Let be a Gorenstein local ring, and let be a finitely generated -module. It is a standard fact that is Cohen-Macaulay iff is zero for all . Moreover, the functor induces an involutive anti-equivalence on the category of Cohen-Macaulay -modules of codimension .
Suppose now that we have a flat local map of local Noetherian rings with Gorenstein, and that is a CM -module which is flat over . It is natural to ask whether is also flat over . The goal of this post is to prove the following partial result towards this question.
Theorem. Notation and setup as above, is flat over if is regular or if .
Before continuing, note that and are automatically Gorenstein. This will be useful later.
The first key lemma is the following.
Lemma 1. Let be a diagram of Noetherian commutative rings, and let be a finitely generated -module flat over . Suppose that the maps are tor-independent (e.g. one of them is flat), and that at least one of the following conditions holds:
2. is of finite tor-dimension.
Then there is a convergent spectral sequence
where and .
This is not so hard to prove. The only real input is Stacks Tag 0A6A – the auxiliary conditions 1. and 2. above corresponds to cases (1) and (4) of that lemma.
Granted this lemma, the auxiliary hypotheses in the theorem let us apply this spectral sequence with chosen as in the theorem and with . Since is Cohen-Macaulay over of some codimension , only contributes, and the spec. seq. degenerates to isomorphisms
Here and in what follows, I write for pullback to the closed fiber of . By the local criterion of flatness, the theorem follows if we can show that , i.e. that . Since is Gorenstein, it clearly suffices to prove that is a Cohen-Macaulay -module of codimension . This is the content of the next lemma, which must be well-known, but which I couldn’t find in ten minutes of googling. Do you know a reference for this?
Lemma 2. If is a flat local map of Gorenstein local rings, and is a finitely generated -module which is flat over , then Cohen-Macaulay over implies Cohen-Macaulay over of the same codimension. Here as above.
To prove this, first note that since and are CM. Then where the first equality holds e.g. by Theorem 23.3 in Matsumura’s Commutative Ring Theory. (One can also read off the equality from the spectral sequence argument above.)
So now we just need to show that . But we can check this last equality with replaced by where is any Artinian quotient. In particular we can assume that where is an -primary ideal generated by a regular sequence. Then
where the first equality holds by definition, the second and fourth equalities are a well-known property of f.g. modules over CM local rings, and the third equality follows from the facts that is Gorenstein and that (thanks to our flatness assumptions) is generated by an -regular sequence which is also a -regular sequence. (To actually prove the third equality, use induction on the number of generators of I.) Thus as desired.
Q1. Does the theorem hold more generally?
Q2. Is there a reference for Lemma 2 in the literature? This can’t be a new observation.
Q3. Can the Gorenstein condition in Lemma 2 be weakened? This seems unlikely to me.