Almost purity

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Section 25 Almost purity

Reference.

[25], section 10.

We deduce a strong form of the almost purity theorem. The statement combines the perfectoid almost purity theorems of Scholze [107] and Kedlaya-Liu [82] (which extend the original almost purity theorem of Faltings) with André's perfectoid Abhyankar lemma [4].

Subsection 25.1 Some initial remarks

To clarify a potential apparent ambiguity in the statement of Theorem 25.2.6, we issue the following reminder.

Remark 25.1.1.

Let $R \to S$ be a ring homomorphism such that $S$ is finitely generated as an $R$-module; such a homomorphism is usually said to be finite, but for added emphasis we will sometimes say that it is module-finite. In any case, under this condition, $S$ is finitely presented as an $R$-module if and only if $S$ is finitely presented as an $R$-algebra ([117], tag 0D46). That is, if we say that $S$ is a “finitely presented, module-finite $R$-algebra”, the two possible interpretations of this statement are equivalent.

We next give an indication of why almost purity is a highly nontrivial statement.

Remark 25.1.2.

A finitely presented, module-finite algebra $S$ over a lens $R$ is not necessarily a lens or even a regular semilens. One rather prosaic reason is that it may not be reduced (e.g., $R[x]/x^2$). Somewhat more serious examples arise from taking quotients, as in Example 19.5.3. See Example 25.1.3 for a different sort of example.

Nonetheless, we will see from the statements of Theorem 25.2.6 and Theorem 25.3.4 that $S$ does inherit some good properties; for instance, its lens coperfection is concentrated in degree $0\text{.}$ We can thus think of a morphism of lenses as being “integral” if it arises by lens coperfection from an integral morphism from a lens to some target.

To begin with, note that if $R$ is itself an integral domain, then we can find some nonzero $f \in R$ such that $R[f^{-1}] \to S[f^{-1}]$ is finite étale. Our strategy will be to do almost commutative algebra using the ideal $J = (f)$ to derive constructions about $R \to S\text{.}$

Example 25.1.3.

Assume $p \neq 2\text{,}$ and take $R = \ZZ_p[x^{p^{-\infty}}]^\wedge_{(p)}$ and $S = R[x^{1/2}]\text{.}$ In this case $S$ is not a lens, but the lens coperfection is easy to describe: it is $\ZZ_p[(x^{1/2})^{p^{-\infty}}]^\wedge_{(p)}$ concentrated in degree $0\text{.}$

Subsection 25.2 Almost purity (first version)

Lemma 25.2.1.

The functor $S \mapsto S_{\lens}$ on derived $p$-complete rings satisfies descent for the arc$_p$-topology.

Proof.

This is a direct consequence of Theorem 22.5.2. Compare [25], Corollary 8.10.

Lemma 25.2.2.

Let $S \to S'$ be a integral morphism of derived $p$-complete rings such that for some derived $p$-complete ideal $J$ of $S\text{,}$ for every $p$-complete eudoxian valuation ring $V\text{,}$ every morphism $S \to V$ which does not kill $J$ extends uniquely to $S'\text{.}$ (Note that we do not allow replacing $V$ with a larger valuation ring!) Then Figure 25.2.3 is a pullback square in the derived category of $S$-modules.

Proof.

By replacing $S'$ with $S' \times S/J\text{,}$ we may reduce to the case where $S \to S'$ is an arc$_p$-covering. By the hypothesis on $J\text{,}$

\begin{equation*} S' \widehat{\otimes}_S S' \to (S'/J \widehat{\otimes}_{S/J} S'/J) \times S' \end{equation*}

is also an arc$_p$-covering. We may then deduce the claim from Lemma 25.2.1 and the universal property of lens coperfection. (Compare [25], Corollary 8.11.)

Remark 25.2.4.

While Lemma 25.2.2 must be stated in the derived category $D(S)$ because that is the best we can prove right now, once we finish the proof of almost purity (Theorem 25.3.4) we will know that all of the objects in Figure 25.2.3 will be concentrated in degree 0. Hence we will also end up with a pullback square in $\Ring\text{.}$

Corollary 25.2.5.

