# Positivity of Serre's intersection multiplicity

*An exposition carrying the complete proof, written to be readable with high-school algebra. The long technical stretches keep every statement precise; the surrounding prose is there so you can follow the shape of the argument even when a single step is heavy.*

## 0. The whole paper in plain language

Two shapes inside a space usually meet in a predictable number of points. Two straight lines in the plane meet in one point. A line and a parabola meet in two points (counted correctly). Two circles meet in two points. This "how many points, counted correctly" is the *intersection number*. Serre, in the 1950s, gave a high-powered formula for the intersection number of any two sub-objects, valid in every dimension and every characteristic:

$$\chi^R(M,N)=\sum_{i=0}^{d}(-1)^i\,\operatorname{length}_R\operatorname{Tor}_i^R(M,N).$$

He observed that in every case he could compute, this number is positive whenever the two objects genuinely cross in a single isolated way. He conjectured it always is. This paper proves that conjecture, in full generality, including the hard "ramified mixed characteristic" cases where $p$ behaves badly.

The subtlety: the formula has *alternating signs*. The $i=0$ term is the naive "how many points", which is positive. But the higher Tor terms subtract and add, and in principle they could overwhelm it and drag the total to zero or below. The whole difficulty is that the alternating signs do not respect the obvious positivity of each term. Serre's conjecture says the geometry secretly forces the total to stay positive.

The proof works by *averaging, twice*. Each averaging turns a difficult signed sum into a sum of manifestly positive pieces. The technical machinery exists to make the two averagings compatible.

## 1. The statement

Let $(R,\mathfrak m)$ be a commutative Noetherian **regular local ring** of dimension $d$: this means $\mathfrak m$ can be generated by exactly $d$ elements, so $R$ is as "smooth" as an $d$-dimensional local ring can be. Let $M,N$ be nonzero finitely generated $R$-modules. Serre's intersection multiplicity is

$$\chi^R(M,N)=\sum_{i=0}^{d}(-1)^i\,\operatorname{length}_R\operatorname{Tor}_i^R(M,N).$$

If the tensor product $M\otimes_R N$ has finite length, every $\operatorname{Tor}$ above also has finite length, and regularity guarantees that all $\operatorname{Tor}_i$ with $i>d$ vanish. So the sum is a finite integer.

**Theorem 1.1 (Serre's positivity conjecture).** *Let $(R,\mathfrak m)$ be a regular local ring of dimension $d$, and let $M,N$ be nonzero finitely generated $R$-modules. If $\operatorname{length}_R(M\otimes_R N)<\infty$ and $\dim_R M+\dim_R N=d$, then $\chi^R(M,N)>0$.*

Two words of caution. "Dimension" means the Krull dimension of the support $\operatorname{Supp}(M)=\{\mathfrak p: M_{\mathfrak p}\neq 0\}$. The hypothesis $\dim M+\dim N=d$ is the "complementary dimension" condition: the two supports are the largest they can be while still meeting in a finite-length way. It is exactly the situation where positivity is not automatic from vanishing theorems.

**Example 1.2.** Let $R=k[x,y]$, $d=2$. Take $M=R/(y)$ (the $x$-axis) and $N=R/(y-x^2)$ (the parabola). Then $M\otimes_R N = R/(y,y-x^2)=R/(y,x^2)\cong k[x]/(x^2)$, which has $k$-length $2$. All higher Tor vanish, so $\chi=2>0$. The parabola is tangent to the axis at the origin, and tangency is worth two points. If instead $N=R/(x)$, we get $R/(x,y)\cong k$, length $1$, the transverse crossing. Positivity is easy here; the whole point of Theorem 1.1 is that it stays true in far more complicated situations.

## 2. What was known, and why it is hard

- **Equicharacteristic, or unramified:** Serre himself reduced to the diagonal and proved positivity whenever the characteristic of the residue field is finite and "unramified" (a lift to an unramified situation exists). This covers most classical geometry.
- **Vanishing:** Roberts, and independently Gillet–Soule, proved that if $\operatorname{length}(M\otimes N)<\infty$ but $\dim M+\dim N0$, the total is positive.

*Second comparison.* Fix one $\sigma$ and consider its number $c_\sigma$. Pick a parameter sequence $x_1,\ldots,x_h\in Q$ forming a regular sequence on $R$ and a parameter system on $D$. The point: $U=C_D/(x)C_D$ has positive finite normalized length, and $U$'s higher Tor with any finite module has *normalized length zero* by the design of $\lambda$ (the "Koszul vanishes in positive degree" property). Approximate $U$ by honest finite-length modules $M_n$. Their minimal resolutions have negligible rank beyond degree $l=\dim E$, so the truncations become compact supported objects in a large "asymptotic" category. Inside that category the same descent theorem applies, over the *other* base $A_E\subset E'$, and gives

$$|G_E|\,c_\sigma \;=\; \sum_{\gamma\in G_E} b_{\sigma,\gamma}\,.$$

The ordinary evaluation of the truncated object is $c_\sigma$ again; the normalized evaluation is $b_{\sigma,\gamma}$; and the second Galois average forces each $b_{\sigma,\gamma}$ to be a strictly positive number (because it dominates the positive normalized generator count of $M_n$ times the positive normalized length of the residue quotient of $C_E$). Hence $c_\sigma>0$ for every $\sigma$, hence the first average is positive.

Neither averaging needs a Galois action on $C_D$ or $C_E$, nor invariance of either normalized length. The two comparisons live on different bases and only meet at the level of alternating sums.

**The two-comparison picture.**

  [ Ordinary supported comparison over A_D subset D' ]
      |G_D| a_D a_E chi^R(D,E) = sum_{sigma in G_D} c_sigma
             prove positivity of each c_sigma separately
  [ Asymptotic supported comparison over A_E subset E' ]
      |G_E| c_sigma = sum_{gamma in G_E} b_{sigma,gamma} > 0

## 4. Conventions

- Universes are chosen large enough for the module and sequence categories used.
- Complexes are indexed homologically.
- A derived algebra $H$ is **connective** if $\pi_i(H)=0$ for $i<0$.
- Perfect modules over associative algebras are right modules.
- All localizations of coefficient algebras are along the central image of the base ring.
- We use nonconnective algebraic $K$-theory of perfect complexes; a subscript $\mathbb Q$ means rationalization.
- For a sequence $a=(a_1,\dots,a_r)$, $K(a;V)$ is its homological Koszul complex with coefficients in $V$, and $H_i(a;V)=H_i(K(a;V))$.
- The notation $a^v$ means $(a_1^v,\dots,a_r^v)$, not the $v$-th power of the ideal.
- Normalized lengths may vanish on nonzero modules; all uses of additivity involve finite values or nonnegative extended values for which the operations are defined.

Several sections follow: normalized length (Section 5), rational descent (Section 6), the asymptotic rank quotient (Section 7), and the positivity proof (Section 8). The short statement of the consequence for symbolic powers and the tangent-cone corollary appears at the end (Section 9).

## 5. Normalized length

This section builds the "substitute length" that powers all the positivity assertions. The idea: ordinary length misbehaves in the presence of ramification and Frobenius. If we pass to a tower of $p$-power root extensions and divide by the growth rate (the *degree* of each finite step), the resulting normalized length has exactly the properties we want:

- a parameter quotient has *prescribed positive* normalized length;
- the positive-degree Koszul homology of any parameter system has normalized length *zero*.

Both properties are needed at once. The first gives us the positive quantity that survives the alternating sum; the second guarantees that in the alternating sum only the degree-zero term has a chance to be nonzero, so nothing is subtracting.

Throughout this section, $(D,\mathfrak d,k)$ is a complete Noetherian local *domain* of dimension $h>0$, with $k$ perfect of characteristic $p>0$. Neither unramifiedness nor Cohen–Macaulayness of $D$ is assumed.

### 5.1 Extending the residue field

Passing to an algebraically closed residue field is a standard trick, but it must preserve regularity, dimensions, lengths and intersection multiplicities. The following lemma does exactly that.

**Lemma 5.1 (Residue-field extension).** *Let $(B,\mathfrak b,k)$ be a complete Noetherian local ring and let $\overline k$ be an algebraic closure of $k$. There is a complete Noetherian local ring $(B',\mathfrak b',\overline k)$ and a flat local map $B\to B'$ with $\mathfrak b B'=\mathfrak b'$. For every finite $B$-module $V$ and every $a\ge1$,*

$$\operatorname{length}_{B'}\big((V\otimes_B B')/(\mathfrak b')^a(V\otimes_B B')\big)=\operatorname{length}_B(V/\mathfrak b^a V).$$

*Consequently finite lengths and module dimensions are preserved. If $B$ is regular, so is $B'$. For finite $B$-modules $M,N$ with finite-length tensor product, every Tor length, and hence the intersection multiplicity, is preserved.*

*Proof.* In equal characteristic a coefficient field gives $B=k[[X_1,\dots,X_t]]/I$; set $B'=\overline k[[X_1,\dots,X_t]]/I\overline k[[X_1,\dots,X_t]]$. The map on power-series rings is flat (local flatness criterion on the variables, plus flatness of the coefficient-field extension), and base change to the quotient gives the stated residue field and maximal ideal.

In mixed characteristic Cohen structure gives $B=V[[X_1,\dots,X_t]]/I$ with $V$ a complete discrete valuation ring of uniformizer $p$ and residue field $k$. To enlarge the residue field while keeping $p$ as a uniformizer, lift a monic irreducible polynomial over $k$ to $f\in V[X]$; then $V[X]/(f)$ is finite free over $V$, reduces mod $p$ to the extension field, and has maximal ideal generated by $p$, hence is again a discrete valuation ring. Iterate along a tower of simple algebraic residue extensions exhausting $\overline k$; at every limit stage an ideal is still generated by a least-valuation element, and the completion $V'$ has uniformizer $p$ and residue field $\overline k$, with $V\to V'$ flat (torsion-free). Set $B'=V'[[X_1,\dots,X_t]]/IV'[[X_1,\dots,X_t]]$; local flatness gives the required flatness. Compare the standard construction in Ma's paper, Lemma 2.2.

