Let be any map of irreducible complex varieties, and let be a perverse sheaf on . Prove that there is a dense open subset such that for any closed point , the shifted restriction is a perverse sheaf on the fiber .

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]]>*The Lithuanian VP, not the French VP.

]]>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:

1. ;

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.

The point of this short post is to note that under a mild assumption, it’s enough to look at the closed fiber only:

**Lemma. **Let be a flat local map of Noetherian local rings, and assume that is *quasi-excellent. *Then the following are equivalent:

- All fiber rings of are geometrically regular, i.e. is a regular ring map.
- is geometrically regular over .
- is formally smooth in the -adic topology.
- is formally smooth in the -adic topology.

*Proof. *1. 2. is trivial. The equivalences 2. 3. 4. are proved in Stacks, Tag 07NQ. The implication 4. 1. is a theorem of M. Andre (*Localisation de la lissite formelle, *Manuscripta Math. 13, pp. 297-307); this is the only place we use the quasi-excellence of .

**Corollary. **Let be a separable extension of fields. Then is regular.

*Proof. *Flatness is trivial (e.g. by the local criterion), and condition 2. in the lemma above is trivial.

Is there a truly easy proof of this corollary? Even if you give yourself access to some big hammers like Popescu’s theorem, I don’t see a simple argument.

**Corollary. **Let be any extension of characteristic zero nonarchimedean fields, and let be any tft -algebra. Then is regular.

*Proof.* It suffices to check that is regular, where is any maximal ideals upstairs which contracts to a maximal ideal downstairs. But now flatness is easy, and condition 2. in the lemma is easy to verify by some standard structural properties of tft algebras.

Andre’s theorem is stated on p. 260 of Matsumura’s *Commutative Ring Theory*, where Matsumura calls it “an extremely strong theorem.” I remember reading this sentence in grad school and completely failing to understand the point (or maybe even the statement…) of Andre’s result. But now I get it.

**Update (April 14).** Here’s another corollary which seems useful.

**Corollary. **Let be a regular map of Noetherian rings, with quasi-excellent. Then

- ( is quasi-excellent and) is a regular ring map.
- For any ideal , (the -adic completion is quasi-excellent and) the induced map on -adic completions is a regular ring map.

*Proof. *By a theorem of Gabber, if is quasi-excellent, then so are and , cf. this nice paper for a detailed discussion. Part 2. is now an easy consequence of 1., by writing as the base change of along a suitable surjection .

For 1., we reduce by an easy induction to the case . Let be any prime ideal, with contractions , , . Then we get a commutative square

of local Noetherian rings where all arrows are flat ring maps. Moreover, the vertical arrows are localizations, and the horizontal arrows are flat local maps. We need to check that the upper horizontal arrow is regular for any . Since the upper row comes from the lower row by localization, it’s enough to check that the lower horizontal arrow is regular. But now we win, because is a flat local map with quasi-excellent source, and the closed fiber of this map is *the same as *the closed fiber of the (local) map . Since is regular by assumption, is geometrically regular over , and now we’re done by applying condition 2. in the Lemma.

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