Let $S \to S'$ be a integral morphism of derived $p$-complete rings such that for some derived $p$-complete ideal $J$ of $S\text{,}$ $\Spec S' \to \Spec S$ is an isomorphism outside $V(J)\text{.}$ Then $S_{\lens} \to S'_{\lens}$ is a $J$-almost isomorphism in $D(S)\text{.}$

Proof.

The hypothesis on $S \to S'$ implies the hypothesis of Lemma 25.2.2, so Figure 25.2.3 is a pullback square in $D(S)\text{.}$ In particular, the cones of the two rows are isomorphic in $D(S)\text{.}$ The bottom row consists of two objects which by construction are $J$-almost zero (Corollary 24.3.6), so its cone is also $J$-almost zero; hence the top row is a $J$-almost isomorphism.

Theorem 25.2.6.

Let $J$ be a finitely generated ideal of a lens $R\text{.}$ Let $S$ be a finitely presented and module-finite $R$-algebra such that $\Spec S \to \Spec R$ is finite étale away from $V(J)\text{.}$

We have $J_{\lens} H^i(S_{\lens}) = 0$ for all $i \gt 0\text{.}$
The map $S \to S_{\lens}$ is an isomorphism away from $V(J)\text{.}$
For every $n \gt 0\text{,}$ the morphism $R/p^n \to H^0(S_{\lens})/p^n$ is almost finite étale with respect to the context $(R/p^n, J_{\lens,n})\text{.}$
Suppose that $S$ admits an action of a finite group $G$ such that $\Spec S \to \Spec R$ is a $G$-Galois cover outside $V(J)\text{.}$ Then $R \to H^0(S_{\lens})$ is a $J$-almost $G$-Galois extension.

Proof.

We may assume from the outset that $p \in J\text{.}$ Suppose first that $S$ admits an action by a finite group $G$ such that $R \to S$ is a $J$-almost $G$-Galois cover. Note that this hypothesis is preserved by a $p$-completely flat base extension, as it can be checked modulo $p$ thanks to derived Nakayama (Remark 6.6.6); moreover, all of the conclusions can also be checked after such a base extension. By Theorem 19.4.4, we may thus assume that $R$ is absolutely integrally closed. By Lemma 19.4.2, we can then find generators $f_1,\dots,f_r$ of $J$ such that $R \to S$ splits outside $V(f_i)$ for $i=1,\dots,r\text{.}$ Since being a $J$-almost isomorphism is equivalent to being an $(f_i)$-almost isomorphism for $i=1,\dots,r\text{,}$ we may reduce to the case where $J = (f)$ and we have an $R$-algebra isomorphism $S[f^{-1}] \cong \prod_{i \in I} R[f^{-1}]$ for some finite index set $I\text{.}$ Put $S' = \prod_{i \in I} R$ and let $S'$ be the integral closure of $R$ in $S[f^{-1}]\text{;}$ we then have maps $S \to S'', S' \to S''$ to which we may apply Corollary 25.2.5. This allows us to equate all of the desired assertions about $S$ to the corresponding statements about $S'\text{,}$ which are self-evident.

Assume next that $R \to S$ has constant degree $r$ outside $V(J)\text{.}$ By Lemma 24.4.5, we can find an $S_r$-Galois covering of $\Spec(R) \setminus V(J)$ which is an $S_{r-1}$-Galois covering of $\Spec(S) \setminus V(J)\text{.}$ Using Corollary 24.4.4, we may reduce to the previous case.

Now consider the general case. In this case, $\Spec(R) \setminus V(J)$ can be partitioned as a finite union $\bigsqcup_i U_i$ of closed-open subsets, on each of which the degree of $\Spec(S) \to \Spec(R)$ is constant. For each $i\text{,}$ let $R_i$ be the image of $R$ in $H^0(U_i, \calO)$ and let $R_{i,\lens}$ be the lens coperfection of $R_i\text{.}$ The map $R \to \prod_i R_{i,\lens}$ satisfies the condition of Lemma 25.2.2, so we may reduce to the previous case.

Subsection 25.3 Almost purity (second version)

It turns out that Theorem 25.2.6 can be formally upgraded by first deducing a statement about lens coperfections of integral extensions of lenses, which amounts to a major upgrade of Corollary 19.3.6. We turn to this next.

Lemma 25.3.1.

