Chapter 0 — Preliminaries

§7. Adic Rings

7.1. Admissible rings

(7.1.1) Recall that in a topological ring $A$ (not necessarily separated), an element $x$ is topologically nilpotent if 0 is a limit of the sequence $(x^n{)}_{n\ge 0}$. A topological ring $A$ is linearly topologized if there is a fundamental system of neighborhoods of 0 in $A$ consisting of (necessarily open) ideals.

Definition (7.1.2). In a linearly topologized ring $A$, an ideal of definition is an open ideal $\mathfrak{J}$ such that for every neighborhood $V$ of 0 there is $n>0$ with ${\mathfrak{J}}^n\subset V$ (by abuse of language: the sequence $({\mathfrak{J}}^n)$ tends to 0). $A$ is preadmissible if there is an ideal of definition in $A$; it is admissible if it is preadmissible, separated, and complete.

Plainly, if $\mathfrak{J}$ is an ideal of definition and $\mathfrak{L}$ an open ideal, then $\mathfrak{J}\cap \mathfrak{L}$ is also an ideal of definition; so the ideals of definition of a preadmissible ring form a fundamental system of neighborhoods of 0.

Lemma (7.1.3). Let $A$ be a linearly topologized ring.

(i) For $x\in A$ to be topologically nilpotent, it is necessary and sufficient that for every open ideal $\mathfrak{J}\subset A$, the canonical image of $x$ in $A/\mathfrak{J}$ be nilpotent. The set $\mathfrak{T}$ of topologically nilpotent elements of $A$ is an ideal. (ii) Suppose $A$ is preadmissible and $\mathfrak{J}$ is an ideal of definition. Then $x$ is topologically nilpotent iff its image in $A/\mathfrak{J}$ is nilpotent; $\mathfrak{T}$ is the inverse image of the nilradical of $A/\mathfrak{J}$, and is therefore open.

Proof. (i) follows from the definitions. For (ii): for every neighborhood $V$ of 0, some ${\mathfrak{J}}^n\subset V$; if $x^m\in \mathfrak{J}$, then $x^{mq}\in V$ for $q\ge n$, so $x$ is topologically nilpotent.

Proposition (7.1.4). Let $A$ be a preadmissible ring with ideal of definition $\mathfrak{J}$.

(i) For ${\mathfrak{J}}^{\prime }\subset A$ to be contained in an ideal of definition, it is necessary and sufficient that ${\mathfrak{J}}^{\prime n}\subset \mathfrak{J}$ for some $n>0$. (ii) For $x\in A$ to be contained in an ideal of definition, it is necessary and sufficient that $x$ be topologically nilpotent.

Corollary (7.1.5). In a preadmissible ring $A$, every open prime ideal contains every ideal of definition.

Corollary (7.1.6). Under the hypotheses of (7.1.4), the following are equivalent for an ideal ${\mathfrak{J}}_0$:

(a) ${\mathfrak{J}}_0$ is the largest ideal of definition; (b) ${\mathfrak{J}}_0$ is a maximal ideal of definition; (c) ${\mathfrak{J}}_0$ is an ideal of definition such that $A/{\mathfrak{J}}_0$ is reduced.

Such ${\mathfrak{J}}_0$ exists if and only if the nilradical of $A/\mathfrak{J}$ is nilpotent; ${\mathfrak{J}}_0$ is then the ideal $\mathfrak{T}$ of topologically nilpotent elements.

When the nilradical of $A/\mathfrak{J}$ is nilpotent, we write $A_{red}$ for the reduced quotient $A/\mathfrak{T}$.

Corollary (7.1.7). A preadmissible Noetherian ring has a largest ideal of definition.

Corollary (7.1.8). If $A$ is preadmissible and the powers ${\mathfrak{J}}^n$ ($n>0$) of some ideal of definition $\mathfrak{J}$ form a fundamental system of neighborhoods of 0, the same holds for ${\mathfrak{J}}^{\prime n}$ for every ideal of definition ${\mathfrak{J}}^{\prime }$.

Definition (7.1.9). A preadmissible ring $A$ is preadic if there is an ideal of definition $\mathfrak{J}$ whose powers ${\mathfrak{J}}^n$ form a fundamental system of neighborhoods of 0 (equivalently, the ${\mathfrak{J}}^n$ are open). An adic ring is a separated and complete preadic ring.

If $\mathfrak{J}$ is an ideal of definition of a preadic (resp. adic) ring $A$, we also say $A$ is 𝔍-preadic (resp. 𝔍-adic), and its topology is the 𝔍-preadic (resp. 𝔍-adic) topology. For an $A$-module $M$, the topology with fundamental system of neighborhoods the submodules ${\mathfrak{J}}^{n}M$ is called the 𝔍-preadic (resp. 𝔍-adic) topology. By (7.1.8), these topologies are independent of the choice of ideal of definition $\mathfrak{J}$.

Proposition (7.1.10). Let $A$ be an admissible ring and $\mathfrak{J}$ an ideal of definition. Then $\mathfrak{J}$ is contained in the radical of $A$.

This is equivalent to any of the following:

Corollary (7.1.11). For every $x\in \mathfrak{J}$, $1+x$ is invertible in $A$.

Corollary (7.1.12). For $f\in A$ to be invertible, it is necessary and sufficient that its image in $A/\mathfrak{J}$ be invertible.

Corollary (7.1.13). For every $A$-module $M$ of finite type, $M=\mathfrak{J}M$ (equivalent to $M{\otimes }_A(A/\mathfrak{J})=0$) implies $M=0$.

Corollary (7.1.14). Let $u:M\to N$ be an $A$-module homomorphism with $N$ of finite type. For $u$ to be surjective, it is necessary and sufficient that $u\otimes 1:M{\otimes }_A(A/\mathfrak{J})\to N{\otimes }_A(A/\mathfrak{J})$ be.

