Almost commutative algebra

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Section 24 Almost commutative algebra

Reference.

[54] (not to be confused with the much longer [55]); [25], section 10.

We introduce the framework of almost commutative algebra in preparation for the discussion of the almost purity theorem in Section 25.

Subsection 24.1 A bit of motivation

We first explain the term purity in this context.

Theorem 24.1.1. Zariski-Nagata purity of the branch locus.

Let $X \to \overline{X}$ be an open immersion of regular noetherian schemes such that $\overline{X} \setminus X$ has codimension at least $2$ in $\overline{X}\text{.}$ Then every finite étale cover of $X$ extends uniquely to a finite étale cover of $\overline{X}\text{.}$

Proof.

See [117], tag 0BMB.

We next give an example where purity of the branch locus does not apply, but something “almost” as good is true.

Proposition 24.1.2.

Let $L/K$ be a finite extension of perfectoid fields (Definition 8.3.1). Let $\frako_K, \frako_L$ be the valuation rings of $K,L$ and let $\frakm_K, \frakm_L$ be the maximal ideals of $\frako_K, \frako_L\text{.}$ Then $\Trace\colon L \to K$ induces a surjection $\frakm_L \to \frakm_K\text{.}$

Proof.

Exercise (Exercise 24.5.2).

Remark 24.1.3.

A closely related phenomenon is the fact that a ramified base change can “weaken” the ramification of a covering. A classical instance of this is Abhyankar's lemma, which can be used to eliminate tame ramification; see [117], tag 0EXT.

Subsection 24.2 A context for almost commutative algebra

The premise of almost commutative algebra is that in certain situations, one would like to treat certain types of “small” modules over a ring as if they were actually zero. For the theory of modules over a ring, this is relatively straightforward to achieve using the notion of the quotient by a thick subcategory. However, we would also like to define “almost” variants of some ring-theoretic concepts, and this is somewhat more involved; we give only the necessary details here, restricted to the minimal level of generality sufficient for our purposes. See [54] for a more comprehensive initial development.

Definition 24.2.1.

By a context (more precisely a context for almost commutative algebra), we will mean a pair consisting of a base ring $V$ and an ideal $\frakm$ such that $\frakm^2 = \frakm\text{.}$

Example 24.2.2.

The pair $(\ZZ, (1))$ is a context for almost commutative algebra. We call this the classical limit, where we expect to recover concepts in ordinary commutative algebra.

Example 24.2.3.

For $V$ a nondiscrete valuation ring with maximal ideal $\frakm\text{,}$ the pair $(V, \frakm)$ is a context for almost commutative algebra. Since $\frakm$ is a colimit of principal ideals, the $V$-module $\frakm \otimes_V \frakm$ is flat; while adding this restriction to the definition of a context is needed for a deeper treatment (for instance, in [54] it is required starting from the end of Chapter 2), we will not need it here.

Definition 24.2.4.

Fix a context $(V, \frakm)$ for almost commutative algebra. A $V$-module $M$ is almost zero if $\frakm M = 0\text{.}$ It is straightforward to check that the subcategory of almost zero $V$-modules is a thick tensor ideal in $\Mod_V\text{.}$ It thus makes sense to say that a morphism in $\Mod_V$ is an almost isomorphism (i.e., its kernel and cokernel are almost zero).

Definition 24.2.5.

Fix a context $(V, \frakm)\text{.}$ Choose $A \in \Ring_V$ and $M \in \Mod_A\text{.}$ The module of almost elements of $M$ is the object

\begin{equation*} M_* = \Hom_A(\frakm, M) \in \Mod_A; \end{equation*}

the natural map

\begin{equation*} M = \Hom_A(A,M) \to \Hom_A(\frakm, M) = M_* \end{equation*}

is an almost isomorphism. Note that for $N \in \Mod_A$ a second object, we have natural isomorphisms

\begin{gather} \Hom_{A_*}(M_*, N_*) \cong \Hom_A(M_*, N_*) \cong \Hom_A(M, N)_*\tag{24.1}\\ M_* \otimes_{A_*} N_* \cong M_* \otimes_A N_* \cong (M \otimes N)_*.\tag{24.2} \end{gather}

To define the category of almost $A$-modules, take objects to be the objects of $\Mod_A\text{,}$ with the morphisms from $M$ to $N$ being $\Hom_A(M, N)_*\text{.}$ This makes sense because by (24.2), composition defines a morphism

\begin{equation*} \Hom_A(M, N)_* \otimes_A \Hom_A(N, P)_* \to \Hom_A(M, P)_*. \end{equation*}

Remark 24.2.6.