For the length claim, tensor a composition series with $B'$: flatness preserves exactness and every simple factor becomes $B'/\mathfrak b'$. Apply this to $V/\mathfrak b^a V$. Equality of Hilbert–Samuel functions gives equality of module dimensions; if $B$ is regular of dimension $d$, its $d$ generators generate $\mathfrak b'$ and $\dim B'=d$, so $B'$ is regular. Flat base change identifies Tor after extension, and lengths are preserved by the first part. $\square$

### 5.2 Regular normalization bases

**Lemma 5.2 (Regular normalization bases).** *Let $(D,\mathfrak d,k)$ be a complete Noetherian local domain of dimension $h>0$ with $k$ perfect of characteristic $p>0$. There is a finite, generically separable inclusion $A\subseteq D$ inducing the identity on residue fields, with*

$$A=k[[t_1,\dots,t_h]]\ \text{if}\ \operatorname{char}D=p,\qquad A=W(k)[[t_2,\dots,t_h]]\ \text{if}\ \operatorname{char}D=0,$$

*$W(k)$ the ring of Witt vectors. In the second case set $t_1=p$. In either case $t_1,\dots,t_h$ is a parameter system of $D$ and a regular parameter system of $A$.*

*Proof.* In characteristic $p$ this is the Cohen–Gabber normalization theorem. In mixed characteristic choose a coefficient ring $W(k)\subseteq D$ and extend $p$ to a parameter system $p,t_2,\dots,t_h$; completeness plus the parameter condition makes the resulting power-series map module finite. Its kernel is zero because source and target both have dimension $h$ and a nonzero ideal in a regular local domain lowers dimension. The fraction-field extension is separable in characteristic zero. $\square$

### 5.3 The $p$-power tower and its length

Fix a base $A$ from Lemma 5.2 and compatible $p$-power roots of its parameters:

$$A_n=A[t_1^{1/p^n},\dots,t_h^{1/p^n}],\qquad \Lambda=\varinjlim_n A_n.$$

The tower is *uncompleted*; only in mixed characteristic do we also use its $p$-adic completion, denoted $A_\infty$.

**Lemma 5.3 (The tower and its length).** *Each $A_n$ is a complete regular local ring with residue field $k$. For $b\ge a$ the map $A_a\to A_b$ is finite free of rank $p^{(b-a)h}$; in particular $\operatorname{rank}_A A_n=p^{nh}$. On the category of $\Lambda$-modules killed by some power of $\mathfrak m_A$ there is a nonnegative additive function $\lambda_\infty$ with three properties:*

1. *a finitely presented module descends to a finite-length $A_n$-module $V_n$, and* $\lambda_\infty(V_n\otimes_{A_n}\Lambda)=p^{-nh}\operatorname{length}_{A_n}(V_n)$;
2. *for a finitely generated module $V$, its length is the infimum of the lengths of finitely presented covers of it;*
3. *for an arbitrary module $V$, $\lambda_\infty(V)=\sup\{\lambda_\infty(V_0): V_0\subseteq V\text{ finitely generated}\}$.*

*The value on an arbitrary module may be infinite. In mixed characteristic, restriction from $A_\infty$ to $\Lambda$ gives the same length on modules killed by a fixed power of $\mathfrak m_A$.*

*Proof.* Stages are power-series rings in the chosen roots (over $W(k)[p^{1/p^n}]$ in mixed characteristic). Monomials in the successive roots with exponents below the root degrees form bases for all transition extensions, proving the regularity, rank, and residue-field statements.

Finite presentations over the union descend to a stage, and the additional relations expressing annihilation by a fixed power of $\mathfrak m_A$ descend after enlarging the stage (finitely many generators and relations). The descended finite-stage module has finite length, and the displayed formula is unchanged on enlarging the stage by flatness and the computed free ranks. The infimum and supremum extension properties, with additivity and compatibility, are Faltings' normalized-length formalism (see Definition 2.2.1 and Propositions 2.2.3, 2.2.6(a) of CLMST). Same residue field gives equality of $A_n$-length and $A$-length on finite-length modules.

In mixed characteristic $\Lambda$ is $p$-torsion-free and $\Lambda/p^a\Lambda\simeq A_\infty/p^a A_\infty$; a module killed by $\mathfrak m_A^a$ is killed by $p^a$, so its submodules and finite generating sets are the same over the two bases. Replacing a cover by $V/\mathfrak m_A^a V$ keeps it finitely presented and over identical quotient rings, proving the last assertion. $\square$

**Lemma 5.4 (Zero-length limits).** *Assume all modules under consideration are killed by one fixed power of $\mathfrak m_A$. Sums, direct sums, and directed colimits of zero-length modules have length zero. For an increasing union $\lambda_\infty(\bigcup_a V_a)=\sup_a\lambda_\infty(V_a)$. If $(V_a)$ is a direct system with $\lambda_\infty(V_a)\to 0$, then $\lambda_\infty(\varinjlim_a V_a)=0$, with no injectivity assumption on the transition maps.*

*Proof.* A finite set of elements of a direct sum lies in a finite subsum; additivity and the supremum formula give the direct-sum assertion, and sums and colimits are quotients of direct sums. For an increasing union, every finite generating set lands in one stage. For the last claim, a finitely generated submodule of the colimit lifts its generators to a common late stage, and monotonicity bounds its length by $\lambda_\infty(V_a)$ there; apply the supremum formula. The common annihilator keeps all modules in the length category. $\square$

### 5.4 The normalized-length algebra $C_D$

For a $\mathfrak d$-primary ideal $I$ of $D$ write $e(I,D)$ for its Hilbert–Samuel multiplicity. Related normalized-length algebras (parameter-quotient length $=$ multiplicity, positive Koszul homology of length zero) appear in OpenAI's Lech paper, Theorem 3.1; here we retain the chosen base $A\subseteq D$ and the finite-stage, infimum, and supremum properties on underlying tower modules.

**Theorem 5.5 (Normalized-length input).** *Let $(D,\mathfrak d,k)$ and a chosen finite generically separable base $A\subseteq D$ be as in Lemma 5.2, with tower $\Lambda$ as above. There is a $D$-algebra $C_D$, carrying a compatible $\Lambda$-action, such that restriction of scalars is an additive nonnegative length $\lambda_D=\lambda_\infty$ on its modules annihilated by a power of $\mathfrak d$. For every parameter system $z_1,\dots,z_h$ of $D$ and every $v\ge1$,*

$$\lambda_D H_i(z_1^v,\dots,z_h^v;C_D)=0\quad(i>0),\qquad \lambda_D\big(C_D/(z_1^v,\dots,z_h^v)C_D\big)=v^h\,e((z_1,\dots,z_h),D).$$

*The right side is positive and finite, and the algebra and length are independent of the parameter system $z$. For the regular parameters of $A$ in particular,*

$$\lambda_D\big(C_D/(t_1^v,\dots,t_h^v)C_D\big)=v^h\operatorname{rank}_A D.$$

*The finite-stage, infimum, and supremum properties of Lemma 5.3 remain available on underlying tower modules; they do not require the relevant submodules to be $C_D$-modules.*

*Proof, characteristic $p$.* Put $C_D=D_{\mathrm{perf}}=\bigcup_n D^{1/p^n}$ inside a fixed algebraic closure of its fraction field, and $\Lambda=A_{\mathrm{perf}}$. Let $K=\operatorname{Frac}A$, $L=\operatorname{Frac}D$, $K'=K_{\mathrm{perf}}$. For $q=p^n$ set $B_q=\Lambda\otimes_{A^{1/q}}D^{1/q}$. Flatness of $\Lambda/A^{1/q}$ and torsion-freeness of $D^{1/q}$ embed this into $K'\otimes_{K^{1/q}}L^{1/q}$, which is a field because $L/K$ is generically separable (linear disjointness from a purely inseparable extension). So $B_q=\Lambda D^{1/q}\subseteq C_D$, increasing to $C_D$.

For a $\mathfrak d$-primary ideal $H$, set $V_q=B_q/HB_q$ and $W_q=\operatorname{im}(V_q\to C_D/HC_D)$. With $H^{[q]}=(a^q:a\in H)$ the Frobenius power, finite-stage normalization gives $\lambda_\infty(V_q)=q^{-h}\operatorname{length}_D(D/H^{[q]})$. The map $V_q\to W_q$ need not be injective; control its kernel by a single trace denominator: there is a nonzero $g\in A$ with

$$g\,C_D\subseteq B_1,\qquad g^{1/q}C_D\subseteq B_q.$$

Indeed, pick a $K$-basis $\{b_j\}$ of $L$ in $D$ and its trace-dual $\{\beta_j\}$; a common denominator $g\in A\setminus\{0\}$ makes every $g\beta_j\in D$. The field $LK'$ is perfect and equals $L_{\mathrm{perf}}$, and $\Lambda$ is normal. For $c\in C_D$, each $\operatorname{Tr}_{LK'/K'}(g\beta_jc)$ is integral over $\Lambda$ and lies in $K'$, hence in $\Lambda$; the trace-dual expansion proves $gC_D\subseteq B_1$, and taking $q$-th roots gives $g^{1/q}C_D\subseteq B_q$.

The kernel of $V_q\to W_q$ is killed by $g^{1/q}$. Hence, using that kernel and cokernel of an endomorphism of a finite-length object have equal length,

$$\lambda_\infty\ker(V_q\to W_q)\le q^{-h}\operatorname{length}_D D/(H^{[q]},g)=O(q^{-1}).$$

For the bound, if $H$ has $s$ generators then $H^{s(q-1)+1}\subseteq H^{[q]}$, so $H^{[q]}$ contains a power of $\mathfrak d$ with exponent $O(q)$, while $\dim D/(g)\le h-1$ for a nonunit $g$; Hilbert–Samuel growth gives the estimate (and for a unit $g$ the quotient is zero). The $W_q$ increase to $C_D/HC_D$ and are uniformly primary-torsion. The Hilbert–Kunz limit together with Lemma 5.4 gives

$$\lambda_D(C_D/HC_D)=e_{\mathrm{HK}}(H,D),$$

using Monsky's existence theorem for Hilbert–Kunz multiplicity (and CLMST/Ma for the notation).

For parameter ideals we need the Hilbert–Samuel value and vanishing of positive Koszul homology. For a parameter system $z$, finite-stage flatness gives $\lambda_\infty H_i(z;B_q)=q^{-h}\operatorname{length}_D H_i(z_1^q,\dots,z_h^q;D)$. The Koszul complex $K(z;D)$ has finite free terms in degrees $0,\dots,h$ and finite-length homology; the Frobenius homology theorem of Roberts (Ma, Theorem 2.4) says the displayed quantity tends to zero for $i>0$. Filtered colimits commute with homology, the homology modules have common annihilator $(z)$, so Lemma 5.4 proves vanishing, also for $z^v$. Finally Koszul Euler formula gives $\sum_i(-1)^i \operatorname{length}_D H_i(z^q;D)=q^h e((z),D)$; dividing by $q^h$, the positive-degree terms go to zero, and with the identification of the quotient length and the power rule this gives $\lambda_D(C_D/(z^v)C_D)=v^h e((z),D)$.