Let $R \to S$ be a module-finite and finitely presented morphism in $\Ring_{\ZZ_{(p)}}\text{.}$ Then there exist elements $g_1,\dots,g_n \in R$ such that for

\begin{equation*} R_i = R/(g_1,\dots,g_{i-1})_{\red}[g_i^{-1}], S_i = S/(g_1,\dots,g_{i-1})_{\red}[g_i^{-1}], \end{equation*}

the following statements hold.

The ideal $(g_1,\dots,g_n)$ of $R$ is the unit ideal. That is, $\bigcup_i \Spec R_i = \Spec R.$
The map $R_i \to S_i$ factors as the composition of a finite étale morphism $R_i \to T_i$ and a universal homeomorphism $T_i \to S_i\text{.}$ Moreover, $T_i[p^{-1}] \to S_i[p^{-1}]$ is an isomorphism.
In each $R_i\text{,}$ either $p=0$ or $p \in R_i^\times\text{.}$

Proof.

Exercise (Exercise 25.6.1), or see [25], Lemma 10.12.

Remark 25.3.2.

In Lemma 25.3.1, condition (2) implies that $R[g_1^{-1}] \to S_{\red}[g_1^{-1}]$ factors as $R[g_1^{-1}] \to T_1 \to S_{\red}[g_1^{-1}]$ where the first map is finite étale and the second map is again a universal homeomorphism. The latter is forced to be an isomorphism if either $p \in R_1^\times$ or $R$ is a lens.

Theorem 25.3.3.

Let $(A,I)$ be a perfect prism with associated lens $R\text{.}$ Let $R \to S$ be the derived $p$-completion of an integral map. Then $\Prism_{S/A,\perf}$ is concentrated in degree $0\text{,}$ where it is a derived $p$-complete perfect $\delta$-ring over $A\text{.}$ Consequently, $S_{\lens}$ is concentrated in degree $0\text{,}$ where it is a lens.

Proof.

By passage to filtered colimits, we may assume that $R \to S$ is module-finite and finitely presented. By Lemma 19.3.4 and Lemma 19.3.5, we know that $\Prism_{S/A,\perf}$ is concentrated in degrees $\geq 0\text{,}$ and everything will follow once we show that it is also concentrated in degrees $\leq 0\text{.}$ For this, we fix a sequence $g_1,\dots,g_n$ as in Lemma 25.3.1 and induct on $n\text{.}$

For the base case $n=1\text{,}$ the map $R \to S_{\red}$ is finite étale, and so $S_{\red}$ is a lens. By arc-descent for lenses (Theorem 22.5.2), $S_{\lens} \to S_{\red,\lens}$ is an isomorphism.

For the induction step $n \gt 1\text{,}$ the induction hypothesis (and arc-descent) implies that $(S/g_1)_{\lens}$ is concentrated in degrees $\leq 0\text{;}$ by Lemma 24.3.7, it is enough to check that $S_{\lens}$ is $g_1$-almost concentrated in degrees $\leq 0\text{.}$ By Remark 25.3.2, $R[g_1^{-1}] \to S_{\red}[g_1^{-1}]$ is a finite étale covering.

Let $S'$ be the integral closure of $R$ in $S_{\red}[g_1^{-1}]\text{.}$ By Lemma 25.2.2, the map $S_{\lens} \to S'_{\lens}$ is a $g_1$-almost isomorphism, so it will be enough to check that $S'_{\lens}$ is $g_1$-almost concentrated in degrees $\leq 0\text{.}$ But this may be deduced from Theorem 25.2.6 by approximating $S'$ with module-finite, finitely presented $R$-algebras. (Compare [25], Theorem 10.11.)

Theorem 25.3.4.

The lens coperfection $S_{\lens}$ is concentrated in degree $0\text{,}$ where it is a lens.
The map $S \to S_{\lens}$ is an isomorphism away from $V(J)\text{.}$
For every $n \gt 0\text{,}$ the morphism $R/p^n \to S_{\lens}/p^n$ is almost finite étale with respect to the context $(R/p^n, J_{\lens,n})$
Suppose that $S$ admits an action of a finite group $G$ such that $\Spec S \to \Spec R$ is a $G$-Galois cover outside $V(J)\text{.}$ Then $R \to S_{\lens}$ is a $J$-almost $G$-Galois extension.

