Lim Cohen–Macaulay Sequences

• Lim Cohen–Macaulay sequences are asymptotic analogues of maximal Cohen–Macaulay modules, defined by the vanishing of higher Koszul homology relative to module size.
• They underpin canonical closure operations with colon-capturing properties and provide a pathway to construct big Cohen–Macaulay modules.
• These sequences arise via Frobenius iterations and perfectoid towers in both prime and mixed characteristic settings, impacting intersection theory and multiplicity problems.

Lim Cohen–Macaulay sequences are asymptotic analogues of maximal Cohen–Macaulay modules for a Noetherian local ring $(R,\mathfrak m,K)$ of dimension $d$. Instead of requiring a single finitely generated $R$-module to have vanishing higher Koszul homology on a system of parameters, one studies sequences of finitely generated full-dimensional modules whose higher Koszul homology becomes negligible compared with the size of the module. In this form, the notion functions as an asymptotic substitute for small Cohen–Macaulay modules and is strong enough to produce closure operations with colon-capturing, big Cohen–Macaulay modules, and consequences for Serre intersection multiplicities; subsequent work places the theory in the mixed-characteristic framework of $\delta$-rings and perfectoid towers (Bhatt et al., 2024).

1. Definition and asymptotic viewpoint

Let $(R,\mathfrak m,K)$ be a Noetherian local ring of dimension $d$, and let $\mathbf x=x_1,\dots,x_d$ be a system of parameters. For a finitely generated $R$-module $M$, write

$$h_i(\mathbf x;M):=\ell(H_i(\mathbf x;M)), \qquad \sigma_i(\mathbf x;M):=\sum_{j=i}^d h_j(\mathbf x;M),$$

where $d$0 is the $d$1-th Koszul homology of $d$2 on $d$3, and let $d$4 denote the least number of generators of $d$5. A sequence $d$6 of finitely generated $d$7-modules with $d$8 is called lim Cohen–Macaulay if there exists a system of parameters $d$9 such that

$R$0

Equivalently,

$R$1

A sequence of module-finite $R$2-algebras satisfying the same asymptotic condition is called a lim Cohen–Macaulay sequence of $R$3-algebras (Bhatt et al., 2024).

The heuristic is direct: a finitely generated module is maximal Cohen–Macaulay precisely when the higher Koszul homology on a system of parameters vanishes, whereas a lim Cohen–Macaulay sequence requires only asymptotic vanishing after normalization by module size. In this sense, the condition replaces exact Cohen–Macaulayness by asymptotic Cohen–Macaulayness. The definition does not require the sequence to form a direct or inverse system, although the principal examples arise from towers of finite algebras or Frobenius iterates (Ishiro et al., 8 Sep 2025).

A weaker notion is weakly lim Cohen–Macaulay, formulated in terms of the first higher Euler characteristic

$R$4

In later refinements, lim Cohen–Macaulayness is the stronger condition controlling each $R$5, while weakly lim Cohen–Macaulayness controls only the total higher-homology correction term (Bhatt et al., 2024).

2. Equivalent formulations and basic properties

A first structural fact is that the lim Cohen–Macaulay condition is independent of the chosen system of parameters. More precisely, if $R$6 and $R$7 are two systems of parameters, then the quantities $R$8 and $R$9 are uniformly comparable, so asymptotic negligibility for one system is equivalent to asymptotic negligibility for any other (Bhatt et al., 2024).

Several useful reformulations follow. A sequence $\delta$0 is lim Cohen–Macaulay if and only if, for some or equivalently every system of parameters $\delta$1,

$\delta$2

where $\delta$3. The weakly lim Cohen–Macaulay condition is equivalently

$\delta$4

These identities make precise that the higher Koszul homology contributes asymptotically nothing to the normalized Hilbert–Samuel count (Bhatt et al., 2024).

Constant sequences recover the ordinary theory. For a fixed finitely generated module $\delta$5, the constant sequence

$\delta$6

is lim Cohen–Macaulay if and only if $\delta$7 is maximal Cohen–Macaulay. Thus the asymptotic notion genuinely extends the classical one rather than replacing it by an unrelated condition (Bhatt et al., 2024).

When rank is defined, normalization by $\delta$8 can often be replaced by normalization by $\delta$9. In particular, if all $(R,\mathfrak m,K)$0 have well-defined rank, then

$(R,\mathfrak m,K)$1

for all $(R,\mathfrak m,K)$2. Over a regular local ring there is an especially concrete criterion: $(R,\mathfrak m,K)$3 where $(R,\mathfrak m,K)$4 is the $(R,\mathfrak m,K)$5-th Betti number (Bhatt et al., 2024).

Reduction by a regular parameter is also exact at the level of lim Cohen–Macaulayness. If $(R,\mathfrak m,K)$6 is a parameter element of $(R,\mathfrak m,K)$7 and is $(R,\mathfrak m,K)$8-regular for all $(R,\mathfrak m,K)$9, then

$d$0

This reduction is central in mixed-characteristic constructions, where one passes to characteristic $d$1 by modding out a nonzerodivisor such as $d$2 (Ishiro et al., 8 Sep 2025).