*Proof, mixed characteristic.* Let $A_\infty$ be the $p$-adic completion of $\Lambda$ and $C_D=(D\otimes_A A_\infty)_{\mathrm{perfd}}$, the perfectoidization of CLMST (built on Bhatt–Scholze, Proposition 8.5); the hypotheses there are exactly that $D$ is a complete mixed-characteristic local domain with perfect residue field over the chosen finite Witt power-series base $A$. Choose $g\in A\setminus\{0\}$ with $A[1/g]\to D[1/g]$ finite etale. CLMST Corollary 4.0.4 annihilates positive Koszul homology for every parameter system of $A_n\otimes_A D$ by $(g)_{\mathrm{perfd}}$; at $n=0$ this covers every parameter system of $D$. CLMST Proposition 4.0.10 makes these uniformly primary-torsion modules have normalized length zero, and Proposition 4.0.14 at $n=0$ identifies the parameter-quotient length with the Hilbert–Samuel multiplicity. Applied to $z^v$, these give the two displayed identities for $C_D$. Lemma 5.3 gives restriction to the uncompleted tower throughout.

In either characteristic, a module killed by a power of $\mathfrak d$ is killed by a power of $\mathfrak m_A$, so all lengths are defined; additivity and tower properties are as established. The multiplicity rank formula and equality of residue fields give $e((t),D)=\operatorname{rank}_A D\,e((t),A)=\operatorname{rank}_A D$, proving the last displayed identity. $\square$

**Corollary 5.6 (Finite-length base change).** *In Theorem 5.5 put $r=\operatorname{rank}_A D$. For every finite-length $A$-module $V$,*

$$\lambda_D\operatorname{Tor}_i^A(V,C_D)=0\ (i>0),\qquad \lambda_D(V\otimes_A C_D)=r\operatorname{length}_A V.$$

*Moreover $\lambda_D(C_D/\mathfrak d C_D)>0$.*

*Proof.* For $V=k$ the first two assertions are the two identities of Theorem 5.5, using the Koszul resolution of $k$ over regular $A$. Induct on a composition series of $V$: the Tor long exact sequences and additivity discard positive-degree terms and show degree-zero length increases by $r$ at each step; all modules have bounded primary torsion.

For the last claim choose a parameter ideal $H$ of $D$; right-exactness of tensoring a composition series of $D/H$ with $C_D$ gives a filtration of $C_D/HC_D$ whose factors are quotients of $C_D/\mathfrak d C_D$, so $00$, additivity of $K$-theory identifies the composite with $a$; this is natural on the whole diagram, so rationally $a^{-1}\operatorname{tr}_f$ makes $\mathcal L$ a retract of $\mathcal L_X$. After base change to $X$, the augmented Cech nerve of $f$ has a section, hence an extra degeneracy, and its image under any presheaf has split descent, so $\mathcal L_X$ satisfies descent for $f$; the comparison for $\mathcal L$ is a retract of an equivalence, hence an equivalence. Finally both sides are Zariski sheaves, so the conclusion glues over the trivializing cover. All constructions commute with base change. $\square$

**Lemma 6.3 (Finite surjection).** *$\mathcal L$ satisfies ordinary Cech descent for every finite surjection of finite-type $A$-schemes.*

*Proof.* Let $\tau$ be the topology generated by cdh covers and finite faithfully flat covers; these families are stable under base change, and $\mathcal L$ is a $\tau$-sheaf by Lemma 6.2 and cdh descent. We show every finite surjection is a $\tau$-cover.

Reductions and finite closed decompositions are cdh-local. If a reduced scheme is $Y=Y_1\cup Y_2$ with $Y_i$ reduced closed subschemes, the abstract blowup square with proper map $Y_1\to Y$ and center $Y_2$ reduces closed unions to their pieces. So reduce to integral $Y$, with induction on dimension for proper closed subsets.

Choose a reduced irreducible component $X_0$ of $X$ dominating $Y$; the map $X_0\to Y$ is finite surjective, flat of positive rank on a dense open $U\subset Y$. Flattening by blowup applied to $\mathcal O_{X_0}$ gives a blowup $Y'\to Y$ isolating $U$ over which the strict transform $X_0'\to Y'$ is flat and finitely presented (Stacks Tag 0815). The strict transform is a closed subscheme of $X_0\times_Y Y'$, hence finite over $Y'$; since $Y'$ is integral and the image of $X_0'$ contains its generic point, that image is closed; so $X_0'\to Y'$ is a finite faithfully flat cover factoring through $X\times_Y Y'$.

Let $Z\subsetneq Y$ be the blowup center; the family $\{Y'\to Y, Z\to Y\}$ is a cdh cover. By induction $X\times_Y Z\to Z$ is a $\tau$-cover. Together with $X_0'\to Y'$, this gives a $\tau$-covering family of $Y$ factors through $X$, so $X\to Y$ generates a covering sieve. Dimension zero starts the induction: a reduced integral base is a field, and any nonzero finite algebra is finite faithfully flat. $\square$

### 6.3 The geometric orbit argument

**Proof of Theorem 6.1.** Write $X=\operatorname{Spec}T$, $S=\operatorname{Spec}A$. On finite-type $S$-schemes consider

$$\mathcal L(Y)\longrightarrow\mathcal L(X\times_S Y)^{hG}.$$

Both sides are sheaves for the topology of Lemma 6.3 (base change preserves covers; homotopy fixed points commute with limits). Applying finite-surjection descent to $X\to S$, it suffices to prove the comparison on nonaugmented terms of its Cech nerve. Over each such term the pulled-back $X$-cover admits a section (projection to an $X$-coordinate), and geometric transitivity is stable under base change. So reduce to covers admitting a section.

Induct on $\dim Y$. Nilinvariance and the closed-union squares reduce to integral reduced $Y$. Let $s:Y\to X\times_S Y$ be a section; its finitely many $G$-translates cover the source on points (extend the residue field at any point to an algebraic closure and use transitivity), each is a closed immersion because the source is separated over $Y$, and each factors through the reduction $V=(X\times_S Y)_{\mathrm{red}}$. Retain the distinct translated sections with indexing set a transitive finite $G$-set $I$. Two distinct sections have equalizer a proper closed subscheme; if they agreed at the generic point of integral reduced $Y$ they would agree everywhere, so their coincidence locus is proper. Let $Z\subsetneq Y$ be the reduced union of the coincidence loci ($Z=\varnothing$ if $I$ is a single point).

The map $q:W=\coprod_{i\in I}Y\to V$ is finite surjective. Over $Y\setminus Z$ the section images are pairwise disjoint and cover $V$, so $q$ is a proper modification, and the square

$$
\begin{array}{ccc}
W\times_Y Z&\to&W\\
\downarrow&&\downarrow\\
V\times_Y Z&\to&V
\end{array}
$$

is an abstract blowup square, $G$-equivariant.

The $G$-action on $W$ permutes its copies of $Y$, and restriction gives a $G$-equivariant equivalence $\mathcal L(W)\simeq\prod_{i\in I}\mathcal L(Y)$; a stabilizer of one section acts identically on its copy (a canonical identity of coefficient-bearing $Y$-schemes). The same holds after restriction to $Z$. Hence, since $\mathcal L$ is rational and averaging over a finite group is exact,

$$\mathcal L(W)^{hG}\simeq\mathcal L(Y),\qquad \mathcal L(W\times_Y Z)^{hG}\simeq\mathcal L(Z).$$

(If $I=G/G_s$, the first fixed-point spectrum is $\mathcal L(Y)^{hG_s}$ for the trivial $G_s$-action, equal to $\mathcal L(Y)$; the trivial $\mathbb Q[G_s]$-module $\mathbb Q$ is projective, so derived and ordinary invariants agree even on unbounded rational complexes.)

Apply $\mathcal L$ to the abstract blowup square and take homotopy $G$-invariants. The inductive hypothesis over $Z$ gives $\mathcal L(V\times_Y Z)^{hG}\simeq\mathcal L(Z)$ (nilinvariance passes between reduced and unreduced pullbacks). The square then gives $\mathcal L(V)^{hG}\simeq\mathcal L(Y)\times_{\mathcal L(Z)}\mathcal L(Z)\simeq\mathcal L(Y)$, and nilinvariance restores $X\times_S Y$ in place of $V$. In dimension zero there are no nonempty coincidence loci and the proof is the disjoint-union calculation. This proves the comparison. Taking $Y=S$ and using the support identification gives the spectral statement, and exact rational invariants give $\pi_0((\mathbb K^J(H\otimes_A^{\mathbf L}T)_\mathbb Q)^{hG})=(K_0^J(H\otimes_A^{\mathbf L}T)\otimes\mathbb Q)^G$, in fact an isomorphism. $\square$

### 6.4 Ordinary and normalized Euler lengths

We isolate the consequence needed before introducing asymptotic categories. Let $(A,\mathfrak m_A,k)$ be a complete regular local $\mathbb Z_{(p)}$-algebra of dimension $e>0$, residue characteristic $p$, with regular parameters $z_1,\dots,z_e$. Let $T$ be a finite local domain extension of $A$ with the same residue field, $r=\operatorname{rank}_A T$. Let $C$ be a $T$-algebra with an additive nonnegative length $\lambda$ on $C$-modules annihilated by powers of $\mathfrak m_A$, $\lambda(0)=0$, infinite values allowed. Suppose for every $v\ge1$ that

$$\lambda H_i(z_1^v,\dots,z_e^v;C)=0\ (i>0),\qquad \lambda\big(C/(z_1^v,\dots,z_e^v)C\big)=v^e r.$$

(Theorem 5.5 supplies exactly this, since the multiplicity of the base parameters on $T$ is its generic rank over the regular base.)

**Lemma 6.4 (Finite-length base change).** *For every finite-length $A$-module $V$,*

$$\lambda\operatorname{Tor}_i^A(V,C)=0\ (i>0),\qquad \lambda(V\otimes_A C)=r\operatorname{length}_A V,$$

*and all these lengths are finite.*

*Proof.* The Koszul resolution on $z_1,\dots,z_e$ resolves $k$ over $A$, so the hypothesis with $v=1$ proves the claim for $V=k$. Induct on a composition series. In the Tor long exact sequence of a short exact sequence with quotient $k$, nonnegativity and additivity make every positive Tor term have length zero and give the factor $r$ in degree zero; finiteness comes for free at every step. $\square$

If a finite group $G$ acts on $T$ over $A$, scalar twisting is $g^*P=P\otimes_{T,g}T$ with the last factor's left $T$-action through $g$; on a free complex this applies $g$ to every differential matrix entry.