Proof of (7.1.10). The equivalences with (7.1.11), (7.1.13), and (7.1.12) follow from standard results of Bourbaki (Alg., Chap. VIII). To prove (7.1.11): since $A$ is separated and complete and $({\mathfrak{J}}^n)$ tends to 0, the series ${\sum }_{n=0}^{\infty }(-1{)}^n{x}^n$ converges in $A$; its sum $y$ satisfies $y(1+x)=1$.

7.2. Adic rings and projective limits

(7.2.1) Every projective limit of discrete rings is plainly a linearly topologized, separated, and complete ring. Conversely, let $A$ be linearly topologized with $({\mathfrak{J}}_{\lambda })$ a fundamental system of open neighborhoods of 0 consisting of ideals. The canonical maps ${\varphi }_{\lambda }:A\to A/{\mathfrak{J}}_{\lambda }$ form a projective system, defining a continuous map $\varphi :A\to \underleftarrow{\lim }A/{\mathfrak{J}}_{\lambda }$; if $A$ is separated, $\varphi$ is a topological isomorphism onto a dense subring; if also complete, $\varphi$ is a topological isomorphism onto $\underleftarrow{\lim }A/{\mathfrak{J}}_{\lambda }$.

Lemma (7.2.2). A linearly topologized ring is admissible if and only if it is isomorphic to a projective limit $A=\underleftarrow{\lim }{A}_{\lambda }$ of discrete rings indexed by a filtered ordered set $L$ with smallest element 0, such that:

1. each $u_{\lambda }:A\to A_{\lambda }$ is surjective;
2. the kernel ${\mathfrak{J}}_{\lambda }$ of $u_{0\lambda }:A_{\lambda }\to A_0$ is nilpotent.

When this holds, the kernel $\mathfrak{J}$ of $u_0:A\to A_0$ equals $\underleftarrow{\lim }{\mathfrak{J}}_{\lambda }$.

(7.2.3) Let $A$ be admissible and $\mathfrak{J}\subset A$ an ideal contained in an ideal of definition (equivalently, $({\mathfrak{J}}^n)$ tends to 0). The ring topology on $A$ having the ${\mathfrak{J}}^n$ ($n>0$) as fundamental system of neighborhoods of 0 — also called the 𝔍-preadic topology — is separated, since ${\bigcap }_n{\mathfrak{J}}^n=0$. Let $\hat{A}=\underleftarrow{\lim }A/{\mathfrak{J}}^n$ (the $A/{\mathfrak{J}}^n$ discrete) be the completion for this topology, and let $u:A\to \hat{A}$ be the (possibly discontinuous) projective limit of $u_n:A\to A/{\mathfrak{J}}^n$. The $\mathfrak{J}$-preadic topology is finer than the given topology $\mathcal{T}$; extending the identity of $A$ (with $\mathfrak{J}$-preadic topology) by continuity to $\mathcal{T}$ gives a continuous map $v:\hat{A}\to A$.

Proposition (7.2.4). If $A$ is admissible and $\mathfrak{J}$ is contained in an ideal of definition, then $A$ is separated and complete for the $\mathfrak{J}$-preadic topology.

Proof. With the notation of (7.2.3), $v\circ u=i{d}_A$ directly, and $u_n\circ v$ is the canonical $\hat{A}\to A/{\mathfrak{J}}^n$, so $u\circ v=i{d}_{\hat{A}}$.

Corollary (7.2.5). Under the hypotheses of (7.2.3), the following are equivalent:

(a) $u$ is continuous; (b) $v$ is bicontinuous; (c) $A$ is $\mathfrak{J}$-adic.

Corollary (7.2.6). Let $A$ be an admissible ring and $\mathfrak{J}$ an ideal of definition. $A$ is Noetherian if and only if $A/\mathfrak{J}$ is Noetherian and $\mathfrak{J}/{\mathfrak{J}}^2$ is an $A/\mathfrak{J}$-module of finite type.

The conditions are clearly necessary. Conversely, by (7.2.4) $A$ is complete for the $\mathfrak{J}$-preadic topology, so it is Noetherian if and only if the associated graded ring $grad(A)$ (for the filtration $({\mathfrak{J}}^n)$) is. Choosing $a_1,\cdots ,a_n\in \mathfrak{J}$ with classes mod ${\mathfrak{J}}^2$ generating $\mathfrak{J}/{\mathfrak{J}}^2$, induction shows their degree-$m$ monomials generate ${\mathfrak{J}}^m/{\mathfrak{J}}^{m+1}$ as $A/\mathfrak{J}$-module. Hence $grad(A)$ is a quotient of $(A/\mathfrak{J})[T_1,\cdots ,T_n]$.

Proposition (7.2.7). Let $(A_i,u_{ij})$ ($i\in \mathbb{N}$) be a projective system of discrete rings, and let ${\mathfrak{J}}_i\subset A_i$ be the kernel of $u_{0i}:A_i\to A_0$. Suppose:

(a) For $i\le j$, $u_{ij}$ is surjective with kernel ${\mathfrak{J}}_j^{i+1}$ (so $A_i\cong A_j/>{\mathfrak{J}}_j^{i+1}$). (b) ${\mathfrak{J}}_1/{\mathfrak{J}}_1^2$ ($={\mathfrak{J}}_1$) is finite-type over $A_0=A_1/{\mathfrak{J}}_1$.