The category of almost $A$-modules can be identified with the localization of $\Mod_A$ at the multiplicative system of almost isomorphisms. The easiest way to check this is not to construct the latter directly, but to check that the former satisfies the universal property that characterizes the latter: the obvious functor from $\Mod_A$ to the category of almost $A$-modules is initial for the property that every almost isomorphism becomes a genuine isomorphism in the target.

We now introduce some definitions which generalize from the classical limit in a perhaps unexpected manner.

Definition 24.2.7.

Fix a context $(V, \frakm)\text{.}$ Choose $A \in \Ring_V$ and $M \in \Mod_A\text{.}$ We say that $M$ is almost finitely generated if for every finitely generated ideal $\frakm_0 \subseteq \frakm\text{,}$ there is a finitely generated A-submodule $M_0 \subseteq M$ with $\frakm_0 M \subseteq M_0\text{.}$

We say that $M$ is almost projective if the functor on $\Mod_A$ given by $N \mapsto \Hom_A(M, N)_*$ is exact.

We write almost finite projective as shorthand for almost finitely generated and almost projective. Note that $M$ is almost finite projective if and only if for each $\eta \in \frakm\text{,}$ there exist a finite free $A$-module $F$ and a pair of morphisms $M \to F \to M$ which compose to multiplication by $\eta\text{.}$ (Compare [54], Proposition 2.3.10, Definition 2.4.4.)

Remark 24.2.8.

While it is true that any $A$-module which is almost isomorphic to a finitely generated $A$-module is almost finitely generated, the converse is not true. Moreover, an almost projective module is not projective in the category of almost modules; see [54], Example 2.4.5.

Definition 24.2.9.

Fix a context $(V, \frakm)\text{.}$ A morphism $A \to B$ in $\Ring_V$ is almost finite étale if $B$ is an almost finite projective $A$-module and also an almost finite projective $(B \otimes_A B)$-module via the multiplication map. (Note that these conditions do characterize a finite étale morphism in the classical limit, by [117], tag 0CKP.)

Remark 24.2.10.

We will use the following limited form of “almost faithfully flat descent”: if $A \to B$ is an almost injective, almost finite étale morphism of rings and $A \to C$ is another morphism of rings, then $A \to C$ is almost finite étale if and only if $B \to B \otimes_A C$ is.

Subsection 24.3 Almost commutative algebra for lenses

It is convenient to make a slightly different set of definitions when working with modules over lenses.

Definition 24.3.1.

For $J$ an ideal of a lens $R\text{,}$ Corollary 19.4.6 implies that the natural map from $R$ to the lens coperfection $(R/J)_{\lens}$ is surjective. Denote its kernel by $J_{\lens}\text{;}$ this means that $R/J_{\lens} = (R/J)_{\lens}\text{,}$ allowing us to omit some parentheses in what follows.

Let $M$ be a derived $p$-complete $R$-module. We say that $M$ is $J$-almost zero if $J_{lens}M = 0\text{.}$ We say that a derived $p$-complete complex $K^\bullet \in D(R)$ is $J$-almost zero if $H^i(M)$ is $J$-almost zero for all $i\text{.}$ (Compare [25], Definition 10.1.)

Example 24.3.2.

In Definition 24.3.1, in the case $J = (p)$ we have $J_{\lens} = \sqrt{pR}\text{.}$

Lemma 24.3.3.

Let $J$ be an ideal of a lens $R\text{.}$ The multiplication maps

\begin{equation*} J_{\lens} \widehat{\otimes}^L_R J_{\lens} \to J_{\lens} \end{equation*}

and

\begin{equation*} R/J_{\lens} \widehat{\otimes}^L_{R} R/J_{\lens} \to R/J_{\lens} \end{equation*}

are quasi-isomorphisms, and moreover

\begin{equation*} J_{\lens} \widehat{\otimes}^L_R R/J_{\lens} = 0. \end{equation*}

Proof.

The second equality is a direct consequence of Proposition 8.4.8, and the others follow from this. (Compare [25], Lemma 10.3.)

Definition 24.3.4.

Let $J$ be an ideal of a lens $R\text{.}$ For each positive integer $n\text{,}$ let $J_{\lens,n}$ be the image of $J_{\lens}$ in $R/p^n\text{.}$ By Lemma 24.3.3, the pair $(R/p^n, J_{\lens,n})$ is a context.