**Lemma 6.5 (Ordinary norm comparison).** *Assume a finite group $G$ acts on $T$ over $A$ and is transitive on every geometric fiber of $\operatorname{Spec}T\to\operatorname{Spec}A$. For a perfect $T$-complex $P$ supported on $V(\mathfrak m_A T)$, the quantities*

$$\chi_T(P)=\sum_i(-1)^i\operatorname{length}_T H_i(P),\qquad \chi_C(P)=\sum_i(-1)^i\lambda H_i(P\otimes_T^{\mathbf L}C)$$

*are finite and define homomorphisms on $K_0^{\mathfrak m_A}(T)$. They agree on every norm sum:*

$$\sum_{g\in G}\chi_T(g^*P)=\sum_{g\in G}\chi_C(g^*P).$$

*More generally they agree on the image of $K_0^{\mathfrak m_A}(A)$ and on every $G$-invariant rational class.*

*Proof.* Establish finiteness and additivity, compare on pullbacks, then apply descent to the norm class. Represent $P$ by a bounded finite free $T$-complex; its ordinary homology is finite and supported at the closed point. For the $C$-evaluation choose a common $v$ with each $z_j^v$ acting nullhomotopically (support criterion). With $K_v=K(z_1^v,\dots,z_e^v;T)$, splitting the nullhomotopic cones gives

$$K_v\otimes_T P\otimes_T C\simeq\bigoplus_{j=0}^e(P\otimes_T C)[j]^{\oplus\binom ej},$$

which respects the module structure on which $\lambda$ is defined. Conversely, taking Koszul homology first gives a finite spectral sequence whose terms are finite sums of $H_j(z_1^v,\dots,z_e^v;C)$, of finite length by hypothesis; so every homology of $K_v\otimes_T P\otimes_T C$ has finite length, and the splitting with nonnegativity gives finiteness for every $H_i(P\otimes_T C)$. The long exact sequence gives additivity.

For $Q$ a perfect $A$-complex supported at $\mathfrak m_A$, the hyper-Tor spectral sequence with Lemma 6.4 gives $\chi_C(Q\otimes_A^{\mathbf L}T)=r\sum_j(-1)^j\operatorname{length}_A H_j(Q)$ (positive rows vanish, zeroth row scaled). For the ordinary evaluation use a bounded finite free $A$-resolution $L_\bullet$ of $T$ of alternating rank $r$ (regular $A$): for finite-length $V$, $\sum_i(-1)^i\operatorname{length}_A\operatorname{Tor}_i^A(V,T)=r\operatorname{length}_A V$, so the hyper-Tor spectral sequence gives $\chi_T(Q\otimes_A^{\mathbf L}T)=r\sum_j(-1)^j\operatorname{length}_A H_j(Q)$ (using that finite-length $T$-modules have the same $A$-length and $T$-length). Hence the two evaluations agree on classes pulled back from $A$.

Apply Theorem 6.1 with $H=A$, $J=\mathfrak m_A$: every invariant class in $K_0^{\mathfrak m_A}(T)\otimes\mathbb Q$ is the pullback of a rational class downstairs. Extending both Euler homomorphisms $\mathbb Q$-linearly, they agree on that invariant subspace, and $\sum_{g\in G}[g^*P]$ is invariant. $\square$

**Remark 6.6.** No invariance of $\lambda$ under $G$ is asserted or needed. The equality is for the sum of conjugate classes, not for each individual class. This lets the two length algebras be chosen independently.

## 7. Supported asymptotic complexes

Section 6 compares ordinary and normalized Euler lengths *before* any rank quotient. The positivity proof (Section 8) needs the same comparison *after* quotienting by things of negligible size. That is the content of this section.

The setting: $(A,J)$ is an excellent regular local $\mathbb Z_{(p)}$-algebra of dimension $e>0$ with residue characteristic $p$, $J$ its maximal ideal, $z_1,\dots,z_e$ a regular system of parameters. Let $T$ be a finite local domain over $A$ with the same residue field, $r=\operatorname{rank}_A T$. Let $C$ be a $T$-algebra with a nonnegative additive length $\lambda$ on its modules annihilated by powers of $J$, satisfying

$$\lambda H_j(z_1^v,\dots,z_e^v;C)=0\ (j>0),\qquad \lambda\big(C/(z_1^v,\dots,z_e^v)C\big)=v^e r\quad(v\ge1).\tag{$*$}$$

(Theorem 5.5 supplies exactly this.) The length may vanish on nonzero modules. Write $K_v=K(z_1^v,\dots,z_e^v;A)$, in homological degrees $0,\dots,e$.

### 7.1 The rank quotient and its support condition

Fix positive integers $s_n$ and a nonprincipal ultrafilter $\mathcal U$ on the positive integers. Rank estimates $O(s_n)$ and $o(s_n)$ always refer to the ordinary limit, not $\mathcal U$. If $(a_n/s_n)$ is bounded, write $\overline a=\lim_{n\to\mathcal U}a_n/s_n$.

For $S=A$ or $T$, consider the additive category of sequences $(S^{a_n})_n$ with $a_n=O(s_n)$ and arbitrary matrix morphisms. Let $\mathcal D_S$ be its perfect hull: complexes in one common bounded degree range for the whole sequence, then idempotent completion. Let $\mathcal N_S$ be the thick subcategory generated by free sequences with $a_n=o(s_n)$, and

$$\mathcal P_S=(\mathcal D_S/\mathcal N_S)^{\natural},$$

$(-)^{\natural}$ denoting idempotent completion; the quotient functor is $q_S$. A thick subcategory is closed under shifts, cones, and direct summands; the Verdier quotient inverts maps whose cones lie in it.

An object $X\in\mathcal D_S$ or $\mathcal P_S$ is **supported on $J$** if some $v\ge1$ has $z_j^v\operatorname{id}_X=0$ for every $j$, with the identity in the category containing $X$. Thus in $\mathcal D_S$ this is a uniform homotopical condition on the sequence, whereas in $\mathcal P_S$ the actions need only vanish after the rank-small quotient. Write $\mathcal S_S\subseteq\mathcal D_S$ and $\mathcal P_S^J\subseteq\mathcal P_S$ for the resulting full subcategories. They are thick: nilpotence of central actions survives shifts, cones, and retracts (larger powers for cones).

**Remark 7.1 (why we cannot just take raw homologies).** Over a discrete valuation ring $A=T$ with parameter $z$, take $s_n=n$ and the rank-one free resolution of $T/(z^n)$. This sequence lies in $\mathcal N_T$, yet its raw normalized degree-zero homology length is $1$. It has no uniform annihilating power before quotienting. But for each fixed $v$ its homology after tensoring with $K(z^v;T)$ is bounded independently of $n$, hence zero after normalization. This is why we first build lengths on uniformly supported lifts, then extend to supported quotient objects.

### 7.2 The supported sub-quotient

**Lemma 7.2 (supported quotient).** *The natural exact functor $\mathcal S_S/(\mathcal S_S\cap\mathcal N_S)\to\mathcal D_S/\mathcal N_S$ is fully faithful, and its idempotent completion is $\mathcal P_S^J$.*

*Proof.* First let $X\in\mathcal S_S$, $Y\in\mathcal N_S$, and $f:X\to Y$. Choose $v$ with $z_j^v\operatorname{id}_X=0$. Centrality gives $z_j^vf=0$, and the exact Hom sequence of the triangle $\operatorname{fib}(z_j^v:Y\to Y)\to Y\xrightarrow{z_j^v}Y$ lifts $f$ through the first object. Repeat for all parameters, always using the zero action on the original source $X$; the result factors $f$ through $K_v\otimes_A Y[-e]$, which lies in $\mathcal N_S$ by thickness and in $\mathcal S_S$ by the usual Koszul homotopies. So every map from a supported object to a small object factors through a supported small object.

Now a map $X\to Y$ in $\mathcal S_S$, computed in $\mathcal D_S/\mathcal N_S$, has a roof $X\xleftarrow{s}Z\xrightarrow{f}Y$ with $\operatorname{cofib}(s)=N_0\in\mathcal N_S$. In the triangle $Z\to X\xrightarrow{u}N_0$, factor $u$ through $N_1\in\mathcal S_S\cap\mathcal N_S$ by the previous paragraph. If $Z_1$ is the fiber of $X\to N_1$, the induced morphism of triangles gives $c:Z_1\to Z$, $s_1:Z_1\to X$ with $sc=s_1$; then $Z_1\in\mathcal S_S$ with $\operatorname{cofib}(s_1)=N_1$, so $X\xleftarrow{s_1}Z_1\xrightarrow{fc}Y$ lives in the supported subquotient. Fullness.

For faithfulness, suppose $f:Z\to Y$ between supported objects becomes zero in $\mathcal D_S/\mathcal N_S$. The calculus of Verdier roofs gives $t:W\to Z$ with small cone and $ft=0$; the triangle of $t$ factors $f$ through that small cone. Factoring its first leg through a supported small object shows $f$ is already zero in the supported subquotient. Apply this to the numerator of any roof.

Finally let $X\in\mathcal P_S^J$. It is a retract of $q_S(Y)$, $Y\in\mathcal D_S$. Choose $v$ with all $z_j^v$ zero on $X$; the cone of each such zero map splits, so $X$ is a retract of $K_v\otimes_A X$, itself a retract of $q_S(K_v\otimes_A Y)$ whose lift is uniformly supported. So $X$ lies in the idempotent completion of the supported subquotient; the reverse inclusion follows because support is closed under retracts. $\square$

### 7.3 Nonnegative homology lengths

Every supported quotient object is now accessible through a uniformly supported lift plus a summand.

For ordinary length, let $\mathcal A_T$ be the abelian category of sequences of finite $T$-modules annihilated by some common power of $J$ with lengths $O(s_n)$. For the $C$ evaluation, let $\mathcal A_C$ be the analogous category of sequences of $C$-modules with finite $\lambda$-lengths $O(s_n)$. Both are abelian, and sequences of normalized ultralimit length zero form a Serre subcategory. Denote the Serre quotients by $\overline{\mathcal A}_T,\overline{\mathcal A}_C$ and their nonnegative additive lengths by $\overline{\operatorname{length}}_T,\overline\lambda$. These lengths are finite on every object ("negligible" means zero ultralimit; the free sequences generating $\mathcal N_T$ satisfy the stronger ordinary estimate $o(s_n)$).

**Proposition 7.3 (evaluations).** *There are homological functors $\mathsf H_i^{\mathrm{ord}}:\mathcal P_T^J\to\overline{\mathcal A}_T$ and $\mathsf H_i^C:\mathcal P_T^J\to\overline{\mathcal A}_C$, whose values on a uniformly supported lift are its ordinary homology sequence and its homology after tensoring with $C$, respectively. For each object $X$ the lengths*

$$b_i^{\mathrm{ord}}(X)=\overline{\operatorname{length}}_T\mathsf H_i^{\mathrm{ord}}(X),\qquad b_i^C(X)=\overline\lambda\mathsf H_i^C(X)$$

*are finite nonnegative real numbers, zero outside a finite range, and the Euler sums define homomorphisms $\chi_{\mathrm{ord}},\chi_C:K_0(\mathcal P_T^J)\to\mathbb R$. If $F=(F_n)$ is a bounded free sequence whose image is supported and $v$ makes all $z_j^v$ act as zero on that image, then with $b_i(F)=b_i(q_TF)$,*

$$(1+t)^e\sum_i b_i(F)t^i=\sum_i t^i\lim_{n\to\mathcal U}\frac{\operatorname{len}H_i(K_v\otimes_A F_n)}{s_n}.\tag{7.3.1}$$

*Here $\operatorname{len}=\operatorname{length}_T$ in the ordinary case; in the $C$ case the complex is further tensored over $T$ with $C$ and $\operatorname{len}=\lambda$. This is an identity of finite Laurent polynomials.*

*Proof.* Bound homology on $\mathcal S_T$, show it kills $\mathcal S_T\cap\mathcal N_T$, extend to the idempotent-completed quotient.