Let $A={\underleftarrow{\lim }}_i{A}_i$, $u_n:A\to A_n$ the canonical map, and ${\mathfrak{J}}^{(n+1)}\subset A$ its kernel. Then:

(i) $A$ is adic, with $\mathfrak{J}={\mathfrak{J}}^{(1)}$ an ideal of definition. (ii) ${\mathfrak{J}}^{(n)}=>{\mathfrak{J}}^n$ for every $n\ge 1$. (iii) $\mathfrak{J}/{\mathfrak{J}}^2$ is isomorphic to ${\mathfrak{J}}_1$, hence finite-type over $A_0=A/\mathfrak{J}$.

Proof. The surjectivity of the $u_{ij}$ makes each $u_n$ surjective; (a) gives ${\mathfrak{J}}_j^{j+1}=0$, so $A$ is admissible by (7.2.2). The ${\mathfrak{J}}^{(n)}$ form a fundamental system of neighborhoods of 0, so (ii) ⟹ (i). Since $\mathfrak{J}={\underleftarrow{\lim }}_i{\mathfrak{J}}_i$ with surjective maps to ${\mathfrak{J}}_i$, (ii) ⟹ (iii). For (ii): ${\mathfrak{J}}^{(n)}$ consists of $(x_k)$ with $x_k=0$ for $k<n$, so ${\mathfrak{J}}^{(n)}\cdot {\mathfrak{J}}^{(m)}\subset {\mathfrak{J}}^{(n+m)}$ — a filtration. Also ${\mathfrak{J}}^{(n)}/{\mathfrak{J}}^{(n+1)}$ projects to ${\mathfrak{J}}_n^n$ (an A_0-module). Choose $r$ elements $a_j=(a_{jk})$ of $\mathfrak{J}$ whose $a_{j1}$ generate ${\mathfrak{J}}_1$ over A_0. We show by induction that monomials of degree $n$ in the $a_j$ generate ${\mathfrak{J}}^{(n)}$; the same argument (passage to graded modules) closes the induction.

Corollary (7.2.8). Under the hypotheses of (7.2.7), $A$ is Noetherian if and only if A_0 is.

Proof. By (7.2.6).

Proposition (7.2.9). Under the hypotheses of (7.2.7), for each $i$ let $M_i$ be an $A_i$-module, and for $i\le j$ let $v_{ij}:M_j\to M_i$ be a $u_{ij}$-homomorphism with $(M_i,v_{ij})$ a projective system. Suppose M_0 is A_0-finite-type and each $v_{ij}$ is surjective with kernel ${\mathfrak{J}}_j^{i+1}{M}_j$. Then $M=\underleftarrow{\lim }{M}_i$ is an $A$-module of finite type, and the kernel of the surjective $v_n:M\to M_n$ is ${\mathfrak{J}}^{n+1}M$ (so $M_n\cong M/{\mathfrak{J}}^{n+1}M=M{\otimes }_A(A/{\mathfrak{J}}^{n+1})$).

Proof. Choose $s$ elements $z_h=(z_{hk})\in M$ such that the $z_{h0}$ generate M_0. Reducing to showing the $z_{hn}$ generate $M_n$ over $A_n$, induction on $n$ using $M_{n-1}=M_n/{\mathfrak{J}}_n^n{M}_n$ plus Nakayama closes the argument. Passage to graded modules then gives ${\mathfrak{J}}^{(n)}M={\mathfrak{J}}^{n}M$, the kernel of $M\to M_{n-1}$.

Corollary (7.2.10). Let $(N_i,w_{ij})$ be a second such projective system, $N=\underleftarrow{\lim }{N}_i$. There is a bijective correspondence between projective systems $(h_i)$ of $A_i$-homomorphisms $M_i\to N_i$ and $A$-homomorphisms $M\to N$ (necessarily continuous for the $\mathfrak{J}$-adic topologies).

Remark (7.2.11). Let $A$ be adic with ideal of definition $\mathfrak{J}$ such that $\mathfrak{J}/{\mathfrak{J}}^2$ is $A/\mathfrak{J}$-finite-type. The $A_i=A/{\mathfrak{J}}^{i+1}$ satisfy (7.2.7), and $A=\underleftarrow{\lim }{A}_i$. Thus (7.2.7) describes all adic rings of this type — in particular all adic Noetherian rings.

Example (7.2.12). Let $B$ be a ring, $\mathfrak{J}\subset B$ an ideal with $\mathfrak{J}/{\mathfrak{J}}^2$ finite-type over $B/\mathfrak{J}$; set $A={\underleftarrow{\lim }}_{n}B/{\mathfrak{J}}^{n+1}$. Then $A$ is the separated $\mathfrak{J}$-preadic completion of $B$. The $A_n=B/{\mathfrak{J}}^{n+1}$ satisfy (7.2.7); so $A$ is adic, the closure $\bar{\mathfrak{J}}$ in $A$ of the image of $\mathfrak{J}$ is an ideal of definition, ${\bar{\mathfrak{J}}}^n$ is the closure of the image of ${\mathfrak{J}}^n$, $A/{\bar{\mathfrak{J}}}^n\cong B/{\mathfrak{J}}^n$, and $\bar{\mathfrak{J}}/{\bar{\mathfrak{J}}}^2$ is isomorphic to $\mathfrak{J}/{\mathfrak{J}}^2$ as $A/\bar{\mathfrak{J}}$-module. Likewise, if $N$ is a $B$-module with $N/\mathfrak{J}N$ finite-type, then $M={\underleftarrow{\lim }}_{i}N/{\mathfrak{J}}^{i+1}N$ is $A$-finite-type, isomorphic to the separated $\mathfrak{J}$-preadic completion of $N$.