3. Existence theorems and constructions

The strongest general existence theorem is in prime characteristic. If $d$3 is an $d$4-finite local ring of characteristic $d$5, $d$6 is any $d$7-module of Krull dimension $d$8, and $d$9 denotes the $\mathbf x=x_1,\dots,x_d$0-fold Frobenius pullback, then

$\mathbf x=x_1,\dots,x_d$1

is a lim Cohen–Macaulay sequence. In particular, if $\mathbf x=x_1,\dots,x_d$2 is reduced, then

$\mathbf x=x_1,\dots,x_d$3

is a lim Cohen–Macaulay sequence. The proof compares the growth of higher Koszul homology under Frobenius with the asymptotic growth of $\mathbf x=x_1,\dots,x_d$4 (Bhatt et al., 2024).

Mixed characteristic is subtler. One class of examples arises from $\mathbf x=x_1,\dots,x_d$5-adically complete, $\mathbf x=x_1,\dots,x_d$6-torsion-free local rings with perfect residue field that admit an endomorphism lifting Frobenius modulo $\mathbf x=x_1,\dots,x_d$7. If $\mathbf x=x_1,\dots,x_d$8 is such a lift, then the sequence

$\mathbf x=x_1,\dots,x_d$9

is lim Cohen–Macaulay; the argument reduces modulo $R$0 and appeals to the characteristic-$R$1 theorem (Bhatt et al., 2024).

A more systematic mixed-characteristic mechanism is provided by $R$2-rings and perfectoid towers. Starting from a $R$3-torsion-free complete local ring with a $R$4-structure and reduced mod-$R$5 reduction, one can iterate the associated Frobenius lift and obtain a tower of finite algebras whose terms form a lim Cohen–Macaulay sequence. More generally, if $R$6 is a complete Noetherian local domain of mixed characteristic with perfect residue field and there exists a perfectoid tower

$R$7

arising from $R$8, then $R$9 is a lim Cohen–Macaulay sequence of algebras (Ishiro et al., 8 Sep 2025).

The perfectoid-tower proof is tilt-theoretic. The tilt of a perfectoid tower is a perfect tower in characteristic $M$0, hence a source of lim Cohen–Macaulay sequences by the Frobenius theorem. One then transfers the asymptotic homological condition back to mixed characteristic using reduction modulo a regular generator of the perfectoid ideal (Ishiro et al., 8 Sep 2025).

These examples show that lim Cohen–Macaulay sequences are strictly weaker than the existence of small Cohen–Macaulay algebras. In mixed characteristic there are section-ring examples admitting lim Cohen–Macaulay sequences but no small Cohen–Macaulay algebra (Bhatt et al., 2024).

4. Closure operations and big Cohen–Macaulay modules

One of the central developments is that a lim Cohen–Macaulay sequence canonically defines a closure operation on submodules of finitely generated modules. Fix a sequence $M$1 and a size function $M$2, such as $M$3 or $M$4. If $M$5 and $M$6 has finite length, then $M$7 is defined as the largest submodule $M$8 such that

$M$9

For arbitrary $$h_i(\mathbf x;M):=\ell(H_i(\mathbf x;M)), \qquad \sigma_i(\mathbf x;M):=\sum_{j=i}^d h_j(\mathbf x;M),$$0, one sets

$$h_i(\mathbf x;M):=\ell(H_i(\mathbf x;M)), \qquad \sigma_i(\mathbf x;M):=\sum_{j=i}^d h_j(\mathbf x;M),$$1

This closure satisfies the expected formal properties of extension, idempotence, order-preservation, and functoriality (Bhatt et al., 2024).

For lim Cohen–Macaulay sequences the associated closure operations satisfy colon-capturing. If $$h_i(\mathbf x;M):=\ell(H_i(\mathbf x;M)), \qquad \sigma_i(\mathbf x;M):=\sum_{j=i}^d h_j(\mathbf x;M),$$2 is part of a system of parameters, then

$$h_i(\mathbf x;M):=\ell(H_i(\mathbf x;M)), \qquad \sigma_i(\mathbf x;M):=\sum_{j=i}^d h_j(\mathbf x;M),$$3

and there is also a higher-power version

$$h_i(\mathbf x;M):=\ell(H_i(\mathbf x;M)), \qquad \sigma_i(\mathbf x;M):=\sum_{j=i}^d h_j(\mathbf x;M),$$4

These are the exact asymptotic analogues of the colon-capturing phenomena central to tight closure theory (Bhatt et al., 2024).

The closure operation attached to a lim Cohen–Macaulay sequence is strong enough to satisfy Dietz’s axioms. Consequently, if a complete local ring admits a lim Cohen–Macaulay sequence, then it admits a big Cohen–Macaulay module. This yields a new route to big Cohen–Macaulay modules from asymptotic finite approximations rather than from a single finitely generated maximal Cohen–Macaulay module (Bhatt et al., 2024).

