(t)/t is a fourth root of unity: applying delta to t^4 = -5
gives delta(t)^4 = -5 = t^4, so (delta(t)/t)^4 = 1. Because t and u differ by
a unit congruent to 1 modulo u, the residue of delta(t)/t equals that of
delta(u)/u. Now delta(u)/u = (zeta^{chi(delta)} - 1)/(zeta - 1), and
zeta^k - 1 = (zeta - 1)(zeta^{k-1} + ... + 1), so this ratio has residue equal
to the integer chi(delta) modulo (zeta - 1). In particular it is a unit, and
delta(t)/t lies in mu_4(V). Hence
delta(t)/t in mu_4(V), residue = chi(delta) (4.13)
in k^x = F_{5^8}^x.
All fourth roots of unity lie in V, since 4 | 5^4 - 1 and V is the unramified
extension with residue field F_{5^8}; Hensel again.
The whole ring of integers. We also have
V[t] = OO. (4.14)
Verification: t satisfies the Eisenstein polynomial T^4 + 5 over V, so V[t] is
finite free over V with fraction field L. Reduction modulo 5 gives T^4 + 5 ->
T^4, so V[t] is local with maximal ideal (5, t) = (t) (since t^4 = -5). Its
dimension is one, hence it is a DVR, hence integrally closed in L. Every
element of OO is integral over V and lies in L, so it lies in V[t]. Therefore
V[t] = OO.
Hensel lifts of the chosen point. Write P = (X_0bar : Y_0bar : 1) for the
point of D(k) constructed in Lemma 4.1. At least one of X_0bar, Y_0bar is
nonzero. If X_0bar != 0, the partial derivative 4X_0bar^3 is a unit of V, so
by Hensel's lemma (V is complete with residue field k) we can lift (X_0bar,
Y_0bar) to (X_0, Y_0) in V^2 with
X_0^4 + Y_0^4 = 1. (4.15)
(The equation is smooth in the X-direction at points with X_0bar != 0;
symmetrically for Y_0bar != 0.)
LEMMA 4.2 (regular local ring and closed divisor).
The ring
A = ( V[x,y] / ( 5 + x^4 + y^4 ) )_(5,x,y) (4.16)
is a regular local domain of dimension two. The assignment
A --> OO, x |-> t X_0, y |-> t Y_0 (4.17)
is a surjective local homomorphism with nonzero kernel of height one.
Proof.
Regularity. The regular local ring V[x,y]_(5,x,y) has dimension 3 (V is a DVR
of dimension one, plus two variables). The equation 5 + x^4 + y^4 is
congruent to 5 modulo the square (5,x,y)^2 of the maximal ideal, hence is
part of a regular system of parameters. Its quotient A is a regular local ring
of dimension 2, and is a domain (Stacks, Tags 00NQ and 00NP).
The map is defined. Equations (4.12) and (4.15) show that the assignment
respects the defining equation: (tX_0)^4 + (tY_0)^4 = t^4 (X_0^4 + Y_0^4) =
(-5)(1) = -5, so 5 + x^4 + y^4 maps to 0. Every denominator inverted in
(4.16) has nonzero constant residue at x = y = 0, hence maps to a unit of OO;
so the homomorphism extends to A -> OO and is local.
Surjectivity. At least one of X_0, Y_0 is a unit of V (because their residues
are not both zero). So the image contains t. By (4.14), the image contains
V[t] = OO.
The kernel has height one. The element Y_0 x - X_0 y belongs to the kernel:
Y_0 (tX_0) - X_0 (tY_0) = 0. Its image in m_A / m_A^2 is nonzero, because
Y_0 X_0bar - X_0 Y_0bar = 0 in the residue field, but as a vector in m/m^2 it
is (Y_0, -X_0) which is not the zero vector. So the kernel is nonzero; it is
not the whole maximal ideal either. The quotient OO is a DVR, so the kernel
is a nonmaximal prime, hence of height exactly one in the two-dimensional local
domain A. Since Y_0 x - X_0 y is a regular parameter (nonzero in m/m^2), it
generates a height-one prime and hence generates the whole kernel.
[end proof]
--------------------------------------------------------------------------------
4.5 The model and the splitting into four points
--------------------------------------------------------------------------------
The closed divisor Spec(OO) inside Spec(A) supports the integral class we push
forward. To detect the pushforward, we identify the generic curve on which the
support separates into conjugate points.