7.3. Preadic Noetherian rings

(7.3.1) Let $A$ be a ring, $\mathfrak{J}\subset A$ an ideal, $M$ an $A$-module. Write $\hat{A}={\underleftarrow{\lim }}_{n}A/{\mathfrak{J}}^n$ (resp. $\hat{M}={\underleftarrow{\lim }}_{n}M/{\mathfrak{J}}^{n}M$) for the separated $\mathfrak{J}$-preadic completion of $A$ (resp. $M$). For an exact sequence $M^{\prime }\stackrel{u}{\to }M\stackrel{v}{\to }{M}^{\prime \prime }\to 0$, the sequence $M^{\prime }/{\mathfrak{J}}^n{M}^{\prime }\to M/{\mathfrak{J}}^{n}M\to M^{\prime \prime }/{\mathfrak{J}}^n{M}^{\prime \prime }\to 0$ is exact for each $n$. Since $v({\mathfrak{J}}^{n}M)={\mathfrak{J}}^n{M}^{\prime \prime }$, the limit $\hat{v}=\underleftarrow{\lim }{v}_n$ is surjective. For $z=(z_k)\in Ker\hat{v}$, lift each $z_k$ to $z_k^{\prime }\in M^{\prime }/{\mathfrak{J}}^k{M}^{\prime }$; we find $z^{\prime }=(z_n^{\prime })\in {\hat{M}}^{\prime }$ whose first $k$ components under û match $z$. So $Im(\hat{u})$ is dense in $Ker\hat{v}$.

If $A$ is Noetherian, so is Â by (7.2.12), and $\mathfrak{J}/{\mathfrak{J}}^2$ is $A$-finite-type. We also have:

Theorem (7.3.2) (Krull). Let $A$ be a Noetherian ring, $\mathfrak{J}\subset A$ an ideal, $M$ an $A$-module of finite type, and $M^{\prime }\subset M$ a submodule. Then the topology on $M^{\prime }$ induced from the $\mathfrak{J}$-preadic topology of $M$ agrees with the $\mathfrak{J}$-preadic topology of $M^{\prime }$.

This follows from:

Lemma (7.3.2.1) (Artin–Rees). Under the hypotheses of (7.3.2), there is $p\ge 0$ with $M^{\prime }\cap {\mathfrak{J}}^{n}M={\mathfrak{J}}^{n-p}(M^{\prime }\cap {\mathfrak{J}}^{p}M)$ for $n\ge p$. (Bourbaki, Alg. comm.)

Corollary (7.3.3). Under the hypotheses of (7.3.2), the canonical map $M{\otimes }_A\hat{A}\to \hat{M}$ is bijective, and $M{\otimes }_A\hat{A}$ is exact in $M$ on $A$-modules of finite type; consequently Â is a flat $A$-module (6.1.1).

Proof. First, $\hat{M}$ is exact on $A$-modules of finite type: for $0\to M^{\prime }\to M\to M^{\prime \prime }\to 0$ exact, Krull (7.3.2) shows the closure of the image of $M^{\prime }$ in $\hat{M}$ is the completion of $M^{\prime }$, so û is injective. The canonical $M{\otimes }_A\hat{A}\to \hat{M}$ is bijective when $M=A^p$; for general $M$ of finite type, take a presentation $A^p\to A^q\to M\to 0$ and apply right exactness of both functors.

Corollary (7.3.4). For $A$ Noetherian, $\mathfrak{J}\subset A$ an ideal, and M, N of finite type, there are canonical functorial isomorphisms

(M ⊗_A N)^∧ ≅ M̂ ⊗_Â N̂,    (Hom_A(M, N))^∧ ≅ Hom_Â(M̂, N̂).

This follows from (7.3.3), (6.2.1), and (6.2.2).

Corollary (7.3.5). Let $A$ be Noetherian and $\mathfrak{J}\subset A$ an ideal. The following are equivalent:

(a) $\mathfrak{J}$ is contained in the radical of $A$; (b) Â is a faithfully flat $A$-module (6.4.1); (c) Every $A$-module of finite type is separated for the $\mathfrak{J}$-preadic topology; (d) Every submodule of an $A$-module of finite type is closed for the $\mathfrak{J}$-preadic topology.

Proof. (b) ⟺ (c) by flatness of Â and (6.6.1). (c) ⟹ (d): if $N\subset M$ with $M$ of finite type, $M/N$ is separated. (d) ⟹ (a): for $\mathfrak{m}\subset A$ maximal, $\mathfrak{m}={\bigcap }_p(\mathfrak{m}+{\mathfrak{J}}^p)$; for large $p$, $\mathfrak{m}+{\mathfrak{J}}^p=\mathfrak{m}$, so ${\mathfrak{J}}^p\subset \mathfrak{m}$, hence $\mathfrak{J}\subset \mathfrak{m}$. (a) ⟹ (b): let $P$ be the closure of {0} in an $M$ of finite type for the $\mathfrak{J}$-preadic topology; by Krull, the induced topology on $P$ is the $\mathfrak{J}$-preadic, so $\mathfrak{J}P=P$; Nakayama gives $P=0$.

The conditions hold when $A$ is local Noetherian and $\mathfrak{J}\ne A$.

Corollary (7.3.6). If $A$ is $\mathfrak{J}$-preadic Noetherian, every $A$-module of finite type is separated and complete for the $\mathfrak{J}$-preadic topology.

Proof. $\hat{A}=A$, so this follows from (7.3.3).

So (7.2.9) describes all finite-type modules over an adic Noetherian ring.

Corollary (7.3.7). Under the hypotheses of (7.3.2), the kernel of $M\to \hat{M}=M{\otimes }_A\hat{A}$ is the set of $x\in M$ killed by an element of $1+\mathfrak{J}$.

7.4. Quasi-finite modules over local rings

Definition (7.4.1). Let $A$ be a local ring with maximal ideal $\mathfrak{m}$. An $A$-module $M$ is quasi-finite (over $A$) if $M/\mathfrak{m}M$ is of finite rank over the residue field $k=A/\mathfrak{m}$.

