15.102 Miscellany
Some results which do not fit anywhere else.
Lemma 15.102.1. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $K \in D(A)$ be pseudo-coherent. Let $a \in \mathbf{Z}$. Assume that for every finite $A$-module $M$ the modules $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M)$ are $I$-power torsion for $i \geq a$. Then for $i \geq a$ and $M$ finite the system $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M/I^ nM)$ is essentially constant with value
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M/I^ nM) \]
Proof.
Let $M$ be a finite $A$-module. Since $K$ is pseudo-coherent we see that $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M)$ is a finite $A$-module. Thus for $i \geq a$ it is annihilated by $I^ t$ for some $t \geq 0$. By Lemma 15.101.4 we see that the image of $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, I^ nM) \to \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M)$ is zero for some $n > 0$. The short exact sequence $0 \to I^ nM \to M \to M/I^ n M \to 0$ gives a long exact sequence
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, I^ nM) \to \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M) \to \mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M/I^ nM) \to \mathop{\mathrm{Ext}}\nolimits ^{i + 1}_ A(K, I^ nM) \]
The systems $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, I^ nM)$ and $\mathop{\mathrm{Ext}}\nolimits ^{i + 1}_ A(K, I^ nM)$ are essentially constant with value $0$ by what we just said (applied to the finite $A$-modules $I^ mM$). A diagram chase shows $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M/I^ nM)$ is essentially constant with value $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, M)$.
$\square$
Lemma 15.102.2. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $N$ be an $A$-module annihilated by $I$. There exists an integer $n > 0$ such that $\text{Tor}^ A_ p(I^ nM, N) \to \text{Tor}^ A_ p(M, N)$ is zero for all $p \geq 0$.
Proof.
By Lemma 15.101.7 we can factor $I^ nM \to M$ as $I^ nM \to M \otimes _ A^\mathbf {L} I \to M$. We claim the composition
\[ I^ nM \otimes _ A^\mathbf {L} N \to (M \otimes _ A^\mathbf {L} I) \otimes _ A^\mathbf {L} N \to M \otimes _ A^\mathbf {L} N \]
is zero. Namely, the diagram
\[ \xymatrix{ (M \otimes _ A^\mathbf {L} I) \otimes _ A^\mathbf {L} N \ar[rr] \ar[rd] & & M \otimes _ A^\mathbf {L} (I \otimes _ A^\mathbf {L} N) \ar[ld] \\ & M \otimes _ A^\mathbf {L} N } \]
commutes (details omitted) and the map $I \otimes _ A^\mathbf {L} N \to N$ is zero as $N$ is annihilated by $I$.
$\square$
Lemma 15.102.3. Let $R$ be a ring. Let $K \in D(R)$ be pseudo-coherent. Let $(M_ n)$ be an inverse system of $R$-modules. Then $R\mathop{\mathrm{lim}}\nolimits K \otimes _ R^\mathbf {L} M_ n = K \otimes _ R^\mathbf {L} R\mathop{\mathrm{lim}}\nolimits M_ n$.
Proof.
Consider the defining distinguished triangle
\[ R\mathop{\mathrm{lim}}\nolimits M_ n \to \prod M_ n \to \prod M_ n \to R\mathop{\mathrm{lim}}\nolimits M_ n[1] \]
and apply Lemma 15.65.5.
$\square$
Lemma 15.102.4. Let $R$ be a Noetherian local ring. Let $I \subset R$ be an ideal and let $E$ be a nonzero module over $R/I$. If $R/I$ has finite projective dimension and $E$ has finite projective dimension over $R/I$, then $E$ has finite projective dimension over $R$ and
\[ \text{pd}_ R(E) = \text{pd}_ R(R/I) + \text{pd}_{R/I}(E) \]
Proof.
We will use that, for a finite module, having finite projective dimension over $R$, resp. $R/I$ is the same as being a perfect module, see discussion following Definition 15.74.1. We see that $E$ has finite projective dimension over $R$ by Lemma 15.74.7. Thus we can apply Auslander-Buchsbaum (Algebra, Proposition 10.111.1) to see that
\[ \text{pd}_ R(E) + \text{depth}(E) = \text{depth}(R),\quad \text{pd}_{R/I}(E) + \text{depth}(E) = \text{depth}(R/I), \]
and
\[ \text{pd}_ R(R/I) + \text{depth}(R/I) = \text{depth}(R) \]
Note that in the first equation we take the depth of $E$ as an $R$-module and in the second as an $R/I$-module. However these depths are the same (this is trivial but also follows from Algebra, Lemma 10.72.11). This concludes the proof.
$\square$
Lemma 15.102.5. Let $A \to B$ be a ring map. There exists a cardinal $\kappa = \kappa (A \to B)$ with the following property: Let $M^\bullet $, resp. $N^\bullet $ be a complex of $A$-modules, resp. $B$-modules. Let $a : M^\bullet \to N^\bullet $ be a map of complexes of $A$-modules which induces an isomorphism $M^\bullet \otimes _ A^\mathbf {L} B \to N^\bullet $ in $D(B)$. Let $M_1^\bullet \subset M^\bullet $, resp. $N_1^\bullet \subset N^\bullet $ be a subcomplex of $A$-modules, resp. $B$-modules such that $a(M_1^\bullet ) \subset N_1^\bullet $. Then there exist subcomplexes
\[ M_1^\bullet \subset M_2^\bullet \subset M^\bullet \quad \text{and}\quad N_1^\bullet \subset N_2^\bullet \subset N^\bullet \]
such that $a(M_2^\bullet ) \subset N_2^\bullet $ with the following properties:
$\mathop{\mathrm{Ker}}(H^ i(M_1^\bullet \otimes _ A^\mathbf {L} B) \to H^ i(N_1^\bullet ))$ maps to zero in $H^ i(M_2^\bullet \otimes _ A^\mathbf {L} B)$,
$\mathop{\mathrm{Im}}(H^ i(N_1^\bullet ) \to H^ i(N_2^\bullet ))$ is contained in $\mathop{\mathrm{Im}}(H^ i(M_2^\bullet \otimes _ A^\mathbf {L} B) \to H^2(N_2^\bullet ))$,
$|\bigcup M_2^ i \cup \bigcup N_2^ i| \leq \max (\kappa , |\bigcup M_1^ i \cup \bigcup N_1^ i|)$.