The model. Let
Dcal = Proj( OO[X,Y,Z] / ( X^4 + Y^4 - Z^4 ) ),
C = Dcal_L, (4.18)
T = Spec( A (tensor)_V L ). (4.19)
Dcal is projective. Over an algebraic closure of either L or k, the three
partial derivatives 4X^3, 4Y^3, -4Z^3 do not vanish simultaneously at any
projective point, so each geometric fiber of Dcal -> Spec(OO) is smooth of
dimension one. Each geometric fiber is also integral (smoothness makes it
reduced; two distinct plane components would intersect and create a
singularity). The same Jacobian calculation, on the relative affine charts
where 4 is invertible, proves Dcal is smooth over OO (Stacks, Tag 00TE). Flatness
over the DVR OO excludes a component supported in the special fiber, and the
generic fiber is integral; so Dcal is an integral surface with special fiber
D_k (base change of the Fermat quartic to k).
LEMMA 4.3 (splitting of the divisor into four points).
The scheme T (4.19) is regular, integral, Noetherian, one-dimensional, and flat
over Spec(A). The substitutions X/Z = x/t, Y/Z = y/t identify T with the
spectrum of a localization of the coordinate ring of the chart Z != 0 of C.
The base change to T of the closed immersion Spec(OO) -> Spec(A) in (4.17) is
the disjoint union of four L-rational closed points p_delta, indexed by delta
in Delta. Their coordinates are
p_delta : ( X/Z, Y/Z ) = ( ( delta(t)/t ) X_0, ( delta(t)/t ) Y_0 ). (4.20)
Their specializations on Dcal are g_{ chi(delta) } P. The delta-factor is
obtained from OO by the ring map a |-> delta(a) into L.
Proof.
Flatness. The V-module L is flat: V -> H_0 is a localization (invert 5), and L
is a vector space over H_0. So A (tensor)_V L is flat over A.
Description of T. The substitution x = t X, y = t Y transforms the defining
equation 5 + x^4 + y^4 = 0 into X^4 + Y^4 = 1 (because t^4 = -5). After
tensoring with L and localizing, T is the spectrum of a localization of the
coordinate ring of the chart Z != 0 of C. Its coordinate ring is therefore a
localization of the coordinate ring of a smooth geometrically integral affine
curve over L: it is regular, integral, Noetherian, of dimension at most one.
Splitting of OO (tensor)_V L. Since 5 is invertible in the second tensor
factor, the coordinate ring of the base-changed divisor is
OO (tensor)_V L = L (tensor)_{H_0} L --> prod_{delta in Delta} L,
a (tensor) b |-> ( delta(a) b )_delta. (4.21)
This is the standard splitting of a finite Galois extension into its
components: OO[1/5] = L, and Delta = Gal(L/H_0).
The four points. The image of x under (4.17) and then in the delta-component
of (4.21), is
x |-> t X_0 |-> delta(t) X_0 in L.
Similarly the image of y is delta(t) Y_0. These directions are nonzero (at
least one of X_0, Y_0 is a unit; and delta(t) is a unit of OO), so the
delta-component defines an actual point of T. Coordinates (4.20) follow by
dividing by t.
The points are distinct. The ratios delta(t) / t are four distinct fourth
roots of unity of V by (4.13); and X_0, Y_0 are V-points not both zero, so the
pairs ((delta(t)/t) X_0, (delta(t)/t) Y_0) are four distinct points. Every
inverted denominator of the localization gives a point (its value at t and the
X_0, Y_0 is a unit of OO). Thus T has closed points and its dimension is
exactly one.
Specializations. The coordinates (4.20) are integral over OO and satisfy the
Fermat equation, so they define sections of Dcal. Reducing modulo 5, by
(4.13) delta(t)/t reduces to chi(delta), and X_0, Y_0 reduce to the chosen
X_0bar, Y_0bar. So the reduction of p_delta is (chi(delta) X_0bar : chi(delta)
Y_0bar : 1), which is g_{chi(delta)} P on D_k. This is the specialisation
r(p_delta) = g_{chi(delta)} P.
The remaining assertion (that the delta-factor is obtained by a |-> delta(a))
is exactly the definition of the isomorphism (4.21).