When $A$ is Noetherian, the $\mathfrak{m}$-preadic completion $\hat{M}$ is then an Â-module of finite type; this follows from (7.2.12) and the hypothesis on $M/\mathfrak{m}M$.

In particular, if $A$ is also complete and $M$ is separated for the $\mathfrak{m}$-preadic topology (i.e. ${\bigcap }_n{\mathfrak{m}}^{n}M=0$), then $M$ is $A$-finite-type: $\hat{M}$ is $A$-finite-type, $M\hookrightarrow \hat{M}$, and so $M$ is also finite-type (and equal to its completion, by (7.3.6)).

Proposition (7.4.2). Let $A$, $B$ be local rings with maximal ideals $\mathfrak{m}$, $\mathfrak{n}$, and suppose $B$ is Noetherian. Let $\varphi :A\to B$ be a local homomorphism, $M$ a $B$-module of finite type. If $M$ is $A$-quasi-finite, then the $\mathfrak{m}$-preadic and $\mathfrak{n}$-preadic topologies on $M$ agree (so both are separated).

Proof. $M/\mathfrak{m}M$ has finite length as $A$-module, hence as $B$-module. Therefore $\mathfrak{n}$ is the unique prime ideal of $B$ containing $Ann(M/\mathfrak{m}M)$: reducing to $M/\mathfrak{m}M$ simple, we have $M/\mathfrak{m}M\cong B/\mathfrak{n}$. By (0.1.7.5), the primes containing $Ann(M/\mathfrak{m}M)$ are those containing $\mathfrak{m}B+\mathfrak{b}$, where $\mathfrak{b}=An{n}_{B}M$. Since $B$ is Noetherian, $\mathfrak{m}B+\mathfrak{b}$ is an ideal of definition for $B$; so for some $k>0$, ${\mathfrak{n}}^k\subset \mathfrak{m}B+\mathfrak{b}\subset \mathfrak{n}$, giving for every $h>0$

𝔫^{hk} ⊂ (𝔪B + 𝔟)^h M = 𝔪^h M ⊂ 𝔫^h M.

Hence the two topologies agree; separation follows from (7.3.5).

Corollary (7.4.3). Under the hypotheses of (7.4.2), if $A$ is also Noetherian and complete for the $\mathfrak{m}$-preadic topology, then $M$ is $A$-finite-type.

(7.4.4) The most important case of (7.4.2) is when $B$ is $A$-quasi-finite — i.e., $B/\mathfrak{m}B$ is a finite-rank $k=A/\mathfrak{m}$-algebra. This breaks into:

(i) $\mathfrak{m}B$ is an ideal of definition for $B$; (ii) $B/\mathfrak{n}$ is a finite-rank extension of $A/\mathfrak{m}$.

Then every $B$-module of finite type is $A$-quasi-finite.

Corollary (7.4.5). Under the hypotheses of (7.4.2), if $\mathfrak{b}=An{n}_B(M)$, then $B/\mathfrak{b}$ is $A$-quasi-finite.

7.5. Rings of restricted formal series

(7.5.1) Let $A$ be a linearly topologized ring, separated and complete; let $({\mathfrak{J}}_{\lambda })$ be a fundamental system of open ideals with $A\cong \underleftarrow{\lim }A/{\mathfrak{J}}_{\lambda }$ (7.2.1). For each $\lambda$, set $B_{\lambda }=(A/{\mathfrak{J}}_{\lambda })[T_1,\cdots ,T_r]$; the $B_{\lambda }$ form a projective system of discrete rings. Set

A{T_1, …, T_r} = lim⃖ B_λ.

This ring is independent of $({\mathfrak{J}}_{\lambda })$. Concretely, let $A^{\prime }$ be the subring of $A[[T_1,\cdots ,T_r]]$ consisting of formal series ${\sum }_{\alpha }{c}_{\alpha }{T}^{\alpha }$ ($\alpha =({\alpha }_1,\cdots ,{\alpha }_r)\in {\mathbb{N}}_r$) with $\lim c_{\alpha }=0$ (along the filter of cofinite subsets of ${\mathbb{N}}_r$); call these restricted formal series in the $T_i$ with coefficients in $A$. With the topology whose fundamental system of neighborhoods of 0 consists of $\sum c_{\alpha }{T}^{\alpha }:c_{\alpha }\in V$ for $V$ a neighborhood of 0 in $A$, $A^{\prime }$ is a separated topological ring. There is a canonical topological isomorphism $A{T}_1,\cdots ,T_r\stackrel{\sim }{\to }{A}^{\prime }$, given by sending $y\in A{T}_1,\cdots ,T_r$ to ${\sum }_{\alpha }{\varphi }_{\alpha }(y)T^{\alpha }$, where ${\varphi }_{\alpha }$ extracts the coefficient of $T^{\alpha }$ (the result is restricted because almost all components vanish in each $A/{\mathfrak{J}}_{\lambda }$).

(7.5.2) Identify $A{T}_1,\cdots ,T_r$ with the ring of restricted formal series via (7.5.1). The canonical isomorphisms $(A/{\mathfrak{J}}_{\lambda })[T_1,\cdots ,T_r][T_{r+1},\cdots ,T_s]\cong (A/{\mathfrak{J}}_{\lambda })[T_1,\cdots ,T_s]$ give a canonical isomorphism

(A{T_1, …, T_r}){T_{r+1}, …, T_s} ≅ A{T_1, …, T_s}.

(7.5.3) Universal property. For every continuous homomorphism $u:A\to B$ to a linearly topologized, separated, complete ring $B$, and every system $(b_1,\cdots ,b_r)$ in $B$, there is a unique continuous homomorphism $\bar{u}:A{T}_1,\cdots ,T_r\to B$ with $\bar{u}|A=u$ and $\bar{u}(T_j)=b_j$, namely

ū(∑_α c_α T^α) = ∑_α u(c_α) b_1^{α_1} ⋯ b_r^{α_r}.