Begin with a bounded free $F$ in $\mathcal S_T$ and uniform powers annihilating its identity. Splitting their cones gives, in the homotopy category,

$$K_v\otimes_A F\simeq\bigoplus_{j=0}^e F[j]^{\oplus\binom ej}.\tag{7.3.2}$$

For ordinary length every Koszul homology $H_j(K_v\otimes_A T)$ has finite length, and taking Koszul homology first in the bounded double complex bounds the homology lengths of $K_v\otimes_A F_n$ by fixed constants times the ranks of the $F_n$ terms, hence by $O(s_n)$. For $C$, the same spectral sequence uses $(*)$: positive Koszul rows have length zero and the zero row finite length; additivity and nonnegativity give the same bound. The splitting (7.3.2) then bounds the homology of $F_n$ themselves; these have a common primary annihilator (a power of every $z_j$ annihilates them).

A general $X\in\mathcal S_T$ may be a retract in $\mathcal D_T$ of a bounded free $F$ that is not supported. Choose $v$ killing the parameter actions on $X$; then $X$ is a retract of $K_v\otimes_A X$, a retract of the uniformly supported $K_v\otimes_A F$. The preceding bounds apply to that sequence and to the homology summands defining $X$ (a retract before the rank quotient). So homology on $\mathcal S_T$ takes values in the indicated abelian length quotients.

Fix $v$. Koszul tensoring a rank-small free generator has homology of normalized length zero by the same bounds; the class of $Y\in\mathcal D_T$ with this property for $K_v\otimes_A Y$ is thick (long exact sequences, nonnegativity, retracts), so it contains $\mathcal N_T$. If $X\in\mathcal S_T\cap\mathcal N_T$, choose $v$ killing its actions; then $X$ is a retract of $K_v\otimes_A X$, so its homology is zero in both length quotients.

The homology functors thus invert maps with supported small cones and descend to $\mathcal S_T/(\mathcal S_T\cap\mathcal N_T)$, preserving long exact sequences. Every object of $\mathcal P_T^J$ is a summand of an object of this subcategory (Lemma 7.2). Extend a homology functor to a summand $(X,a)$ by taking $\operatorname{im}\mathsf H_i(a)$ in its abelian target: this needs no sequence of chain projectors lifting $a$. Distinguished triangles in an idempotent completion are summands of distinguished triangles in the original category (Balmer–Schlichting, Definition 1.10 and Theorem 1.12), and taking images of compatible idempotents preserves exact sequences, so the extensions are homological.

Each object is a summand of a fixed bounded uniformly supported Koszul lift, so its homology objects vanish outside a finite range and are bounded by those of that lift; this gives finiteness, nonnegativity, and boundedness of the $b_i$. Short exact sequences prove additivity of the Euler sums.

For (7.3.1), the splitting (7.3.2) now holds in $\mathcal P_T$ (the powers act as zero there), and the extended homological functors give the left side. Its Koszul tensor has a uniformly supported lift whose actual homology computes the right side; this defines the coefficients without formal division. $\square$

In particular, (7.3.1) assigns zero to the quotient object of Remark 7.1, as required, and its nonnegative coefficients come from homological functors, not from assumed positivity of a quotient.

### 7.4 Scalar extension and nonlifting idempotents

Construct $\overline{\mathcal A}_A$ and its length using ordinary finite $A$-modules. The construction over $A$ with ordinary length defines $\chi_A$ on $\mathcal P_A^J$, and scalar extension gives an exact functor $\mathcal P_A^J\to\mathcal P_T^J$. We must show the evaluations agree on all classes from $\mathcal P_A^J$, including idempotent summands.

**Lemma 7.4.** *For every finite-length $A$-module $V$, $\lambda\operatorname{Tor}_j^A(V,C)=0$ for $j>0$ and $\lambda(V\otimes_A C)=r\operatorname{length}_A V$. All these modules are annihilated by a power of $J$.*

*Proof.* The Koszul and composition-series proof of Lemma 6.4 applies, using regularity and $(*)$ but not completeness; a power of $J$ annihilating $V$ also annihilates its derived tensor homology. $\square$

**Proposition 7.5 (base equality).** *For every $X\in\mathcal P_A^J$,*

$$\chi_{\mathrm{ord}}(X\otimes_A^{\mathbf L}T)=r\chi_A(X)=\chi_C(X\otimes_A^{\mathbf L}T).$$

*This holds for summands defined by idempotents in the Verdier quotient, without assuming the idempotents lift to chain maps before localization.*

*Proof.* The ordinary evaluation recovers $r$ from the alternating rank of a resolution of $T$; the normalized one from vanishing of positive Tor lengths. To make both on an arbitrary summand, descend their finite spectral sequences through supported Verdier roofs before taking idempotent images.

Let $Y$ be a uniformly supported bounded free sequence over $A$ and $a$ an idempotent endomorphism of its image in $\mathcal S_A/(\mathcal S_A\cap\mathcal N_A)$; every supported object of $\mathcal P_A$ is such a summand by Lemma 7.2 (enlarging the initial lift to a bounded free sequence before Koszul tensoring). Put $V_b=\operatorname{im}\mathsf H_b^A(a)$ in $\overline{\mathcal A}_A$; these images exist even without a chain-level projector.

Choose a fixed bounded finite free $A$-resolution $P_\bullet$ of $T$ (regular $A$). The double complex $Y\otimes_A P_\bullet$ computes $Y\otimes_A^{\mathbf L}T$, and filtering by the degree of $P$ gives a finite spectral sequence with first page $E^1_{u,b}=H_b(Y)\otimes_A P_u$. It is natural in homotopy-category maps from this page onward (homotopies in $Y$ preserve the filtration, act trivially on the first page). On retracts in the prequotient category the construction is defined by taking the corresponding idempotent images on every page and on the abutment filtration; its terms are $A$-modules and the abutment's ordinary $A$-length equals its $T$-length (same residue field). A map with supported small cone is an isomorphism on $H_b(Y)$ in $\overline{\mathcal A}_A$, hence on the first page, all later pages, and the finite filtered abutment in the length quotient. The spectral sequence descends through all Verdier roofs in the supported subquotient, and the natural augmentation $\operatorname{Tot}(Y\otimes_A P_\bullet)\to Y\otimes_A T$ identifies the abutment with the homology of scalar extension, also descending through roofs. So the action of $a$ on the abutment is precisely the scalar-extended idempotent.

Thus $a$ acts as an actual idempotent on every page and on the abutment, and the differentials commute with it. Taking images of an idempotent commutes with homology (an idempotent splits a complex into image and complementary image) and preserves the finite exact sequences of the filtration (for a filtration preserved by $a$, $\operatorname{im}(a)\cap F_u=a(F_u)$, so $\operatorname{gr}_u\operatorname{im}(a)=\operatorname{im}(\operatorname{gr}_u a)$). Hence the Euler length on the scalar-extended summand is the Euler length of the images on the first page, which are $\operatorname{rank}_A(P_u)$ copies of $V_b$:

$$\Big(\sum_u(-1)^u\operatorname{rank}_A P_u\Big)\sum_b(-1)^b\overline{\operatorname{length}}_A V_b=r\sum_b(-1)^b\overline{\operatorname{length}}_A V_b,$$

the alternating rank being $r$ after tensoring with $\operatorname{Frac}A$.

For $C$, use the finite hyper-Tor spectral sequence ${}^CE^2_{u,b}=\operatorname{Tor}_u^A(H_b(Y),C)\Rightarrow H_{u+b}(Y\otimes_A^{\mathbf L}C)$, constructed from the functorial finite Postnikov filtration of $Y$ (bounded $b$, $0\le u\le e$, no convergence issue). Lemma 7.4 makes every positive Tor term zero in $\overline{\mathcal A}_C$, and implies that $V\mapsto V\otimes_A C$ induces an exact functor $\overline{\mathcal A}_A\to\overline{\mathcal A}_C$ multiplying length by $r$ (failure of left exactness is the image of a positive Tor module of zero length; negligible sequences map to negligible ones). A supported-small-cone map induces isomorphisms on the second page and the finite filtered abutment in the length quotient, so this spectral sequence also descends through the supported roofs. Images of the same $a$ give only the zero Tor row, the images of $V_b$ under this exact tensor functor, with Euler length $r\sum_b(-1)^b\overline{\operatorname{length}}_A V_b$, which equals $\chi_A(Y,a)$ by construction. This proves both equalities for every summand. $\square$

### 7.5 Connective coefficients and the norm comparison

We now put the rank quotient into the form required by Theorem 6.1; the coefficient algebra must be connective and the support condition must be central-localization support.

**Lemma 7.6 (connective coefficients).** *There is a connective associative derived $A$-algebra $H$ and compatible equivalences $\mathcal P_A\simeq\operatorname{Perf}(H)$ and $\mathcal P_T\simeq\operatorname{Perf}(H\otimes_A^{\mathbf L}T)$, respecting central localization by elements of $A$.*

*Proof.* The sequence $E_S=(S^{s_n})_n$ is an additive generator: if $a_n=O(s_n)$ then one integer $c$ has $a_n\le cs_n$ for all $n$ (increase $c$ for the finitely many exceptions), and coordinate inclusions and projections make $(S^{a_n})_n$ a summand of $E_S^c$. Hence $\mathcal D_S\simeq\operatorname{Perf}(B_S)$ with $B_S=\prod_n M_{s_n}(S)$.

The underlying $A$-module of $B_A$ is flat by the ideal criterion: every finitely generated ideal of the Noetherian $A$ is finitely presented, so tensoring with this product of finite free modules commutes with the product and each required injection is a product of injections. Since $T$ is finitely presented over $A$, tensoring with $T$ also commutes with the product, so $B_A\otimes_A^{\mathbf L}T\simeq B_A\otimes_A T\cong B_T$.