[end proof]
--------------------------------------------------------------------------------
4.6 Summary of what Section 4 provides
--------------------------------------------------------------------------------
We now have:
* a smooth proper model Dcal -> Spec(OO) with generic fiber C over L and
special fiber the Fermat quartic D over k;
* a Picard homomorphism psi : Pic(D_k) -> E(k)/5E(k) that kills Z_inf
(the four points over Z = 0), by (4.3);
* a weighted orbit P in D(k), Q = h(P) in E(k_0), with
4Q != 0 in E(k)/5E(k), by (4.8);
* four points p_delta in T specializing to g_{ chi(delta) } P, by Lemma 4.3,
so that
sum_{ delta in Delta } chi(delta)^2 * psi( [ g_{ chi(delta)} P ] )
= 4Q != 0 in E(k) / 5 E(k), (4.22)
using the counts of Lemma 4.1 (the four values of chi(delta) are exactly
the four elements of F_5^x).
These are precisely the inputs demanded by Proposition 3.1, and they are what
Section 5 assembles.
--------------------------------------------------------------------------------
5. ASSEMBLING THE COUNTEREXAMPLE
--------------------------------------------------------------------------------
The proof now has all three inputs:
(i) the integral cyclotomic class c from Proposition 2.3;
(ii) the closed divisor Spec(OO) inside Spec(A) and its four split points
p_delta from Lemmas 4.2 and 4.3;
(iii) the Picard specialization criterion, Proposition 3.1, together with the
elliptic detector of Section 4.
We first push forward integrally to obtain a class gamma in K_5(A) that is
killed by restriction to Frac A. We then prove gamma is nonzero, by reducing
mod 5 and applying the criterion.
PROOF OF THEOREM (main).
--------------------------------------------------------------------------------
5.1 The integral class gamma
--------------------------------------------------------------------------------
Let c in K_5(S) be the class from Proposition 2.3, and let c_L in K_5(L) be its
restriction along the fixed embedding iota : S -> L (zeta |-> zeta).
There is no ring map S -> OO, because OO has characteristic zero and 5 is not
invertible there; so we cannot restrict c directly to OO. Instead we use the
DVR OO. Integral localization for OO (a complete DVR with residue field k =
F_{5^8}) gives the segment
K_5(OO) -> K_5(L) -> K_4(k) -> K_4(OO). (5.1)
By Quillen's computation for finite fields, K_{2n}(k) = 0 for n >= 1, so
K_4(k) = 0. Hence the map K_5(OO) -> K_5(L) is surjective, and we may choose a
lift c-tilde in K_5(OO) of c_L.
Take the closed immersion
i : Spec(OO) -> Spec(A)
supplied by Lemma 4.2 and define
gamma = i_*( c-tilde ) in G_5(A). (5.2)
Because A is regular, the canonical resolution theorem identifies
G_5(A) = K_5(A); so gamma lives in K_5(A).
gamma is killed by generic restriction. Restriction to Frac A is the direct
image functor along the open immersion Spec(Frac A) -> Spec A. Every coherent
module supported on the proper closed subscheme Spec(OO) restricts to zero on
Spec(Frac A). So every class in the image of i_* dies under restriction to
Frac A:
gamma in ker( K_5(A) -> K_5(Frac A) ). (5.3)
This is the arithmetic half: an integral class, pushed forward from a divisor,
with zero generic image. It remains to show gamma != 0.
--------------------------------------------------------------------------------
5.2 Flat base change and the four points
--------------------------------------------------------------------------------
To detect gamma, make the flat base change V -> L. Put
B_T = A (tensor)_V L, T = Spec( B_T ).
By Lemma 4.3 the diagram
Coprod_{delta in Delta} Spec L --(Coprod p_delta)--> T
| |
v v
Spec OO ----- i ---------> Spec A
is Cartesian. For every finitely generated OO-module M, scalar extension gives
natural isomorphisms of B_T-modules
B_T (tensor)_A i_* M
= ( OO (tensor)_V L ) (tensor)_OO M
= (direct sum over delta in Delta) L (tensor)_{OO, delta} M. (5.4)
The delta-summand is a B_T-module through the quotient defining p_delta. The
functors are exact: B_T is flat over A, and each OO -> L (followed by an
automorphism delta) is a localization composed with an automorphism. So (5.4)
is an identity on integral K-theory.
The image of c-tilde on the delta-factor. Write res_iota : S -> L for the
fixed embedding. Since c-tilde restricts to c_L = res_iota(c) under the
inclusion OO -> L,
its image on the delta-factor (the map OO -> L, a |-> delta(a)) is
res_delta( c-tilde ) = res_{iota} ( delta c ).
Indeed, delta is an automorphism of OO extending to L that commutes with the
inclusion OO -> L, so pushing c-tilde along delta and then along the identity
factor equals restricting the delta-image of c along iota. This uses no
equivariance of the chosen lift c-tilde.