This characterizes $A{T}_1,\cdots ,T_r$ up to unique isomorphism.

Proposition (7.5.4).

(i) If $A$ is admissible, so is $A^{\prime }=A{T}_1,\cdots ,T_r$. (ii) If $A$ is $\mathfrak{J}$-adic with $\mathfrak{J}/{\mathfrak{J}}^2$ finite-type over $A/\mathfrak{J}$, then setting ${\mathfrak{J}}^{\prime }=\mathfrak{J}{A}^{\prime }$, $A^{\prime }$ is ${\mathfrak{J}}^{\prime }$-adic with ${\mathfrak{J}}^{\prime }/{\mathfrak{J}}^{\prime 2}$ finite-type over $A^{\prime }/{\mathfrak{J}}^{\prime }$. If $A$ is Noetherian, so is $A^{\prime }$.

Proof. (i) For an ideal of definition $\mathfrak{J}\subset A$, let ${\mathfrak{J}}^{\prime }\subset A^{\prime }$ consist of $\sum c_{\alpha }{T}^{\alpha }$ with all $c_{\alpha }\in \mathfrak{J}$. Then ${\mathfrak{J}}^{\prime n}\subset ({\mathfrak{J}}^n{)}^{\prime }$, so ${\mathfrak{J}}^{\prime }$ is an ideal of definition. (ii) Apply (7.2.7) to the projective system $A_i^{\prime }=A_i[T_1,\cdots ,T_r]$ with $A_i=A/{\mathfrak{J}}^{i+1}$. (One checks the kernels match ${\mathfrak{J}}_j^{\prime i+1}$ using induction; details omitted.) Noetherianness follows from (7.2.8).

Proposition (7.5.5). Let $A$ be a Noetherian $\mathfrak{J}$-adic ring and $B$ an admissible topological ring; let $\varphi :A\to B$ be a continuous homomorphism making $B$ an $A$-algebra. The following are equivalent:

(a) $B$ is Noetherian and $\mathfrak{J}B$-adic, and $B/\mathfrak{J}B$ is a finite-type algebra over $A/\mathfrak{J}$. (b) $B$ is topologically $A$-isomorphic to $\underleftarrow{\lim }{B}_n$, where $B_n=B_m/{\mathfrak{J}}^{n+1}{B}_m$ for $m\ge n$, and B_1 is a finite-type $A_1=A/{\mathfrak{J}}^2$-algebra. (c) $B$ is topologically $A$-isomorphic to a quotient of some $A{T}_1,\cdots ,T_r$ by a (necessarily closed) ideal.

Proof sketch. (c) ⟹ (a): $A^{\prime }=A{T}_1,\cdots ,T_r$ is Noetherian (7.5.4); $\mathfrak{J}{A}^{\prime }$ is an ideal of definition of $A^{\prime }$, and $B/\mathfrak{J}B$ is a quotient of $(A/\mathfrak{J})[T_1,\cdots ,T_r]$. (a) ⟹ (b): by (7.2.11), $B\cong \underleftarrow{\lim }B/{\mathfrak{J}}^{n+1}B$. (b) ⟹ (c): choose generators $(c_i)$ of the $A/\mathfrak{J}$-algebra $B/\mathfrak{J}B$ and apply (7.5.3) to get a continuous $A$-homomorphism $u:A^{\prime }\to B$; surjectivity is checked passing to associated graded modules.

7.6. Completed rings of fractions

(7.6.1) Let $A$ be linearly topologized, $({\mathfrak{J}}_{\lambda })$ a fundamental system of open neighborhoods of 0 consisting of ideals, and $S\subset A$ a multiplicative subset. With $A_{\lambda }=A/{\mathfrak{J}}_{\lambda }$, set $S_{\lambda }=u_{\lambda }(S)$; the $u_{\lambda \mu }$ give surjective $S_{\mu }^{-1}{A}_{\mu }\to S_{\lambda }^{-1}{A}_{\lambda }$, a projective system. Write

A{S⁻¹} = lim⃖ S_λ⁻¹ A_λ.

This is independent of $({\mathfrak{J}}_{\lambda })$:

Proposition (7.6.2). $A{S}^{-1}$ is topologically isomorphic to the separated completion of $S^{-1}A$ for the topology with fundamental system of neighborhoods of 0 the $S^{-1}{\mathfrak{J}}_{\lambda }$.

Corollary (7.6.3). If $S^{\prime }$ is the canonical image of $S$ in Â, then $A{S}^{-1}\cong \hat{A}{S}^{\prime -1}$.

If $A$ is separated and complete, $S^{-1}A$ need not be: take $S=f^n$ with $f$ topologically nilpotent but not nilpotent; then $S^{-1}A\ne 0$ but $S^{-1}{\mathfrak{J}}_{\lambda }=S^{-1}A$ for each $\lambda$.

Corollary (7.6.4). If $0\notin S$ in $A$, then $A{S}^{-1}\ne 0$.

(7.6.5) Call $A{S}^{-1}$ the completed ring of fractions of $A$ with denominators in $S$. There is a canonical continuous $A\to A{S}^{-1}$.

(7.6.6) Universal property. For every continuous $u:A\to B$ to a separated, complete linearly topologized ring $B$ with $u(S)$ consisting of invertible elements, $u$ factors uniquely as $A\to A{S}^{-1}\stackrel{u^{\prime }}{\to }B$ with $u^{\prime }$ continuous.

