The Stacks project

Remark 51.21.9. Let $I$ be an ideal of a Noetherian ring $A$. Say $I = (f_1, \ldots , f_ r)$. Denote $K_ n^\bullet $ the Koszul complex on $f_1^ n, \ldots , f_ r^ n$ as in More on Algebra, Situation 15.91.15 and denote $K_ n \in D(A)$ the corresponding object. Let $M^\bullet $ be a bounded complex of finite $A$-modules and denote $M \in D(A)$ the corresponding object. Consider the following inverse systems in $D(A)$:

  1. $M^\bullet /I^ nM^\bullet $, i.e., the complex whose terms are $M^ i/I^ nM^ i$,

  2. $M \otimes _ A^\mathbf {L} A/I^ n$,

  3. $M \otimes _ A^\mathbf {L} K_ n$, and

  4. $M \otimes _ P^\mathbf {L} P/J^ n$ (see below).

All of these inverse systems are isomorphic as pro-objects: the isomorphism between (2) and (3) follows from More on Algebra, Lemma 15.94.1. The isomorphism between (1) and (2) is given in More on Algebra, Lemma 15.100.3. For the last one, see below.

However, we can ask if these isomorphisms of pro-systems are “strict”; this terminology and question is related to the discussion in [pages 61, 62, quillenhomology]. Namely, given a category $\mathcal{C}$ we can define a “strict pro-category” whose objects are inverse systems $(X_ n)$ and whose morphisms $(X_ n) \to (Y_ n)$ are given by tuples $(c, \varphi _ n)$ consisting of a $c \geq 0$ and morphisms $\varphi _ n : X_ n \to Y_{n - c}$ for all $n \geq c$ satisfying an obvious compatibility condition and up to a certain equivalence (given essentially by increasing $c$). Then we ask whether the above inverse systems are isomorphic in this strict pro-category.

This clearly cannot be the case for (1) and (3) even when $M = A[0]$. Namely, the system $H^0(K_ n) = A/(f_1^ n, \ldots , f_ r^ n)$ is not strictly pro-isomorphic in the category of modules to the system $A/I^ n$ in general. For example, if we take $A = \mathbf{Z}[x_1, \ldots , x_ r]$ and $f_ i = x_ i$, then $H^0(K_ n)$ is not annihilated by $I^{r(n - 1)}$.1

It turns out that the results above show that the natural map from (2) to (1) discussed in More on Algebra, Lemma 15.100.3 is a strict pro-isomorphism. We will sketch the proof. Using standard arguments involving stupid truncations, we first reduce to the case where $M^\bullet $ is given by a single finite $A$-module $M$ placed in degree $0$. Pick $N, c \geq 0$ as in Proposition 51.21.7. The proposition implies that for $n \geq N$ we get factorizations

\[ M \otimes _ A^\mathbf {L} A/I^ n \to \tau _{\geq -t}(M \otimes _ A^\mathbf {L} A/I^ n) \to M \otimes _ A^\mathbf {L} A/I^{n - c} \]

of the transition maps in the system (2). On the other hand, by More on Algebra, Lemma 15.27.3, we can find another constant $c' = c'(M) \geq 0$ such that the maps $\text{Tor}_ i^ A(M, A/I^{n'}) \to \text{Tor}_ i(M, A/I^{n' - c'})$ are zero for $i = 1, 2, \ldots , t$ and $n' \geq c'$. Then it follows from Derived Categories, Lemma 13.12.5 that the map

\[ \tau _{\geq -t}(M \otimes _ A^\mathbf {L} A/I^{n + tc'}) \to \tau _{\geq -t}(M \otimes _ A^\mathbf {L} A/I^ n) \]

factors through $M \otimes _ A^\mathbf {L}A/I^{n + tc'} \to M/I^{n + tc'}M$. Combined with the previous result we obtain a factorization

\[ M \otimes _ A^\mathbf {L}A/I^{n + tc'} \to M/I^{n + tc'}M \to M \otimes _ A^\mathbf {L} A/I^{n - c} \]

which gives us what we want. If we ever need this result, we will carefully state it and provide a detailed proof.

For number (4) suppose we have a Noetherian ring $P$, a ring homomorphism $P \to A$, and an ideal $J \subset P$ such that $I = JA$. By More on Algebra, Section 15.60 we get a functor $M \otimes _ P^\mathbf {L} - : D(P) \to D(A)$ and we get an inverse system $M \otimes _ P^\mathbf {L} P/J^ n$ in $D(A)$ as in (4). If $P$ is Noetherian, then the system in (4) is pro-isomorphic to the system in (1) because we can compare with Koszul complexes. If $P \to A$ is finite, then the system (4) is strictly pro-isomorphic to the system (2) because the inverse system $A \otimes _ P^\mathbf {L} P/J^ n$ is strictly pro-isomorphic to the inverse system $A/I^ n$ (by the discussion above) and because we have

\[ M \otimes _ P^\mathbf {L} P/J^ n = M \otimes _ A^\mathbf {L} (A \otimes _ P^\mathbf {L} P/J^ n) \]

by More on Algebra, Lemma 15.60.1.

A standard example in (4) is to take $P = \mathbf{Z}[x_1, \ldots , x_ r]$, the map $P \to A$ sending $x_ i$ to $f_ i$, and $J = (x_1, \ldots , x_ r)$. In this case one shows that

\[ M \otimes _ P^\mathbf {L} P/J^ n = M \otimes _{A[x_1, \ldots , x_ r]}^\mathbf {L} A[x_1, \ldots , x_ r]/(x_1, \ldots , x_ r)^ n \]

and we reduce to one of the cases discussed above (although this case is strictly easier as $A[x_1, \ldots , x_ r]/(x_1, \ldots , x_ r)^ n$ has tor dimension at most $r$ for all $n$ and hence the step using Proposition 51.21.7 can be avoided). This case is discussed in the proof of [Proposition 3.5.1, BS].

[1] Of course, we can ask whether these pro-systems are isomorphic in a category whose objects are inverse systems and where maps are given by tuples $(r, c, \varphi _ n)$ consisting of $r \geq 1$, $c \geq 0$ and maps $\varphi _ n : X_{rn} \to Y_{n - c}$ for $n \geq c$.

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