Consequently, integrally,
gamma|_T = sum_{ delta in Delta } (p_delta)_* res_iota( delta c ). (5.5)
Reducing mod 5. Naturality of coefficient reduction and Proposition 2.3 turn
(5.5) into
( gamma|_T ) mod 5
= sum_{ delta in Delta } (p_delta)_* ( chi(delta)^2 * b^2[ delta u ]_5 ),
(5.6)
a class in G_5(T;5). Here b is multiplication by the restriction of beta to
the field L, and the character factors chi(delta)^2 come from Proposition 2.3.
No choice of the lift c-tilde affects this formula.
--------------------------------------------------------------------------------
5.3 Applying the Picard specialization criterion
--------------------------------------------------------------------------------
We apply Proposition 3.1 with:
R = OO (valuation ring of L, residue k = F_{5^8});
X = Dcal -> Spec(OO) (the Fermat model of Lemma 4.3);
C = Dcal_L (generic fiber: Fermat quartic over L);
D = Dcal_k = D_k (special fiber: Fermat quartic over k);
U = T (the localization of Lemma 4.3);
Z = Z_inf (the four points over Z = 0 on D_k);
B = E(k) / 5E(k) (killed by 5);
psi = the elliptic detector (4.2).
We must verify the three hypotheses of Proposition 3.1.
(H1). By Lemma 4.3, T is the spectrum of a localization of the coordinate ring
of the affine chart Z != 0 of C. The points p_delta are L-rational and
distinct by Lemma 4.3. So U = T and p_1, ..., p_4 = p_delta satisfy (H1),
with m = 4.
(H2). We must show: every closed point q of C whose specialization r(q) is not
in Z_inf belongs to U = T. This follows from the definition of A. Let q be
such a closed point, and let OO_q be the valuation ring of the finite residue
field extension kappa(q)/L. If r(q) is not in Z_inf, then r(q) lies in the
chart Z != 0, and the proper extension Spec(OO_q) -> Dcal is also contained in
that chart, because an open set containing the closed point of a DVR contains
its generic point. Hence X/Z and Y/Z belong to OO_q. The original coordinates
x = tX/Z and y = tY/Z then have positive valuation in OO_q. Every denominator
inverted outside (5, x, y) in the definition (4.16) of A has nonzero constant
residue at the closed point, hence is a unit at q. So q lies in the spectrum of
the localization, i.e. q lies in T. This proves (H2), with Z = Z_inf.
(H3). B = E(k)/5E(k) is killed by 5. The homomorphism psi is defined by (4.2).
And by (4.3), psi([r]) = 0 for every r in Z_inf. So (H3) holds.
The test sum. In the notation of Proposition 3.1, set
a_i = delta u (delta in Delta), e_i = chi(delta)^2,
r_i = r(p_delta) = g_{ chi(delta) } P.
Each delta u is a uniformizer of L (because u is), so v_L(delta u) = 1.
The test sum of (3.3) is therefore
sum_{ delta in Delta } chi(delta)^2 * v_L(delta u) * psi( [ g_{chi(delta)} P ] )
= sum_{ delta in Delta } chi(delta)^2 * psi( [ g_{chi(delta)} P ] ). (5.7)
The map delta |-> chi(delta) is an isomorphism Delta -> F_5^x, so the four
values chi(delta) run over all of F_5^x exactly once. Re-indexing the sum by
a = chi(delta) in F_5^x, (5.7) becomes
sum_{ a in F_5^x } a^2 * psi( [ g_a P ] )
= 4 Q != 0 in E(k)/5E(k), (5.8)
by Lemma 4.1. So the condition (3.3) holds.
Conclusion of the criterion. Proposition 3.1 now gives
sum_{ delta in Delta } (p_delta)_* ( chi(delta)^2 * b^2[ delta u ]_5 )
!= 0 in G_5(T;5). (5.9)
By (5.6), the left-hand side is exactly ( gamma|_T ) mod 5. Hence
( gamma|_T ) mod 5 != 0 in G_5(T;5). (5.10)
--------------------------------------------------------------------------------
5.4 gamma is nonzero
--------------------------------------------------------------------------------
Since (gamma|_T) mod 5 is nonzero, gamma|_T itself is nonzero in G_5(T). The
class gamma|_T is the flat pullback of gamma along the flat map T -> Spec A.
A flat pullback of a zero class is zero. Therefore gamma != 0 in G_5(A) =
K_5(A).