The same product argument makes every morphism module between the original free sequences flat over $A$, so the explicit dg quotient of this degree-zero category by its full rank-small subcategory computes the derived quotient (Drinfeld, Section 3.1 and Theorem 3.4). Its morphisms are sums of paths formed from tensor products of the original morphism modules and degree-$1$ contracting homotopies; they are concentrated in nonnegative degrees and degreewise $A$-flat, hence $K$-flat. Passing to the perfect hull and idempotent completion kills exactly $\mathcal N_A$. The image of $E_A$ remains a generator, and its endomorphism algebra $H=\operatorname{End}_{\mathcal P_A}(q_AE_A)$ is connective, giving $\mathcal P_A\simeq\operatorname{Perf}(H)$ (a statement about the chosen generator, not arbitrary shifts).

Since $T$ is finitely presented over $A$, tensoring each morphism module with $T$ gives the morphism module between free $T$-sequences and the path formula tensors to the $T$-linear dg quotient by rank-small free sequences. The same flatness criterion applies over Noetherian $T$, so $H\otimes_A T\simeq H\otimes_A^{\mathbf L}T$, whose perfect hull and idempotent completion are $\mathcal P_T$ (agreeing with preservation of exact quotients under scalar extension, Land–Tamme Lemma 3.12). Everything is centrally $A$-linear and preserves central localization. $\square$

Under these equivalences the support condition of this section is exactly the one in Theorem 6.1: for perfect $X$ and central $z_j$, localization is the telescope of multiplication by $z_j$, and compactness makes $X[1/z_j]=0$ iff some $z_j^v$ kills $\operatorname{id}_X$; a common power works for the finitely many parameters. The same holds for $\operatorname{Perf}(B_S)$ and $\mathcal S_S$. No uniformly supported lift of an object of $\mathcal P_S^J$ is needed for this.

Now suppose a finite group $G$ acts on $T$ over $A$, $\operatorname{Spec}T\to\operatorname{Spec}A$ is surjective, and $G$ is transitive on every geometric fiber. It acts on $\mathcal P_T^J$ by scalar twisting $gX$ ($g^*P=P\otimes_{T,g}T$, with the convention of Section 6.4).

**Theorem 7.7 (asymptotic norm comparison).** *Let $A,J,T,C,\lambda$ be as in this section and $(*)$. Suppose a finite group $G$ acts on $T$ over $A$, $\operatorname{Spec}T\to\operatorname{Spec}A$ is surjective, and $G$ is transitive on every geometric fiber. Then for every $X\in\mathcal P_T^J$,*

$$\sum_{g\in G}\chi_{\mathrm{ord}}(gX)=\sum_{g\in G}\chi_C(gX).$$

*Neither invariance of $\lambda$ under $G$ nor equality of the two evaluations on individual classes is assumed.*

*Proof.* Lemma 7.6 identifies the supported categories with the supported perfect categories of $H$ and $H\otimes_A^{\mathbf L}T$, $H$ connective; the identification over $T$ is compatible with the $G$-action because scalar twisting acts on the $T$ factor. Theorem 6.1 gives a surjection $K_0(\mathcal P_A^J)\otimes\mathbb Q\to(K_0(\mathcal P_T^J)\otimes\mathbb Q)^G$. The norm class $\sum_g[gX]$ is invariant, and $\chi_{\mathrm{ord}}-\chi_C$ vanishes on the image from $A$ by Proposition 7.5. Applying it to a preimage of the norm class proves the assertion. $\square$

## 8. Proof of strict positivity

We now put the two comparisons together. Fix a pair of prime quotients and reduce; then for each Galois conjugate, replace one module by the normalized-length algebra, and use the two norm comparisons from Sections 6 and 7 to force positivity.

Throughout, "reduction" means: pass to the complete regular local ring with algebraically closed residue field, and to the prime quotients.

### 8.1 Reduction and the first norm comparison

**Lemma 8.1 (reduction).** *To prove Theorem 1.1 it suffices to prove it for*

$$D=R/P,\qquad E=R/Q,\qquad h=\dim D>0,\qquad l=\dim E>0,\qquad h+l=d,$$

*where $R$ is complete with algebraically closed residue field $k$ of characteristic $p>0$, $P,Q$ are prime, and $P+Q$ is primary to $\mathfrak m_R$.*

*Proof.* The equicharacteristic case is Serre's theorem. Assume positive residue characteristic. Completion and then Lemma 5.1 reduce to complete $R$ with algebraically closed residue field (a flat local extension with $\mathfrak m_RR_1=\mathfrak m_{R_1}$ preserves finite-length lengths via composition series, identifies Tor after extension, preserves the Hilbert–Samuel function and hence module dimensions, and preserves regularity).

Use the Roberts–Gillet–Soule vanishing theorem in its general regular local form: $\chi^R(V,W)=0$ if $\operatorname{length}_R(V\otimes_R W)<\infty$ and $\dim V+\dim W0$. $\square$

Fix a pair as in Lemma 8.1. By Lemma 5.2 choose regular complete normalization bases $A_D\subset D$, $A_E\subset E$ of dimensions $h,l$, same residue field $k$, with the extensions finite and generically separable. Let $D',E'$ be integral closures in finite Galois closures of the fraction-field extensions; they are finite over the bases by excellence, and over $D,E$ as well, and they are local: a finite algebra over a complete local ring is a product of complete local rings, and a domain has one factor (Stacks Tag 03QH); their residue fields are finite over $k$, hence equal $k$. Put

$$G_D=\operatorname{Gal}(\operatorname{Frac}D'/\operatorname{Frac}A_D),\qquad G_E=\operatorname{Gal}(\operatorname{Frac}E'/\operatorname{Frac}A_E).$$

Normality of the bases gives $(D')^{G_D}=A_D$, $(E')^{G_E}=A_E$. Geometric transitivity holds: over an algebraically closed $A$-field $\Omega$, $S=T\otimes_A\Omega$ with $u=\sum_i t_i\otimes\omega_i$ has $\prod_{g\in G}g(u)$ scalar (the coefficients of $\prod_g(\sum_i g(t_i)X_i)$ lie in $T^G=A$, then specialize $X_i=\omega_i$); if the finite Artinian $\Omega$-algebra $S$ had more than one orbit of points, a nontrivial $G$-invariant idempotent $e$ (equal to $1$ on one orbit, $0$ elsewhere) would satisfy $e^{|G|}=e$ scalar, forcing $e=0$ or $1$, a contradiction. So every geometric fiber is transitive, without asserting that invariants commute with base change (Stacks Tags 0BRI, 0BRJ). Thus Theorem 6.1 applies to both extensions.

By Theorem 5.5 choose the algebras and lengths

$$(C_D,\lambda_D)=(C_{D'},\lambda_{D'}),\qquad (C_E,\lambda_E)=(C_{E'},\lambda_{E'}),$$

fixed throughout. Set $a_D=\operatorname{rank}_D D'$, $a_E=\operatorname{rank}_E E'$. For $\sigma\in G_D$ define $\iota_\sigma:R\to D'\xrightarrow{\sigma}D'$, give $D'$ this $R$-action, and give $C_D$ the action obtained by composing $\iota_\sigma$ with $D'\to C_D$. Define

$$c_\sigma=\sum_{i=0}^d(-1)^i\lambda_D\big(\operatorname{Tor}_i^R(C_D,E')\big),$$

with this $R$-action on $C_D$ understood.

**Lemma 8.2 (first average).** *The lengths in the definition of $c_\sigma$ are finite, and*

$$|G_D|\,a_D\,a_E\,\chi^R(D,E)=\sum_{\sigma\in G_D}c_\sigma.$$

*Proof.* Take a finite $R$-free resolution of $E'$ and extend it to $D'$. The resulting bounded free complex is supported at the closed point: after localizing its homology away from $\mathfrak m_R$, the supports of $D'$ and $E'$ are disjoint; the same holds after every twist, whose kernel on $R$ is still $P$. Lemma 6.5 gives finite normalized homology lengths and equality of ordinary and normalized Euler evaluations after summing all conjugates.

Conjugating the complex entrywise by $\sigma$ gives a semilinear isomorphism from the untwisted complex to its conjugate, so all ordinary Euler characteristics in that sum equal $\chi^R(D',E')$. Ordinary lengths over $D'$ and over $R$ agree here because both residue fields are $k$.

Choose an injection $D^{a_D}\to D'$ with torsion cokernel; the cokernel has dimension less than $h$, so its intersection with $E'$ vanishes by vanishing. Additivity gives $\chi^R(D',E')=a_D\chi^R(D,E')$, and similarly $\chi^R(D,E')=a_E\chi^R(D,E)$. The norm comparison gives the displayed equation. $\square$

It remains to show $c_\sigma>0$ for each $\sigma$. Fix one $\sigma$ until the final paragraph; write $C=C_D$, $\lambda=\lambda_D$ with the chosen $R$-action.

### 8.2 A parameter quotient and its Tor groups

Choose $x_1,\dots,x_h\in Q$ forming an $R$-regular sequence whose images are a parameter system of $D$. (Simultaneous prime avoidance: after $i-1$ choices, the associated primes of $R/(x_1,\dots,x_{i-1})$ have height $i-1$ (so $R$ is Cohen–Macaulay and none contains $Q$ of height $h\ge i$), and we also avoid the primes in $D/(x_1,\dots,x_{i-1})$ obstructing the next dimension drop — each has positive dimension, and if its inverse image contained $Q$ it would contain $P+Q$, hence be the maximal ideal, contradiction. So prime avoidance inside $Q$ allows the choice at every step.)

Put

$$B=R/(x_1,\dots,x_h),\qquad I_\sigma=(\iota_\sigma(x_1),\dots,\iota_\sigma(x_h))D',\qquad U=C/I_\sigma C.$$

The ideal $I_\sigma$ is a parameter ideal of $D'$ (finiteness and the automorphism $\sigma$ preserve this), so Theorem 5.5 gives

$$0<\lambda(U)=e(I_\sigma,D')<\infty,\qquad \lambda\big(H_q(x;C)\big)=0\quad(q>0).$$

Some integer $a\ge1$ satisfies $\mathfrak m_B^a(D'/I_\sigma)=0$; it annihilates $U$ and every $D'/I_\sigma$-module.

**Lemma 8.3 (Tor bound).** *For every finite $B$-module $L$, all the following lengths are finite, and*

$$\tau_i(L):=\lambda\big(\operatorname{Tor}_i^B(U,L)\big)=\lambda\big(\operatorname{Tor}_i^R(C,L)\big),\qquad \tau_i(L)=0\quad(i>l).$$

*Proof.* The complexes have compatible $C$-actions. The Koszul homology $H_q(x;C)$ is annihilated by $x$ and by $P$, hence bounded-primary-torsion; for $q>0$ its length is zero. A free $B$-resolution of $L$ (each term finite rank) gives $\lambda(\operatorname{Tor}_p^B(H_q(x;C),L))=0$ for $q>0$ (a subquotient of a finite sum of a length-zero module), while $q=0$ is finite because $\lambda(U)<\infty$. The change-of-rings spectral sequence $\operatorname{Tor}_p^B(\operatorname{Tor}_q^R(C,B),L)\Rightarrow\operatorname{Tor}_{p+q}^R(C,L)$ gives equality of lengths; only finitely many terms enter each total degree.