Proposition 24.3.5.

Let $J$ be an ideal of a lens $R\text{.}$ Within the category of derived $p$-complete $R$-modules, the subcategory of $J$-almost zero modules is stable under kernels, cokernels, and extensions in the ambient category, and is an ideal under $\otimes\text{.}$ It is also equivalent to the category of derived $p$-complete $R/J_{\lens}$-modules.

Proof.

For $M,N$ two derived $p$-complete $(R/J)_{\lens}$-modules,

\begin{equation*} R\Hom_R(M,N) = R\Hom_{R/J_{\lens}}(M \widehat{\otimes}^L_R R/J_{\lens}, N) \end{equation*}

and by Lemma 24.3.3 we have $M \widehat{\otimes}^L_R R/J_{\lens} \cong M$ (reducing to the case $M = R/J_{\lens}$). It follows that the restriction functor from derived $p$-complete $R/J_{lens}$-modules to derived $p$-complete $R$-modules, which evidently factors through the subcategory in question, defines an equivalence to this subcategory and preserves $\Ext^1\text{;}$ this yields all of the claims.

Corollary 24.3.6.

Let $J$ be an ideal of a lens $R\text{.}$ A derived $p$-complete complex $K^\bullet \in D(R)$ is $J$-almost zero if and only if $J_{\lens} \widehat{\otimes}^L_R K^\bullet = 0\text{.}$ Within the category of derived $p$-complete complexes of $R$-modules, the subcategory of $J$-almost zero complexes forms a thick tensor ideal which is equivalent to the category of derived $p$-complete complexes of $R/J_{\lens}$-modules.

Proof.

This is a direct consequence of Proposition 24.3.5. (Compare [25], Proposition 10.4.)

Lemma 24.3.7.

Let $J$ be an ideal of a lens $R\text{.}$ Let $K^\bullet$ be a derived $p$-complete complex of $R$-modules. Then $K$ is concentrated in degrees $\leq 0$ if and only if $K^\bullet \widehat{\otimes}^L_R R/J_{\lens}$ is concentrated in degrees $\leq 0$ and $H^i(K^\bullet)$ is $J$-almost zero for all $i \gt 0\text{.}$

Proof.

The “only if” is clear because tensor products are right exact. For the converse, note that in the distinguished triangle

\begin{equation*} J_{\lens} \widehat{\otimes}^L_R \tau^{\leq 0} K^\bullet \to J_{\lens} \widehat{\otimes}^L_R K^\bullet \to J_{\lens} \widehat{\otimes}^L_R \tau^{>0} K^\bullet \to \end{equation*}

the first term is concentrated in degrees $\leq 0$ and the last term is zero by Proposition 24.3.5. Combining this with the distinguished triangle

\begin{equation*} J_{\lens} \widehat{\otimes}^L_R K^\bullet \to K^\bullet \to R/J_{\lens} \widehat{\otimes}^L_R K^\bullet \to \end{equation*}

yields the claim. (Compare [25], Lemma 10.5.)

Subsection 24.4 Almost Galois extensions of rings

Just as it is sometimes useful to study field extensions using Galois theory (see Remark 23.1.2 for an example that we encountered recently), we would like to study finite étale maps of rings using Galois actions.

Definition 24.4.1.

Fix a context $(V, \frakm)\text{.}$ Let $A \to B$ be a morphism in $\Ring_V\text{.}$ Let $G$ be a finite group acting $A$-linearly on the ring $B\text{.}$ We say that $A \to B$ is an almost $G$-Galois extension if the map $A \to B^G$ is an almost isomorphism and the canonical map

\begin{equation} B \otimes_A B \to \prod_{g \in G} B, \qquad b \otimes b' \mapsto (\gamma(b)b')_{\gamma \in G}\tag{24.3} \end{equation}

is an almost isomorphism. Note that this property persists under base change on $A\text{.}$

Lemma 24.4.2.

With notation as in Definition 24.4.1, the following statements hold.

The morphism $A \to B$ is almost finite étale.
Let $C$ be the fixed subring of $B$ under a subgroup $H$ of $G\text{,}$ and suppose that $C \to B$ is an almost $H$-Galois extension. Then $A \to C$ is almost finite étale.

Proof.