Combined with (5.3), this proves
gamma in ker( K_5(A) -> K_5(Frac A) ), gamma != 0,
which is the statement of the Theorem. The proof of the main theorem is
complete.
[end proof]
--------------------------------------------------------------------------------
5.5 Remarks
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Why a principal divisor can give a nonzero pushforward. The divisor
Spec(OO) is cut out by the single regular parameter Y_0 x - X_0 y, so it is a
principal divisor in the local ring A. One might expect its pushforward to
vanish. It does not: the self-intersection formula gives i^* i_* = 0 on
assembled classes, and the projection formula gives i_* i^* = 0, but neither
formula asserts i_* = 0 (that would be the composite in the wrong direction).
The pushforward of an arbitrary class on the divisor need not be zero in higher
K-theory, and in our case it is not. In particular, the proof never assumes
that c-tilde is the restriction to OO of a class on A.
Where the position of the four points matters. If one forgets the four signed
positions and retains only the four normalized valuations (each equal to 1), the
scalar sum 1 + 4 + 4 + 1 = 10 = 0 in F_5. So a purely scalar functional would
see nothing. The Picard test keeps the actual points r_delta = g_{chi(delta)} P
on the special fiber and computes 4Q, which is nonzero. This is the whole
content of the elliptic detector: it can distinguish the points, whereas the
scalar reduction cannot. (The scalar computation is one specific forgetting; it
makes no assertion about all possible scalar functionals.)
What the proof does not give. Detecting the mod-5 reduction of gamma identifies
gamma as nonzero, but it does not determine the exact order of gamma in K_5(A),
nor whether the rationalization of gamma is nonzero. The theorem settles the
unrestricted integral form of Gersten's conjecture and leaves the
rational-coefficient variant untouched.
--------------------------------------------------------------------------------
A. SUPPLEMENTARY: ON A CLAIMED VANISHING THEOREM
--------------------------------------------------------------------------------
This appendix is independent of the main proof. It records a defect in a
separate preprint (Mochizuki, Local Gersten's conjecture for regular system of
parameters, arXiv:1503.07966v8), in the piece of that paper that would, if
correct, annihilate the very pushforward we use. The Theorem (main) does not
rely on any argument here.
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A.1 The claim and why it concerns us
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Mochizuki's Theorem 5 (in version 8) asserts, for a commutative Noetherian
local ring B and a nonzerodivisor g: the inclusion from finitely generated
projective B/gB-modules into finitely generated B-modules of projective
dimension at most one induces zero on K-theory.
This statement reaches the exact support of our construction. Lemma 4.2 gives
B = A, g = Y_0 x - X_0 y, and A / gA = OO, with g a regular parameter.
Composing the claimed zero map with the inclusion into coherent A-modules would
annihilate i_*(c-tilde), the class gamma.
We give an explicit counterexample to the functor assertion in
Mochizuki's Lemma 13(III), which is used in the proof of that Theorem. The
counterexample already has both B and B/gB regular. It concerns the supplied
proof construction; a failure of this construction alone does not refute the
Theorem 5 statement. And the main theorem of this document is proved without
any input from the preprint.
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A.2 The complexes and the proposed operation
--------------------------------------------------------------------------------
Let B be a regular local ring and let g be a nonzero regular parameter. In
particular g is a nonzerodivisor in the maximal ideal, so 1 - g is a unit.
The exact category C used in the cited argument consists of two-term complexes
of finite free B-modules, in homological degrees one and zero, whose
differential is injective and whose cokernel is killed by g. Exact sequences
are degreewise exact. Write, for n, m >= 0,
(n,m)_B = [ B^n + B^m --diag(g I_n, I_m)--> B^n + B^m ]. (A.1)
We use only these objects and split exact sequences between them, so no
classification of other objects of C is needed.
For X = (1,1)_B, put D = diag(g, 1). Following the block convention in
Mochizuki (Conventions 11, equations (15)-(16)), an encoded endomorphism
M = [[ a, b ],
[ c, d ]]
has actual chain matrices
M_1 = [[ a, b ], M_0 = [[ a, gb ],
[ gc, d ]], [ c, d ]]. (A.2)
These satisfy D M_1 = M_0 D. Conversely, the chain-map condition together with
g a nonzerodivisor gives this presentation. Multiplying the actual matrices
shows that composition of encoded matrices is
[[ a', b' ], [[ a, b ], [[ a'a + g b'c, a'b + b'd ],
[ c', d' ]] * [ c, d ]] = [ c'a + d'c, g c'b + d'd ]].