Proof.

Combine Theorem 25.2.6 with Theorem 25.3.3.

Remark 25.3.5.

In the case where $R$ is a $p$-torsion-free lens and $J = (p)\text{,}$ Theorem 25.3.4 recovers the almost purity theorem for perfectoid spaces, as in [107], [82]; the conclusion in this case includes the statement that $S[p^{-1}] \cong S_{\lens}[p^{-1}]\text{.}$ The case where $J \neq (p)$ incorporates the perfectoid Abhyankar lemma of [4].

Subsection 25.4 An application to cohomological dimension

Lemma 25.4.1.

Let $R$ be a $p$-torsion-free lens. Let $R \to S$ be the derived $p$-completion of a finitely presented morphism. Let $J$ be a finitely generated ideal of $S\text{.}$ Then $\Cone(S_{\lens} \to (S/J)_{\lens})$ is concentrated in degrees $\leq 0\text{.}$

Proof.

By Theorem 25.3.3, $S_{\lens}$ and $(S/J)_{\lens}$ are both lenses concentrated in degree $0\text{.}$ By Corollary 19.4.6, the map $S_{\lens} \to (S_{\lens}/J S_{\lens})_{\lens} = (S/J)_{\lens}$ is surjective. This proves the claim: the cone is actually concentrated in degrees -1 and 0.

Corollary 25.4.2.

With notation as in Lemma 25.4.1, write $R$ as the slice of the perfect prism $(A,I)\text{.}$ Then $\Cone(\Prism_{S/A,\perf} \to \Prism_{(S/J)/A,\perf})$ is concentrated in degrees $\leq 0\text{.}$

Proof.

By derived Nakayama (Proposition 6.6.2), this reduces to the corresponding statement after reduction modulo $I\text{,}$ which is Lemma 25.4.1.

Corollary 25.4.3.

Let $R$ be a $p$-torsion-free lens. Let $R \to S$ be a finitely presented morphism of rings. Let $J$ be a finitely generated ideal of $S\text{.}$ Put $Y = \Spec S[p^{-1}]\text{,}$ let $U \subseteq Y$ be the complement of $V(J)\text{,}$ and let $j\colon U \to Y$ be the canonical open immersion. Then

\begin{equation*} R\Gamma_{\et}(Y, j_! \underline{\FF_p}) \in D^{\leq 1}(\FF_p). \end{equation*}

Proof.

Combine Corollary 25.4.2 with the étale comparison theorem (Theorem 22.6.1).

Theorem 25.4.4.

Let $R$ be a $p$-torsion-free lens and put $X = \Spec R[p^{-1}]\text{.}$ Then for every etale $\FF_p$-sheaf $\calF$ on $X\text{,}$ we have $H^i(X, \calF) = 0$ for all $i \gt 0\text{.}$ That is, the $\FF_p$-étale cohomological dimension of $X$ is at most $1\text{.}$

Proof.

This reduces to Corollary 25.4.3 using the “method of the trace” ([117], tag 03SH) as in [25], Theorem 11.1.

Remark 25.4.5.

Echoing a remark from [25], we point out that Theorem 25.4.4 fails completely if we replace the scheme $X$ with the Huber adic spectrum of the ring $R[p^{-1}]\text{;}$ for example, the homotopy type of this space can contribute to cohomology in higher degrees.

Remark 25.4.6.

In connection with Theorem 25.4.4, we should mention some results of Achinger ([1]). First, every connected affine scheme over $\FF_p$ is a $K(\pi, 1)$ space for the étale topology. Second, every noetherian adic affinoid space over $\QQ_p\text{,}$ and every perfectoid space over $\QQ_p\text{,}$ is a $K(\pi, 1)$ space. Both of these results can be interpreted as saying that the fundamental groups of these space are so large as to “absorb” all higher homotopy groups.

Subsection 25.5 The direct summand conjecture

The following application of almost purity to Hochster's direct summand conjecture is given in [5], [16]. This has various consequences in commutative algebra which we do not discuss here; see instead [66].

Theorem 25.5.1.

Put $R = \ZZ_p\llbracket x_1,\dots,x_r\rrbracket$ and let $R \to S$ be an injective, module-finite ring homomorphism. Then this map splits in $\Mod_R\text{.}$

Proof.

