On homological dimensions of finitely generated modules over a local ring and its completion

Question

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $\hat R$ be the $\mathfrak m$-adic completion of $R$ so we have a canonical map $R \to \hat R$, which makes $\hat R$ into a $R$-algebra, so every $\hat R$-module has a canonical $R$-module structure. My questions are the following:

(1) Let $M$ be a finitely generated $\hat R$-module such that inj$\dim_{\hat R} M <\infty$ , then is it true that inj$\dim_{R} M <\infty$ ?

(2) Let $M$ be a finitely generated $\hat R$-module such that proj$\dim_{ R} M <\infty$ , then is it true that proj$\dim_{\hat R} M <\infty$ ?

If need be, I'm willing to assume $R$ is Cohen-Macaulay and $M$ is a maximal Cohen-Macaulay $\hat R$-module .

Answer 1

As stated in the comments, (2) is true irrespective of an assumption of finite generation.

I cannot find a reference anywhere for (1) in total generality, but as Ben said in the comments, it holds over Gorenstein rings. This is the key step in my proof:

Theorem (4.3.1 Weibel). Let $f:R\rightarrow S$ be a ring homomorphism. If $M$ is an $S$-module, then as an $R$-module $$\text{pd}_{R}(M)\leq \text{pd}_{S}(M)+\text{pd}_{R}(S).$$

Returning to your question, we have the following:

Corollary. (1) holds if $R$ is Gorenstein, without the assumption of finite generation.

Proof. $R$ is Gorenstein $\iff \hat{R}$ is Gorenstein, so if $M$ is an $\hat{R}$-module such that $\text{id}_{\hat{R}}(M)<\infty$, then $\text{pd}_{\hat{R}}(M)<\infty$. We therefore have $\text{pd}_{R}(M)<\infty$ by the theorem, since $R$ is Gorenstein and $\hat{R}$ is flat over $R$. Using $R$ being Gorenstein again, we have $\text{id}_{R}(M)<\infty$.

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For certain modules (1) also holds over non-Gorenstein local rings:

Lemma. Let $(R,\mathfrak{m},k)$ be a noetherian local ring and $\hat{M}$ a finitely generated $\hat{R}$-module that is the completion of a finitely generated $R$-module, $M$. Then $\text{id}_{R}(M)<\infty \iff \text{id}_{\hat{R}}(\hat{M})<\infty.$

Proof. If $A$ is any $R$-module, then for every $j\geq 0$ we have $$\text{Ext}_{R}^{j}(k,A)=0 \iff \text{Ext}_{\hat{R}}^{j}(k,\hat{R}\otimes A)=0,$$ from the fact that $\hat{R}$ is faithfully flat over $R$ and $k\otimes \hat{R}\simeq \hat{R}/\mathfrak{m}\hat{R}\simeq k$. Now, if $M$ is a finitely generated $R$-module we have \begin{align} \text{id}_{R}(M)<\infty&\iff \text{Ext}_{R}^{j}(k,M)=0 \text{ for all }j>\text{depth}\,R \\ &\iff \text{Ext}_{\hat{R}}^{j}(k,\hat{M})=0 \text{ for all }j>\text{depth}\,R=\text{depth}\,\hat{R} \\ &\iff \text{id}_{\hat{R}}(\hat{M})<\infty. \end{align}

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Here are some related results that relate to injective dimension and injective modules under flat base change.

This first result is due to H.-B. Foxby and can be found in Injective modules under flat base change:

Theorem. Let $f:R\rightarrow S$ be a morphism of commutative rings such that $S$ is a flat $R$-module. Then if $E$ is an injective $R$-module, one has $$\text{id}_{S}\,S\otimes_{R}E=\sup_{\mathfrak{p}\in\text{Ass}\,E}\text{id}_{F(\mathfrak{p})}\,F(\mathfrak{p})$$

Here $F(\mathfrak{p})=k(\mathfrak{p})\otimes_{R}S$ is the fibre of $f$ at $\mathfrak{p}$.

In particular, if one of the fibres is not Gorenstein, then $S\otimes -$ doesn't preserve finite injective dimension. Foxby also gives the example of localisation:

Example. Let $R$ be the homomorphic image of a Gorenstein ring. Then the fibres of $R\rightarrow\hat{R}$ are trivial, and $\text{dim}\,F(\mathfrak{p})\leq\max\{0,\text{dim}\,A/\mathfrak{p}-1\}.$

The local case is just CM rings with canonical module. This next result is due to Foxby and A. Thorup, found in Minimal injective resolutions under flat base change. This is a corollary to the main result, but I'll call it a theorem here.

Theorem Let $(R,\mathfrak{m})$ and $S$ be local rings and $f:R\rightarrow S$ a flat ring morphism of local rings. If $M$ is a finitely generated nonzero $R$-module, then $$\text{id}_{S}(M\otimes_{R}S)= \text{id}_{R}(M)+\text{id}_{C}(C),$$ where $C=S/\mathfrak{m}S$. In particular, the left hand side is finite if and only if $\text{id}_{R}(M)<\infty$ and $C$ is Gorenstein.

Another paper with relevant results in is Injective Modules under Faithfully Flat Ring Extensions by L.W. Christensen and F. Koksal.

I'm sure there are many more that I have not mentioned or discovered that have equally interesting things to say about injective modules and dimension in this situation. Feel free to make edits to include anything further.