To prove (1), we only need to check that $B$ is an almost finite projective $A$-module, as (24.3) already implies that $B$ is an almost finite projective $B \otimes_A B$-module. By (24.3), the idempotent element of $\prod_{g \in G} B$ that picks out the identity component is an almost element of $B \otimes_A B\text{.}$ Consequently, for each $\eta \in \frakm\text{,}$ we may multiply by $\eta$ to get a genuine element $e_\eta \in B \otimes_A B$ satisfying $e_\eta^2 = e_\eta$ that kills the kernel of $\mu$ and projects to $\eta \in B\text{.}$ Write $e_\eta = \sum_{i=1}^n b_i \otimes b'_i$ for some $b_i, b_i' \in B\text{;}$ we then have $\sum_{i=1}^n \gamma(b_i) b_i' = 0$ for $\gamma \in G \setminus \{e\}$ and $\sum_{i=1}^n b_i b_i' = \eta\text{.}$

Define the trace map $t_{B/A}\colon B \to A$ as the sum over $G$-conjugates. Then

\begin{equation*} \sum_i t_{B/A}(bb_i) b'_i = \eta b \qquad (b \in B). \end{equation*}

In other words, the composition

\begin{equation*} B \stackrel{b \mapsto (t_{B/A}(b b_i))_i}{\longrightarrow} A^n \stackrel{(a_i) \mapsto \sum a_i b'_i}{\longrightarrow} B \end{equation*}

is multiplication by $\eta\text{;}$ since $\eta \in \frakm$ was arbitrary, this proves that $B$ is an almost finite projective $A$-module.

To prove (2), we first apply (1) to deduce that $C \to B$ is almost finite étale. We then check that the canonical map $C \otimes_A B \to \prod_{G/H} B$ is an almost isomorphism: we can check this after tensoring over $C$ with $B\text{,}$ in which case we have almost isomorphisms

\begin{equation*} B \otimes_C (C \otimes_A B) = B \otimes_A B \to \prod_G B \to \prod_{G/H} (B \otimes_C B) = B \otimes_C \prod_{G/H} B. \end{equation*}

Thus the map $A \to C$ becomes almost finite étale after tensoring over $A$ with $B\text{,}$ and so by Remark 24.2.10 is itself almost finite étale. (Compare [4], Proposition 9.1.)

Definition 24.4.3.

Let $J$ be an ideal of a lens $R\text{.}$ We define a $J$-almost $G$-Galois extension of $R$-algebras by analogy with Definition 24.4.1.

Corollary 24.4.4.

Let $J$ be an ideal of a lens $R\text{.}$ Let $S$ be a derived $p$-complete $R$-algebra with an action by a finite group $G\text{,}$ such that $R \to S$ is a $J$-almost $G$-Galois cover. Let $n$ be a positive integer.

The morphism $R/p^n \to S/p^n$ is almost finite étale for the context $(R/p^n, J_{\lens,n})\text{.}$
Let $S'$ be the fixed subring of $S$ under a subgroup $H$ of $G$ and suppose that $S' \to S$ is a $J$-almost $H$-Galois cover. Then $R/p^n \to S'/p^n$ is almost finite étale for the context $(R/p^n, J_{\lens,n})\text{.}$

Proof.

The map $R/p^n \to S/p^n$ is again a $J$-almost $G$-Galois cover, so we may apply Lemma 24.4.2 to conclude. (Compare [25], Proposition 10.8.)

The construction of Galois closures of field extensions has the following analogue in this context. (One can give a version of this for almost commutative algebra, but we will only need the classical limit.)

Lemma 24.4.5.

Let $R \to S$ be a finite étale morphism of constant rank $r\text{.}$ Then there exists an $S_r$-Galois extension $R \to T$ (for the classical context) factoring through an $S_{r-1}$-Galois extension $S \to T\text{.}$

Proof.

Let $\Spec T$ be the closed-open subscheme of the $r$-fold fiber product of $\Spec S$ over $\Spec R$ which is the complement of all of the partial diagonals; this has the desired effect. (Compare [4], Lemma 1.9.2.)

Exercises 24.5 Exercises

1.

Prove that in the notation of Definition 24.4.1, the induced map $A \to R\Gamma(G, B)$ is also an almost isomorphism; that is, the groups $H^i(G, B)$ are almost zero for all $i \gt 0\text{.}$

Hint.

In the classical limit, $B$ is an induced $A[G]$-module.

2.

Prove Proposition 24.1.2.

Hint.

Reduce to the case of characteristic $p\text{.}$