It suffices to check that $R/p^n \to S/p^n$ splits in $\Mod_{R/p^n}$ for every $n\text{,}$ as then an application of the Artin-Rees lemma shows that $R \to S$ splits (see [16], Lemma 5.3). That is, we must show that the boundary class $\alpha \in \Ext^1_R(S/R, R)$ vanishes modulo $p^n$ for all $n \geq 2\text{.}$

Define the lens

\begin{equation*} R_1 = \ZZ_p[p^{p^{-\infty}}]\llbracket x_1^{p^{-\infty}},\dots,x_r^{p^{-\infty}}\rrbracket^\wedge_{(p)}. \end{equation*}

The apparent map $R \to R_1$ is faithfully flat because $R_1$ is the $p$-completion of a free $R$-module. By Theorem 19.4.4, there exists a $p$-completely faithfully flat morphism $R_1 \to R_2$ of lenses such that $R_2$ is AIC. Put $S_i = S \otimes_R R_i$ and let $\alpha_i \in \Ext^1_{R_i}(S_i/R_i, R_i)$ be the image of $\alpha\text{;}$ by faithfully flat descent, it is enough to check that $\alpha_2$ vanishes modulo $p^n$ for all $n \geq 3\text{.}$

Choose a nonzero element $f \in R$ such that $R[f^{-1}] \to S[f^{-1}]$ is finite étale, and define the ideal $J = (p, f)R_2\text{.}$ By Theorem 25.3.4, $R_2/p^n \to S_{2,\lens}/p^n$ is almost finite étale for the context $(R_2/p^n, J_{\lens}/p^n)\text{.}$ Consequently, $\alpha_2/p^n$ is $(J_{\lens}/p^n)$-almost zero; in particular, it is killed by $(p f)^{p^{-m}} \in R_2$ for all $m \ge 0\text{.}$ (Note that $f^{p^{-m}}$ makes sense in $R_2$ because the latter is AIC; this is why we didn't stop at $R_1\text{.}$ Also, we are using that $S_2$ maps to $S_{2,\lens}$ but not any closer relationship between these two objects.)

Now suppose that $\alpha/p^n \neq 0$ for some $n \geq 2\text{.}$ By Krull's intersection theorem ([117], tag 00IP), $pf \notin (\Ann_{R/p^n}(\alpha/p^n))^{p^m}$ for $m \gg 0\text{.}$ Since $R \to R_2$ is $p$-completely faithfully flat, we also have $pf \notin (\Ann_{R_2 /p^n}(\alpha/p^n))^{p^m}\text{;}$ but this contradicts the previous paragraph. This conclusion yields the desired result. (Compare [16], Theorem 5.4.)

Remark 25.5.2.

A similar argument (see [16], Theorem 6.1) yields the derived direct summand conjecture: if $X \to \Spec R$ is a proper surjective morphism, then $R \to R\Gamma(X, \calO)$ splits in $D(A_0)\text{.}$

Remark 25.5.3.

Another result that can be deduced from almost purity is a mixed-characteristic analogue of the Kunz criterion of regularity in positive characteristic (Remark 19.1.2): a classically $p$-complete noetherian ring is regular if and only if it admits a faithfully flat morphism to some lens. See [20].

Remark 25.5.4.

Yet another result in this context (but which requires methods beyond the scope of these notes) is the following. Let $A$ be an excellent noetherian integral domain. Let $A^+$ be an absolute integral closure of $A$ (that is, take the integral closure of $A$ in some algebraic closure of $\Frac A$). Then for every positive integer $n\text{,}$ the $A/p^n$-module $A^+/p^n$ is Cohen-Macaulay ([17], Theorem 1.1).

Remark 25.5.5.

See [25], Remark 10.13 for an indication of how to apply Theorem 25.3.4 to recover some additional results in commutative algebra, such as the results of [63].

Exercises 25.6 Exercises

1.

Prove Lemma 25.3.1.

Hint.

We can ignore condition (3), as we may enforce it at the end by refining the stratification. To handle (1) and (2), by noetherian approximation we may reduce to the case where $R$ is a finitely generated $\ZZ_{(p)}$-algebra; in that case, see [25], Lemma 10.12.