(7.6.7) Functoriality. For $\varphi :A\to B$ continuous with $\varphi (S)\subset T$, there is a unique continuous ${\varphi }^{\prime }:A{S}^{-1}\to B{T}^{-1}$ extending $\varphi$. For $B=A$, $\varphi =id$, and $S\subset T$, this gives ${\rho }^{T,S}:A{S}^{-1}\to A{T}^{-1}$ with ${\rho }^{U,S}={\rho }^{U,T}\circ {\rho }^{T,S}$ for $S\subset T\subset U$.

(7.6.8) For multiplicative subsets $S_1,S_2\subset A$ with $S_2^{\prime }$ the image of S_2 in $A{S}_1^{-1}$,

$$A(S_1{S}_2{)}^{-1}\cong A{S}_1^{-1}{S}_2^{\prime -1}.$$

(7.6.9) Let $\mathfrak{a}\subset A$ be an open ideal. Then $S^{-1}\mathfrak{a}$ is open in $S^{-1}A$; its separated completion $\mathfrak{a}{S}^{-1}=\underleftarrow{\lim }(S^{-1}\mathfrak{a}/S^{-1}{\mathfrak{J}}_{\lambda })$ is an open ideal of $A{S}^{-1}$, and $A{S}^{-1}/\mathfrak{a}{S}^{-1}\cong S^{-1}A/S^{-1}\mathfrak{a}=S^{-1}(A/\mathfrak{a})$. Conversely, every open ideal of $A{S}^{-1}$ is of the form $\mathfrak{a}{S}^{-1}$ for a unique open $\mathfrak{a}\supset {\mathfrak{J}}_{\lambda }\subset A$.

Proposition (7.6.10). The map $\mathfrak{p}\mapsto \mathfrak{p}{S}^{-1}$ is an increasing bijection between open prime ideals of $A$ not meeting $S$ and open prime ideals of $A{S}^{-1}$; the residue field of $A{S}^{-1}/\mathfrak{p}{S}^{-1}$ is canonically the field of fractions of $A/\mathfrak{p}$.

Proposition (7.6.11).

(i) If $A$ is admissible, so is $A^{\prime }=A{S}^{-1}$, and ${\mathfrak{J}}^{\prime }=\mathfrak{J}{S}^{-1}$ is an ideal of definition for $A^{\prime }$ whenever $\mathfrak{J}$ is for $A$. (ii) If $A$ is $\mathfrak{J}$-adic with $\mathfrak{J}/{\mathfrak{J}}^2$ finite-type over $A/\mathfrak{J}$, then $A^{\prime }$ is ${\mathfrak{J}}^{\prime }$-adic with ${\mathfrak{J}}^{\prime }/{\mathfrak{J}}^{\prime 2}$ finite-type over $A^{\prime }/{\mathfrak{J}}^{\prime }$. If $A$ is Noetherian, so is $A^{\prime }$.

Corollary (7.6.12). Under (7.6.11)(ii), $(\mathfrak{J}{S}^{-1}{)}^n={\mathfrak{J}}^n{S}^{-1}$.

Proposition (7.6.13). Let $A$ be an adic Noetherian ring and $S\subset A$ multiplicative. Then $A{S}^{-1}$ is a flat $A$-module.

Proof. $A{S}^{-1}$ is the completion of the Noetherian $S^{-1}A$ for its $S^{-1}\mathfrak{J}$-preadic topology, hence flat over $S^{-1}A$ (7.3.3); transitivity (6.2.1) plus flatness of $S^{-1}A$ over $A$ (6.3.1) finishes the proof.

Corollary (7.6.14). Under (7.6.13), if $S^{\prime }\subset S$, then $A{S}^{-1}$ is flat over $A{S}^{\prime -1}$.

(7.6.15) For $f\in A$, write $A_f=A{S}_f^{-1}$ with $S_f=f^n:n\ge 0$, and ${\mathfrak{a}}_f=\mathfrak{a}{S}_f^{-1}$ for an open ideal $\mathfrak{a}$. For $g\in A$, there is a canonical $A_f\to A_{fg}$ (7.6.7). For $S$ multiplicative, set $A_S={\underrightarrow{\lim }}_{f\in S}{A}_f$; there is a canonical $A_S\to A{S}^{-1}$.

Proposition (7.6.16). If $A$ is Noetherian, $A{S}^{-1}$ is flat over $A_S$.

Proof. By (7.6.14), $A{S}^{-1}$ is flat over each $A_f$ ($f\in S$); conclude by (6.2.3).

Proposition (7.6.17). Let $\mathfrak{p}$ be an open prime ideal in an admissible ring $A$, and $S=A-\mathfrak{p}$. Then $A{S}^{-1}$ and $A_S$ are local rings, the canonical $A_S\to A{S}^{-1}$ is local, and both residue fields are canonically the field of fractions of $A/\mathfrak{p}$.

Corollary (7.6.18). If moreover $A$ is adic Noetherian, then $A{S}^{-1}$ and $A_S$ are Noetherian local rings, and $A{S}^{-1}$ is a faithfully flat $A_S$-module.

7.7. Completed tensor products

(7.7.1) Let $A$ be linearly topologized and M, N two linearly topologized $A$-modules. Let $\mathfrak{J}\subset A$, $V\subset M$, $W\subset N$ be open submodules with $\mathfrak{J}M\subset V$, $\mathfrak{J}N\subset W$. The $(M/V){\otimes }_{A/\mathfrak{J}}(N/W)$ form a projective system; their limit is an Â-module, the completed tensor product, written $(M{\otimes }_{A}N{)}^{\wedge }$. In terms of completions, $(M{\otimes }_{A}N{)}^{\wedge }\cong \hat{M}{\hat{\otimes }}_{\hat{A}}\hat{N}$.