For the sharper bound, choose an $R$-free resolution $P_L$ of $L$ in degrees $0,\dots,d$ and consider $K(x;R)\otimes_R C\otimes_R P_L$. Koszul first: all positive rows have length zero and the zero row is $U\otimes_R P_L$, so total homology has length zero above degree $d$. Since $xL=0$, in the derived category $K(x;R)\otimes_R^{\mathbf L}L\simeq\bigoplus_{j=0}^h L[j]^{\binom hj}$; tensoring with $C$ preserves the splitting, so total homology in degree $i+h$ contains $\operatorname{Tor}_i^R(C,L)$ as a summand. For $i>l=d-h$ that degree exceeds $d$, proving the bound; this also shows finiteness directly. $\square$

### 8.3 Finite approximations preserving the actions

We need finite-length $B$-modules whose normalized Tor lengths approach those of $U$, retaining the single annihilator $\mathfrak m_B^a$; there is no finite generation or presentation of $U$ assumed. The approximation lives among modules carrying both the tower action and the $B$-action. In this subsection $\lambda$ denotes $\lambda_\infty$ on underlying tower modules (as permitted by Theorem 5.5); the approximating modules need not carry a $C$-action.

Let $A=A_D$, let $A_j$ be its flat regular tower from Lemma 5.3, $\Lambda=\bigcup_j A_j$, so $s(j)=\operatorname{rank}_A A_j=p^{jh}$. Set

$$S_j=A_j\otimes_A(D'/I_\sigma),\qquad S=\Lambda\otimes_A(D'/I_\sigma).$$

The commuting $\Lambda$- and $D'/I_\sigma$-actions on $U$ combine into an $S$-action; working with $S$-modules retains both. Descent to $S_j$ retains the existing $B$-action by restriction along $B\to D'/I_\sigma$ (induced by $\iota_\sigma$); no new $B$-action is chosen at a finite stage.

**Lemma 8.4 (approximation).** *There are finite-length $B$-modules $M_n$ and positive integers $s_n\to\infty$ with $\mathfrak m_B^aM_n=0$ such that*

$$\frac{\operatorname{length}_B(M_n)}{s_n}\to\lambda(U),\qquad \frac{\operatorname{length}_B\operatorname{Tor}_i^B(M_n,L)}{s_n}\to\tau_i(L)\quad(L=k,E',\,i\ge0).$$

*Writing $\beta_0(M_n)=\dim_k(M_n/\mathfrak m_BM_n)$, also*

$$\frac{\beta_0(M_n)}{s_n}\to\nu_\sigma,\qquad \nu_\sigma\ge\frac{\lambda(U)}{\operatorname{length}_B(B/\mathfrak m_B^a)}>0.$$

*Proof.* The algebra $S$ is finitely presented as a $\Lambda$-module and has finite normalized length. Every $S_j$ is finite over the twisted image of $B$ ($D'$ finite over the twisted image of $R/P$, $A_j$ finite free over $A$), is annihilated by $\mathfrak m_B^a$, and is Artinian local with residue field $k$ (the tower roots are nilpotent over the Artinian $D'/I_\sigma$, reduction leaves the common residue field $k$). Thus finite $S_j$-modules have finite lengths over both $B$ and $A_j$, and these lengths agree (a composition series has factors $k$ of length one for either ring); unchanged when a residue-field identification is composed with an automorphism.

Given $\epsilon>0$, choose a finitely generated $S$-submodule $V\subset U$ with $\lambda(U/V)<\epsilon$ (up to the supremum property; the $S$-span of a finitely generated $\Lambda$-submodule is finitely generated over $\Lambda$). For finite presentation, choose a surjection $S^r\to V$ with kernel $K$ (finite length, bounded by $r\lambda(S)$), and a finitely generated $S$-submodule $K_0\subset K$ with $\lambda(K/K_0)<\epsilon$. Then $P_\epsilon=S^r/K_0\to U$ has kernel and cokernel of length less than $\epsilon$, is finitely presented over $S$ and over $\Lambda$, and is still annihilated by $\mathfrak m_B^a$.

Fix $B$-resolutions of $k$ and $E'$ with finite free terms; let $r_i(L)$ be the rank in degree $i$, $r_{-1}(L)=0$. If $K_\epsilon,W_\epsilon$ are kernel and cokernel of $P_\epsilon\to U$, the two long exact homology sequences give

$$\Big|\lambda\operatorname{Tor}_i^B(P_\epsilon,L)-\lambda\operatorname{Tor}_i^B(U,L)\Big|\le\big(r_i(L)+r_{i-1}(L)\big)\lambda(K_\epsilon)+\big(r_{i+1}(L)+r_i(L)\big)\lambda(W_\epsilon),$$

each homology of the tensor with $K_\epsilon$ (resp. $W_\epsilon$) having length at most the corresponding rank times $\lambda(K_\epsilon)$ (resp. $\lambda(W_\epsilon)$). All quantities are finite, and $|\lambda(P_\epsilon)-\lambda(U)|<2\epsilon$.

A finite $S$-presentation of $P_\epsilon$ descends to an $S_j$-module $P_j$ for some $j$, with $P_\epsilon=\Lambda\otimes_{A_j}P_j$; flatness of $\Lambda/A_j$ gives $\operatorname{Tor}_i^B(P_\epsilon,L)=\Lambda\otimes_{A_j}\operatorname{Tor}_i^B(P_j,L)$. The finite-stage homology on the right is finite over $S_j$, and normalization (Theorem 5.5) together with the length comparison gives

$$\lambda\operatorname{Tor}_i^B(P_\epsilon,L)=\frac{\operatorname{length}_{A_j}\operatorname{Tor}_i^B(P_j,L)}{s(j)}=\frac{\operatorname{length}_B\operatorname{Tor}_i^B(P_j,L)}{s(j)},$$

and similarly for $P_\epsilon$.

At step $n$ choose $\epsilon$ so that all errors above, for $i\le n$ and $L=k,E'$, plus the length error, are less than $1/n$. Descend the presentation, enlarge the stage so $j\ge n$; further flat tower extension leaves every normalized value unchanged. Take $M_n=P_j$, $s_n=s(j)$. This proves the two limits, the common annihilator, and $s_n\to\infty$.

The case $i=0,L=k$ gives the limit in the generator bound. A minimal generating set and the annihilator give a surjection $(B/\mathfrak m_B^a)^{\beta_0(M_n)}\to M_n$, so $\operatorname{length}_B(M_n)\le\beta_0(M_n)\operatorname{length}_B(B/\mathfrak m_B^a)$; divide by $s_n$ and take limits. $\square$

### 8.4 Bounded complexes and their ordinary evaluation

The approximations give two kinds of control. In minimal resolutions the Tor bound $\tau_i(k)=0$ for $i>l$ makes the corresponding ranks negligible; the common annihilator then makes the truncated sequences supported after quotienting by negligible ranks, and has already given the positive normalized generator bound that will make the second evaluation strictly positive.

Choose minimal free $B$-resolutions $F_n^\infty$ of $M_n$, indexed homologically in nonnegative degrees. Let $F_n$ be their brutal truncations to degrees $0,\dots,l$. Minimality and the approximation Tor limit give, in each fixed degree,

$$\operatorname{rank}_B F_{n,i}^\infty=O(s_n),\qquad \operatorname{rank}_B F_{n,i}^\infty=o(s_n)\quad(i>l),$$

because these ranks equal $\operatorname{length}_B\operatorname{Tor}_i^B(M_n,k)$.

Let $z_1,\dots,z_l$ be regular parameters of $A_E$. For $\gamma\in G_E$ extend $F_n$ through $B\to E\to E'\xrightarrow{\gamma}E'$, getting $F_{n,\gamma}'$; use normalization $s_n$ in the asymptotic category over $E'$. These $s_n$ come from the $D$-side approximation; Theorem 7.7 allows any positive integer sequence, so no compatibility between the parameter-root towers of $A_D$ and $A_E$ is needed.

**Lemma 8.5 (supported truncation).** *The sequence $(F_{n,\gamma}')_n$ is supported at the closed point in that category, and its ordinary Euler evaluation equals $c_\sigma$ for every $\gamma\in G_E$.*

*Proof.* Every $b\in\mathfrak m_B^a$ acts nullhomotopically on $F_n^\infty$ (it annihilates $M_n$). Restrict a chosen homotopy to the brutal truncation; its only remaining error is in degree $l$, $F_{n,l}\xrightarrow{h_l}F_{n,l+1}^\infty\xrightarrow{d_{l+1}}F_{n,l}$, a factorization by chain maps through $F_{n,l+1}^\infty[l]$ (the map from $d_{l+1}$ is a chain map because $d_ld_{l+1}=0$; from $h_l$ because the truncation has no degree $l+1$ term). The middle module has rank $o(s_n)$ and is killed by the asymptotic quotient. These identities persist after extension to $E'$, with no flatness over $B$.

The ideal $\mathfrak m_BE'$ for each structural map is primary to $\mathfrak m_{E'}$. Choose $v$ so large that every $z_j^v$ lies in the extension of $\mathfrak m_B^a$ for all $\gamma$ (one fixed choice, the group finite). Writing $z_j^v$ as a finite $E'$-linear combination of elements of $\mathfrak m_B^a$ proves it acts by zero on the sequence in the quotient; the same combinations give genuine nullhomotopies on the full extended resolutions. So the sequence is supported.

Apply the ordinary case of (7.3.1). One may replace truncated extended resolutions by the full ones in each fixed homological degree: the quotient of the full resolution by its brutal truncation has only terms of degrees $\ge l+1$, all of rank $o(s_n)$ in each fixed degree; after tensoring that tail with $K(z^v;E')$ and taking Koszul homology first, each fixed-degree homology is bounded by finite sums of $\operatorname{rank}_B F_{n,i}^\infty\cdot\operatorname{length}_{E'}H_q(z^v;E')=o(s_n)$ (fixed finite Koszul homology lengths, finitely many terms per total degree); the long exact sequence proves the replacement.

On each full extended resolution the $z_j^v$ act nullhomotopically, so Koszul tensoring gives the direct sum of shifts with multiplicities $\binom lj$. Its ordinary homology lengths are $\operatorname{length}_B\operatorname{Tor}_i^B(M_n,E')$, unchanged by conjugating the $E'$-action (applying $\gamma$ entrywise is a semilinear isomorphism). The approximation Tor limit and the Tor identification, followed by coefficient comparison in (7.3.1), identify the ordinary homology-length coefficients with $\tau_i(E')=\lambda\operatorname{Tor}_i^R(C,E')$, whose alternating sum is exactly $c_\sigma$. $\square$

### 8.5 A positive normalized evaluation

The ordinary evaluation recovered $c_\sigma$ on every conjugate; we now evaluate the same supported object with $C_E$. The parameter Koszul identities leave only the nonnegative degree-zero coefficient, and minimality of the resolutions bounds that coefficient below using the generator bound.