Proof.
Let $\kappa = \max (|A|, |B|, \aleph _0)$. Set $|M^\bullet | = |\bigcup M^ i|$ and similarly for other complexes. With this notation we have
\[ \max (\kappa , |\bigcup M_1^ i \cup \bigcup N_1^ i|) = \max (\kappa , |M_1^\bullet |, |M_2^\bullet |) \]
for the quantity used in the statement of the lemma. We are going to use this and other observations coming from arithmetic of cardinals without further mention.
First, let us show that there are plenty of “small” subcomplexes. For every pair of collections $E = \{ E^ i\} $ and $F = \{ F^ i\} $ of finite subsets $E^ i \subset M^ i$, $i \in \mathbf{Z}$ and $F^ i \subset N^ i$, $i \in \mathbf{Z}$ we can let
\[ M_1^\bullet \subset M_1(E, F)^\bullet \subset M^\bullet \quad \text{and}\quad N_1^\bullet \subset N_1(E, F)^\bullet \subset N^\bullet \]
be the smallest subcomplexes of $A$ and $B$-modules such that $a(M_1(E, F)^\bullet ) \subset N_1(E, F)^\bullet $ and such that $E^ i \subset M_1(E, F)^ i$ and $F^ i \subset M_2(E, F)^ i$. Then it is easy to see that
\[ |M_1(E, F)^\bullet | \leq \max (\kappa , |M_1^\bullet |) \quad \text{and}\quad |M_2(E, F)^\bullet | \leq \max (\kappa , |M_2^\bullet |) \]
Details omitted. It is clear that we have
\[ M^\bullet = \mathop{\mathrm{colim}}\nolimits _{(E, F)} M_1(E, F)^\bullet \quad \text{and}\quad N^\bullet = \mathop{\mathrm{colim}}\nolimits _{(E, F)} N_1(E, F)^\bullet \]
and the colimits are (termwise) filtered colimits.
There exists a resolution $\ldots \to F^{-1} \to F^0 \to B$ by free $A$-modules $F_ i$ with $|F_ i| \leq \kappa $ (details omitted). The cohomology modules of $M_1^\bullet \otimes _ A^\mathbf {L} B$ are computed by $\text{Tot}(M_1^\bullet \otimes _ A F^\bullet )$. It follows that $|H^ i(M_1^\bullet \otimes _ A^\mathbf {L} B)| \leq \max (\kappa , |M_1^\bullet |)$.
Let $i \in \mathbf{Z}$ and let $\xi \in H^ i(M_1^\bullet \otimes _ A^\mathbf {L} B)$ be an element which maps to zero in $H^ i(N_1^\bullet )$. Then $\xi $ maps to zero in $H^ i(N^\bullet )$ and hence $\xi $ maps to zero in $H^ i(M^\bullet \otimes _ A^\mathbf {L} B)$. Since derived tensor product commutes with filtered colimits, we can find finite collections $E_\xi $ and $F_\xi $ as above such that $\xi $ maps to zero in $H^ i(M_1(E_\xi , F_\xi )^\bullet \otimes _ A^\mathbf {L} B)$.
Let $i \in \mathbf{Z}$ and let $\eta \in H^ i(N_1^\bullet )$. Then the image of $\eta $ in $H^ i(N^\bullet )$ is in the image of $H^ i(M^\bullet \otimes _ A^\mathbf {L} B) \to H^ i(N^\bullet )$. Hence as before, we can find finite collections $E_\eta $ and $F_\eta $ as above such that $\eta $ maps to an element of $H^ i(N_1(E_\eta , F_\eta )$ which is in the image of the map $H^ i(M_1(E_\eta , F_\eta )^\bullet \otimes _ A^\mathbf {L} B) \to H^ i(N_1(E_\eta , F_\eta )$.
Now we simply define
\[ M_2^\bullet = \sum \nolimits _\xi M_1(E_\xi , F_\xi )^\bullet + \sum \nolimits _\eta M_1(E_\eta , F_\eta )^\bullet \]
where the sum is over $\xi $ and $\eta $ as in the previous two paragraphs and the sum is taken inside $M^\bullet $. Similarly we set
\[ N_2^\bullet = \sum \nolimits _\xi N_1(E_\xi , F_\xi )^\bullet + \sum \nolimits _\eta N_1(E_\eta , F_\eta )^\bullet \]
where the sum is taken inside $N^\bullet $. By construction we will have properties (1) and (2) with these choices. The bound (3) also follows as the set of $\xi $ and $\eta $ has cardinality at most $\max (\kappa , |M_1^\bullet |, |N_1^\bullet |)$.
$\square$
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