(7.7.2) $(M{\otimes }_{A}N{)}^{\wedge }$ is the separated completion of $M{\otimes }_{A}N$ for the topology whose fundamental system of neighborhoods of 0 consists of $Im(V{\otimes }_{A}N)+Im(M{\otimes }_{A}W)$ (V, W open in M, N); this is the tensor product topology.

(7.7.3) For continuous $u:M\to M^{\prime }$ and $v:N\to N^{\prime }$, there is a continuous $u\hat{\otimes }v:(M{\otimes }_{A}N{)}^{\wedge }\to (M^{\prime }{\otimes }_A{N}^{\prime }{)}^{\wedge }$. Thus $(M{\otimes }_{A}N{)}^{\wedge }$ is bifunctorial in M, N.

(7.7.4) Similarly for any finite number of factors, with associativity and commutativity.

(7.7.5) For $A$-algebras B, C linearly topologized, $B{\otimes }_{A}C$ carries a tensor-product ring topology whose fundamental system of neighborhoods of 0 consists of ideals $Im(\mathfrak{K}{\otimes }_{A}C)+Im(B{\otimes }_A\mathfrak{L})$ ($\mathfrak{K},\mathfrak{L}$ open ideals of B, C). $(B{\otimes }_{A}C{)}^{\wedge }$ is a topological Â-algebra.

(7.7.6) Universal property. For every separated, complete $A$-algebra $D$ and every pair $(u,v)$ of continuous $A$-homomorphisms $u:B\to D$, $v:C\to D$, there is a unique continuous $A$-homomorphism $w:(B{\otimes }_{A}C{)}^{\wedge }\to D$ with $u=w\circ \rho$, $v=w\circ \sigma$ (where $\rho ,\sigma$ are the canonical maps from B, C).

Proposition (7.7.7). If B, C are preadmissible $A$-algebras, then $(B{\otimes }_{A}C{)}^{\wedge }$ is admissible; if $\mathfrak{K},\mathfrak{L}$ are ideals of definition of B, C, the closure in $(B{\otimes }_{A}C{)}^{\wedge }$ of the canonical image of $Im(\mathfrak{K}{\otimes }_{A}C)+Im(B{\otimes }_A\mathfrak{L})$ is an ideal of definition.

Proof. Use ${\mathfrak{h}}^{2n}\subset Im({\mathfrak{K}}^n{\otimes }_{A}C)+Im(B{\otimes }_A{\mathfrak{L}}^n)$.

Proposition (7.7.8). Let $A$ be preadic with ideal of definition $\mathfrak{J}$, $M$ an $A$-module of finite type with the $\mathfrak{J}$-preadic topology. For every adic Noetherian $A$-algebra $B$, $B{\otimes }_{A}M$ is identified with $(B{\otimes }_{A}M{)}^{\wedge }$.

7.8. Topologies on modules of homomorphisms

(7.8.1) Let $A$ be a Noetherian $\mathfrak{J}$-adic ring, M, N two $A$-modules of finite type with $\mathfrak{J}$-preadic topology. By (7.3.6), they are separated and complete; every $A$-homomorphism $M\to N$ is continuous, and ${\mathrm{Hom}}_A(M,N)$ is $A$-finite-type. With $A_i=A/{\mathfrak{J}}^{i+1}$, $M_i=M/{\mathfrak{J}}^{i+1}M$, $N_i=N/{\mathfrak{J}}^{i+1}N$, the ${\mathrm{Hom}}_{A_i}(M_i,N_i)$ form a projective system, and by (7.2.10) there is a canonical φ : Hom_A(M, N) ⥲ lim⃖_i Hom_{A_i}(M_i, N_i).

Proposition (7.8.2). Under the hypotheses of (7.8.1), the submodules ${\mathrm{Hom}}_A(M,{\mathfrak{J}}^{i+1}N)$ form a fundamental system of neighborhoods of 0 in ${\mathrm{Hom}}_A(M,N)$ for the $\mathfrak{J}$-adic topology, and $\varphi$ is a topological isomorphism.

Proof. Reduce to $M=L=A^m$; then ${\mathrm{Hom}}_A(L,N)=N^m$ and ${\mathrm{Hom}}_A(L,{\mathfrak{J}}^{i+1}N)={\mathfrak{J}}^{i+1}{N}^m$.

Proposition (7.8.3). Under the hypotheses of (7.8.2), the set of injective (resp. surjective, bijective) homomorphisms $M\to N$ is open in ${\mathrm{Hom}}_A(M,N)$.

Proof. For surjectivity: by (7.3.5) and (7.1.14), $u$ is surjective iff $u_0:M/\mathfrak{J}M\to N/\mathfrak{J}N$ is, so the set is the preimage of a discrete subset under the continuous ${\mathrm{Hom}}_A(M,N)\to {\mathrm{Hom}}_{A_0}(M_0,N_0)$. For injectivity: let $v:M\to N$ be injective and set $M^{\prime }=v(M)$. By Artin–Rees (7.3.2.1), there is $k\ge 0$ with $M^{\prime }\cap {\mathfrak{J}}^{m+k}N\subset {\mathfrak{J}}^m{M}^{\prime }$ for $m>0$. For $w\in {\mathfrak{J}}^{k+1}{\mathrm{Hom}}_A(M,N)$, $u=v+w$ is injective: if $u(x)=0$ and $x\in {\mathfrak{J}}^{i}M$, then $w(x)\in {\mathfrak{J}}^{i+k+1}N$, so $v(x)=-w(x)\in M^{\prime }\cap {\mathfrak{J}}^{i+k+1}N\subset {\mathfrak{J}}^{i+1}{M}^{\prime }$, hence $x\in {\mathfrak{J}}^{i+1}M$. By induction, $x\in \bigcap {\mathfrak{J}}^{i}M=0$.