Corollary 5.6 gives $0<\eta_E:=\lambda_E(C_E/\mathfrak m_{E'}C_E)<\infty$. Set $q_E=(z_1,\dots,z_l)E'$, $r_E=\operatorname{rank}_{A_E}E'$, $t_E=\operatorname{length}_{E'}(E'/q_E)$. The length input gives $\lambda_E(C_E/q_EC_E)=r_E>0$, and a composition series of $E'/q_E$, tensored right-exactly with $C_E$, shows

$$r_E\le t_E\eta_E,$$

since at each step the image of the tensor of the submodule has length at most the tensor's; injectivity after tensoring is not needed. Finiteness of $\eta_E$ follows from the surjection $C_E/q_EC_E\to C_E/\mathfrak m_{E'}C_E$.

**Lemma 8.6 (positive evaluation).** *The $\lambda_E$ Euler evaluation of $(F_{n,\gamma}')_n$ is a number $b_{\sigma,\gamma}$ satisfying*

$$b_{\sigma,\gamma}\ge\nu_\sigma\eta_E\ge\frac{\lambda_D(U)}{\operatorname{length}_B(B/\mathfrak m_B^a)}\,\frac{r_E}{t_E}>0.$$

*Proof.* Let $b_i$ be the nonnegative homology-length coefficients of this evaluation in (7.3.1). Consider the finite double complex $K(z^v;E')\otimes_{E'}F_{n,\gamma}'\otimes_{E'}C_E$. Koszul first, Theorem 5.5 makes all positive Koszul rows have length zero; the remaining row has complex degrees $0,\dots,l$. So the normalized homology lengths vanish in total degrees $>l$ and in negative degrees. Equation (7.3.1) says their polynomial is $(1+t)^l\sum_i b_it^i$, all $b_i\ge0$. Boundedness and comparison of the least degree rule out negative-degree coefficients; any $b_i>0$ with $i>0$ would give a positive coefficient in degree $i+l>l$, contradiction. So only $b_0$ can be nonzero and the Euler evaluation equals $b_0$.

In degree zero the double complex has homology

$$C_E^{\beta_0(M_n)}\Big/\Big(\operatorname{im}(d_1)+(z_1^v,\dots,z_l^v)C_E^{\beta_0(M_n)}\Big).$$

Since $l>0$, $d_1$ is the degree-one presentation differential of $M_n$; minimality over $B$ puts all its entries in $\mathfrak m_B$, whose image under any twist lies in $\mathfrak m_{E'}$, and the base parameter powers are also in $\mathfrak m_{E'}$. There is a surjection from this degree-zero homology onto $(C_E/\mathfrak m_{E'}C_E)^{\beta_0(M_n)}$, whose normalized length is at least $\beta_0(M_n)\eta_E/s_n$. The degree-zero coefficient of (7.3.1) is $b_0$, so taking the ultralimit and the ordinary limit in the generator bound gives $b_0\ge\nu_\sigma\eta_E$. The generator bound and $r_E\le t_E\eta_E$ give the displayed chain. Define $b_{\sigma,\gamma}=b_0$. $\square$

### 8.6 Assembly: proof of the main theorem

**Proof of Theorem 1.1.** By Lemma 8.1 use the fixed domain pair. For each $\sigma\in G_D$ carry out the construction with its specified $R$-action. Apply Theorem 7.7 to the supported sequence of Lemma 8.5 over $A_E\subset E'$: it equates the sums of ordinary and normalized evaluations of all $G_E$-conjugates. The ordinary evaluation is $c_\sigma$ for every conjugate; the normalized is $b_{\sigma,\gamma}$. Hence

$$|G_E|\,c_\sigma=\sum_{\gamma\in G_E}b_{\sigma,\gamma}>0$$

by Lemma 8.6. Finally Lemma 8.2 gives

$$|G_D|\,a_D\,a_E\,\chi^R(D,E)=\frac{1}{|G_E|}\sum_{\sigma\in G_D}\sum_{\gamma\in G_E}b_{\sigma,\gamma}>0,$$

since both ranks and both group orders are positive. Thus $\chi^R(D,E)>0$, and the reduction proves the theorem. Neither norm comparison needed a Galois action on $C_D$ or $C_E$, nor invariance of either normalized length under a twist. $\square$

## 9. Consequences

The main theorem is a statement about intersection multiplicities, but it has two clean downstream consequences worth recording, because they show strict positivity doing real work.

### 9.1 Symbolic powers

**Corollary 9.1 (symbolic-power containment).** *Let $(R,\mathfrak m)$ be a ramified mixed-characteristic regular local ring (residue characteristic $p\in\mathfrak m^2$), and let $P,Q$ be prime ideals with $\sqrt{P+Q}=\mathfrak m$ and $\operatorname{ht}P+\operatorname{ht}Q=\dim R$. Then, with $Q^{(r)}=Q^rR_Q\cap R$ the $r$-th symbolic power,*

$$P\cap Q^{(r)}\subseteq\mathfrak m^{r+1}\qquad(r\ge1).$$

*Proof.* This is the implication of Kurano–Roberts (Theorem 3.2 of their paper), applied to the positivity statement of Theorem 1.1. Kurano–Roberts show that if the intersection multiplicity of $R/P$ and $R/Q$ at $\mathfrak m$ is strictly positive whenever $\operatorname{ht}P+\operatorname{ht}Q=\dim R$ and $\sqrt{P+Q}=\mathfrak m$, then the symbolic-power containment holds. Before this paper the implication applied only where positivity was known (equicharacteristic or unramified); now it applies unconditionally in ramified mixed characteristic. $\square$

This is a direct application, not an extra ingredient in the proof of Theorem 1.1.

### 9.2 The tangent-cone corollary

**Corollary 9.2 (finite strict-transform intersection).** *Let $(A,\mathfrak m)$ be a commutative Noetherian regular local ring with residue field $k$, and let $P,Q$ be prime ideals such that $A/(P+Q)$ has finite length and $\dim(A/P)+\dim(A/Q)=\dim A$. Let $e_\mathfrak m$ be Hilbert–Samuel multiplicity and*

$$T=\operatorname{gr}(A/P)\otimes_{\operatorname{gr}A}\operatorname{gr}(A/Q),$$

*all associated graded rings taken for the $\mathfrak m$-adic filtrations. If $\dim T\le1$, then*

$$\chi^A(A/P,A/Q)\ge e_\mathfrak m(A/P)\,e_\mathfrak m(A/Q),$$

*with equality if and only if $\dim T=0$.*

*Proof.* The case $\dim A=0$: then $A=k$, $P=Q=0$, both sides are $1$, and $T=k$.

Otherwise let $\widetilde X$ be the blowup of $X=\operatorname{Spec}A$ at its closed point and $\widetilde Y,\widetilde Z$ the strict transforms of $Y=\operatorname{Spec}(A/P)$, $Z=\operatorname{Spec}(A/Q)$. By Skalit (Lemma 3.5 of the unramified paper), the underlying set $S$ of $\widetilde Y\cap\widetilde Z$ identifies with that of $\operatorname{Proj}T$; so $S$ is a finite set of closed points of the exceptional divisor, empty exactly when $\dim T=0$. Fulton's blowup formula and Skalit's finite-support calculation (Theorem 3.4 and the proof of Proposition 3.10 of the unramified paper) give

$$\chi^A(A/P,A/Q)-e_\mathfrak m(A/P)e_\mathfrak m(A/Q)=\sum_{x\in S}[\kappa(x):k]\,\chi^{B_x}(M_x,N_x),$$

where $B_x=\mathcal O_{\widetilde X,x}$, $M_x=\mathcal O_{\widetilde Y,x}$, $N_x=\mathcal O_{\widetilde Z,x}$. Each residue degree is finite and positive because the exceptional divisor is projective over $k$. The blowup $\widetilde X$ is regular (Jannsen, Proposition A.4(1)). The modules $M_x,N_x$ are nonzero and finite over $B_x$, with finite-length tensor product because their intersection is supported on $S$. The local rings $A/P$, $A/Q$, $A$ are formally equidimensional and universally catenary, so Skalit (Lemma 3.8 and the proof of Proposition 3.10 of the unramified paper) gives $\dim M_x=\dim(A/P)$, $\dim N_x=\dim(A/Q)$, and $\dim B_x=\dim A$. Theorem 1.1 then makes every local intersection multiplicity in the sum strictly positive, so the sum is nonnegative and vanishes exactly when $S$ is empty, i.e. when $\dim T=0$. $\square$

## 10. What was proved, and what to remember

You have just read the complete proof of Serre's positivity conjecture. The skeleton:

1. **Reduce** to two complete local domains $D=R/P$, $E=R/Q$ of complementary dimensions over a complete regular local ring with algebraically closed residue field of characteristic $p>0$.
2. **Build normalized length** $(\lambda_D,C_D)$ from a $p$-power root tower plus (in mixed characteristic) perfectoidization; it makes parameter quotients positive and positive-degree parameter Koszul homology zero.
3. **Prove rational descent** $\mathbb K^J(H)_\mathbb Q\simeq(\mathbb K^J(H\otimes_A^{\mathbf L}T)_\mathbb Q)^{hG}$ for finite surjections that are transitive on geometric fibers, with arbitrary connective coefficients.
4. **First averaging:** express $|G_D|\,a_D\,a_E\,\chi^R(D,E)$ as a sum of the twisted normalized Euler characteristics $c_\sigma$ of $(C_D,E')$ over $R$.
5. **Second averaging (asymptotic):** for each $\sigma$, approximate the parameter quotient $U=C_D/(x)C_D$ by finite-length modules $M_n$, truncate their minimal resolutions at degree $\dim E$, and let the resulting compact supported asymptotic object carry both evaluations; the ordinary one is $c_\sigma$, the normalized one is positive.
6. **Conclude:** $|G_E|\,c_\sigma>0$ for every $\sigma$, so the first sum is positive, so $\chi^R(D,E)>0$.

Three ideas do the real work: the normalized length that kills the alternating-sign obstruction; the descent theorem applied twice, on two different bases; and the uniform annihilator $\mathfrak m_B^a$, which simultaneously makes the truncated sequence supported and makes the normalized evaluation strictly positive.

## 11. Sources

The argument rests on the following results; full bibliographic details are in the original paper's reference list.

- J.-P. Serre, *Local Algebra*, Chapter V: intersection multiplicities, the diagonal reduction, equicharacteristic positivity.
- P. Roberts, and independently H. Gillet, C. Soulé: vanishing $\chi^R(V,W)=0$ when $\dim V+\dim W