The small Cohen-Macaulay modules conjecture

The question

The small Cohen-Macaulay modules conjecture. If $R$ is a complete Noetherian local ring, then there is a finitely generated $R$-module $M\ne 0$ such that some system of parameters of $R$ is a regular sequence on $M$.

Hochster stated the conjecture for complete local domains [Ho17, Conjecture 2.1]. The two forms are equivalent: for a minimal prime $P$ of $R$ with $\dim R/P=\dim R$, a small Cohen-Macaulay module over the complete local domain $R/P$ is one over $R$. In the 2000s Hochster conjectured the opposite, that some complete local domain has no small Cohen-Macaulay module [Ho17, Conjecture 2.2], so this statement may well be false.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of Krull dimension $d$. A system of parameters of $R$ is a sequence $x_1,\dots ,x_d$ of $d$ elements of $R$ with $\sqrt{(x_1,\dots ,x_d)}=\mathfrak{m}$, equivalently with $R/(x_1,\dots ,x_d)$ Artinian.

A small Cohen-Macaulay module, also called a maximal Cohen-Macaulay module, is a finitely generated $R$-module $M\ne 0$ such that some system of parameters of $R$ is a regular sequence on $M$. A balanced big Cohen-Macaulay module is a module $W$, not necessarily finitely generated, with $\mathfrak{m}W\ne W$ and on which every system of parameters is a regular sequence. For finitely generated modules the two notions agree.

Small Cohen-Macaulay modules conjecture. If $R$ is complete, then $R$ has a small Cohen-Macaulay module.

Hochster conjectured this for complete local domains in the early 1970s. In the 2000s he conjectured the opposite, that there are complete local domains with no small Cohen-Macaulay module [Ho17]. Small Cohen-Macaulay modules are known to exist when $\dim R\le 2$, and when $R$ is $\mathbb{N}$-graded over a perfect field of characteristic $p$ with an isolated non-Cohen-Macaulay point at the origin, a case first observed by Hartshorne and rediscovered by Peskine and Szpiro [Ho75b]. In dimension at least three the conjecture is open in every characteristic, although classes of examples keep being found there, such as three-dimensional $F$-pure complete local ${\mathbb{F}}_p$-algebras, due to Schoutens, and three-dimensional completions of $\mathbb{N}$-graded rings over a field of characteristic $p$, due to Hochster; both are listed, with references, in [ST23]. The analogous statement for algebras is false: Bhatt constructed complete local normal domains of characteristic $p$ admitting no module-finite extension ring that is Cohen-Macaulay [Bh14].

Balanced big Cohen-Macaulay modules, by contrast, are now known to exist over every Noetherian local ring, by Hochster in equal characteristic and by André in mixed characteristic.

Systems of parameters and Cohen-Macaulay modules are defined in FormalConjecturesForMathlib.RingTheory.SystemOfParameters and FormalConjecturesForMathlib.RingTheory.CohenMacaulayModule.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.SmallCohenMacaulayModules.

theorem exists_isSmallCohenMacaulay [IsAdicComplete (maximalIdeal R) R] :
    ∃ (M : Type u) (_ : AddCommGroup M) (_ : Module R M), IsSmallCohenMacaulay R M

What counts as progress

• A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• [Wikipedia, Homological conjectures in commutative algebra](https://en.wikipedia.org/wiki/Homological_conjectures_in_commutative_algebra), conjectures 8 and 14.
• [Ho75a] M. Hochster, Topics in the homological theory of modules over commutative rings, C.B.M.S. Regional Conf. Ser. in Math. 24, Amer. Math. Soc., 1975.
• [Ho75b] M. Hochster, *Big Cohen-Macaulay modules and algebras and embeddability in rings of Witt vectors*, Queen's Papers in Pure and Applied Math. 42, 1975, 106-195.
• [Ho17] M. Hochster, Homological conjectures and lim Cohen-Macaulay sequences, in Homological and Computational Methods in Commutative Algebra, Springer INdAM Ser. 20, Springer, 2017. PDF, Conjectures 2.1 and 2.2.
• [An18] Y. André, La conjecture du facteur direct, Publ. Math. IHÉS 127 (2018), 71-93. arXiv:1609.00345. This is the paper [Ho17] credits with the existence of big Cohen-Macaulay algebras; it rests on the perfectoid Abhyankar lemma of Le lemme d'Abhyankar perfectoïde, Publ. Math. IHÉS 127 (2018), 1-70.
• Stacks, Tag 00N6, on regular sequences in a Cohen-Macaulay module.
• [ST23] K. Shimomoto, E. Tavanfar, *Remarks on the Small Cohen-Macaulay conjecture and new instances of maximal Cohen-Macaulay modules*, J. Algebra 634 (2023), 667-697. arXiv:2203.10368
• [Bh14] B. Bhatt, On the non-existence of small Cohen-Macaulay algebras, J. Algebra 411 (2014), 1-11. arXiv:1207.5413

Source and licence

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