(A.3)
Thus M' * M represents M' (compose) M. "Upper triangular" means c = 0, and
"lower triangular" means b = 0, referring to the encoded matrices.
The assignments mu'_1, mu'_2 of the cited paper (preceding Definition 12)
replace (n,m)_B by Typ(g)^{+n} or Typ(1)^{+n} respectively, where
Typ(h) = [ B --h--> B ]. On a morphism they retain its upper-left block in
both chain degrees. On all morphisms this is not a functor, because the
upper-left entry of (A.3) contains the term g b' c. The cited argument
therefore restricts the diagrams and the maps.
Here is the part of that restriction relevant to the example. An S_2-object in
an exact category is an admissible short exact sequence
X_01 >-> X_02 ->> X_12, together with the zero objects on its diagonal;
morphisms are commuting maps of these sequences. Mochizuki's Definition 12
requires upper triangular structure arrows and allows lower triangular
componentwise isomorphisms between the diagrams. Lemma 13(III) asserts that
mu'_1, mu'_2 induce functors on this modified S-construction. The following
example violates the naturality required of an image morphism already in
degree two.
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A.3 The counterexample
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LEMMA A.1 (triangular naturality failure).
For every regular local ring B with a nonzero regular parameter g, the
upper-left-block assignments just described fail to send all lower triangular
isomorphisms between upper triangular S_2-objects to morphisms of the target
diagrams.
Proof.
Take the encoded endomorphisms of X = (1,1)_B:
U = [[ 1, 1 ], U' = [[ 1-g, 1 ],
[ 0, 1 ]], [ 0, 1 ]],
L = [[ 1, 0 ], L' = [[ 1, 0 ],
[ 1, 1 ]], [ 1, 1-g ]].
Their actual chain matrices are
U_1 = [[ 1, 1 ], U_0 = [[ 1, g ],
[ 0, 1 ]], [ 0, 1 ]],
U'_1 = [[ 1-g, 1 ], U'_0 = [[ 1-g, g ],
[ 0, 1 ]], [ 0, 1 ]],
L_1 = [[ 1, 0 ], L_0 = [[ 1, 0 ],
[ g, 1 ]], [ 1, 1 ]],
L'_1 = [[ 1, 0 ], L'_0 = [[ 1, 0 ],
[ g, 1-g ]], [ 1, 1-g ]].
In each degree the determinants of U, U', L, L' are respectively
1, 1-g, 1, 1-g. All are units, so all four are chain isomorphisms. The first
two are upper triangular and the last two lower triangular.
A direct calculation with (A.3):
L' * U = [[ 1, 1 ],
[ 1, 1 ]] = U' * L.
Equivalently, both degree-one composites are [[1,1],[g,1]] and both
degree-zero composites are [[1,g],[1,1]]. Hence the diagram
X --U--> X --> 0
| |
L L'
v v
X --U'-> X --> 0
commutes. Each row is split exact (the first arrow is an isomorphism). Besides
that first arrow, all structure arrows of the corresponding S_2-object are zero
or identity, hence upper triangular. The vertical maps (L, L', id_0) are lower
triangular isomorphisms. So the diagram is a morphism in the proposed source
category.
Now consider the image under either assignment. Both upper-left entries of L
and L' are 1, whereas those of U and U' are 1 and 1-g. Either assignment would
send the left square to
Typ(h) --1--> Typ(h)
| |
1 1
v v
Typ(h) -1-g-> Typ(h), h = g or h = 1.
Its two composites are 1 and 1-g in both degrees. These are unequal, because
g != 0. So the proposed image is not a morphism of S_2-diagrams.
[end proof]
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A.4 What the calculation identifies
--------------------------------------------------------------------------------
The computation isolates the precise condition missing from the cited argument:
preservation of upper-upper and lower-lower compositions does not imply
preservation of mixed upper-lower commuting squares.
In version 8 the relevant locators are: the encoded composition and assignments
on page 9, Lemma 13(III) on page 10, and the subsequent additivity argument on
page 11. That argument uses the asserted functors in Lemma 13(IV) and in the
proof of assertion (beta). Although the two projected composites can be chain
homotopic, the target category requires strictly commuting natural
transformations; the later homotopy comparison does not define replacement
functors on this square. A construction that supplies the required higher
compatibilities and additivity would be an additional argument.
This appendix does not bear on the main theorem: the main proof never invokes
Mochizuki's Theorem 5, and the counterexample class gamma is detected by the
elliptic quotient of Section 4, not by any vanishing statement.
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END OF DOCUMENT
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