In characteristic $$h_i(\mathbf x;M):=\ell(H_i(\mathbf x;M)), \qquad \sigma_i(\mathbf x;M):=\sum_{j=i}^d h_j(\mathbf x;M),$$5, the theory recovers tight closure exactly. If $$h_i(\mathbf x;M):=\ell(H_i(\mathbf x;M)), \qquad \sigma_i(\mathbf x;M):=\sum_{j=i}^d h_j(\mathbf x;M),$$6 is reduced, equidimensional, $$h_i(\mathbf x;M):=\ell(H_i(\mathbf x;M)), \qquad \sigma_i(\mathbf x;M):=\sum_{j=i}^d h_j(\mathbf x;M),$$7-finite, and local of characteristic $$h_i(\mathbf x;M):=\ell(H_i(\mathbf x;M)), \qquad \sigma_i(\mathbf x;M):=\sum_{j=i}^d h_j(\mathbf x;M),$$8, then the closure operation defined from

$$h_i(\mathbf x;M):=\ell(H_i(\mathbf x;M)), \qquad \sigma_i(\mathbf x;M):=\sum_{j=i}^d h_j(\mathbf x;M),$$9

using either $d$00 or rank agrees with ordinary tight closure. More generally, when rank is defined, these asymptotic closures are always contained in integral closure: $d$01 for every ideal $d$02 of a formally equidimensional local ring (Bhatt et al., 2024).

5. Intersection theory and asymptotic multiplicity problems

A major application is to Serre intersection multiplicities. Let $d$03 be a regular local ring of dimension $d$04, let $d$05 be prime ideals such that

$d$06

and write

$d$07

If both $d$08 and $d$09 admit lim Cohen–Macaulay sequences, then Serre’s intersection multiplicity is positive: $d$10 More precisely,

$d$11

where $d$12 and $d$13 are lim Cohen–Macaulay sequences for $d$14 and $d$15, respectively (Bhatt et al., 2024).

The argument is asymptotic. Higher $d$16-terms between $d$17 and $d$18 are shown to be negligible after normalization by the product of ranks, so only the normalized tensor-product length survives in the limit. This reproduces the classical positivity mechanism of Cohen–Macaulay modules in an asymptotic form (Bhatt et al., 2024).

The resulting conditional reduction is sharp. If complete local domains with algebraically closed or perfect residue field are known to admit lim Cohen–Macaulay sequences up to dimension $d$19, then Serre’s positivity conjecture follows up to dimension $d$20. In this sense, the existence problem for lim Cohen–Macaulay sequences controls a major open problem in intersection theory (Bhatt et al., 2024).

A related asymptotic multiplicity refinement is given by lim Ulrich sequences. A sequence $d$21 is lim Ulrich or weakly lim Ulrich if it is lim Cohen–Macaulay or weakly lim Cohen–Macaulay and additionally

$d$22

Weakly lim Ulrich sequences imply Lech’s conjecture for flat local extensions, and such sequences exist for all standard graded domains over perfect fields of positive characteristic (Ma, 2020).

6. Variants, extensions, and neighboring notions

A stronger local-cohomological variant also appears. A sequence $d$23 is strongly lim Cohen–Macaulay if there exists a finite set of Artinian modules $d$24 such that for every $d$25,

$d$26

where $d$27 denotes the filtration length relative to $d$28. Strongly lim Cohen–Macaulay sequences are lim Cohen–Macaulay, and over an $d$29-finite local ring the Frobenius sequences $d$30 are strongly lim Cohen–Macaulay (Bhatt et al., 2024).

The mixed-characteristic theory of $d$31-rings and perfectoid towers extends the original framework by producing lim Cohen–Macaulay sequences from Frobenius lifts, $d$32-stable ideals, and perfectoid tilting. In particular, monomial, binomial, determinantal, and geometric section-ring constructions all fit into the same mixed-characteristic architecture once an appropriate perfectoid tower exists (Ishiro et al., 8 Sep 2025).

The terminology can be confused with several filtration-based notions that are formally different. Sequentially Cohen–Macaulay modules are defined by requiring every successive quotient in the dimension filtration to be Cohen–Macaulay, and recent work characterizes them by the existence of suitable sequential parameter sequences rather than by asymptotic module families (Linh et al., 21 Jun 2025). Initially Cohen–Macaulay modules are defined by the equality

$d$33

where $d$34 is the minimum coheight among associated primes, and this notion again belongs to the theory of single modules and dimension filtrations rather than to asymptotic sequences (Namiq, 30 Oct 2025).

Lim Cohen–Macaulay sequences therefore occupy a distinct position in the modern Cohen–Macaulay landscape. They are not filtration conditions on one module, but asymptotic families of finite modules or finite algebras whose homological defects vanish after normalization. That asymptotic perspective is precisely what allows the theory to bridge small Cohen–Macaulay approximations, closure operations, perfectoid constructions, and intersection-theoretic applications.