# Lower bounds on Loewy lengths of modules of finite projective dimension

Nawaj KC  Department of Mathematics, University of Nebraska, Lincoln, NE
68588, U.S.A<EMAIL_ADDRESS>and  Josh Pollitz  Mathematics Department,
Syracuse University, Syracuse, NY 13244 U.S.A<EMAIL_ADDRESS>
###### Abstract.

In this article we study nonzero modules of finite length and finite
projective dimension over a local ring. We show the Loewy length of such a
module is larger than the regularity of the ring whenever the ring is strict
Cohen-Macaulay, extending work of Avramov–Buchweitz–Iyengar–Miller beyond the
Gorenstein setting. Applications include establishing a conjecture of
Corso–Huneke–Polini–Ulrich and verifying a Lech-like conjecture, comparing
generalized Loewy length along flat local extensions, for strict Cohen-
Macaulay rings. We also give significant improvements on known lower bounds
for Loewy lengths of modules of finite projective dimension without any
assumption on the associated graded ring. The strongest general bounds are
achieved over complete intersection rings.

###### Key words and phrases:

Loewy length, finite projective dimension, (Castelnouvo-Mumford) regularity,
Cohen-Macaulay, Lech’s conjecture, free resolutions, socle, Koszul complex

###### 2020 Mathematics Subject Classification:

13C14 (primary), 13B10, 13D05, 13H10, 13H15

## Introduction

This work is concerned with modules of finite length and finite projective
dimension over a local ring. Such modules have received a great deal of
attention and encode the singularity of the ring; for instance, a consequence
of Roberts’ New Intersection Theorem [34] is that if a ring admits a nonzero
module of finite length and finite projective dimension, then the ring must be
Cohen-Macaulay. There are a number of interesting questions that remain open
regarding these modules. In this article we revisit one on a uniform lower
bound for their Loewy lengths.

Fix a local ring $R$ with maximal ideal $\mathfrak{m}$. Recall that the Loewy
length of an $R$-module $M$ is $\ell\ell_{R}(M)=\inf\\{i\geqslant
0\mid\mathfrak{m}^{i}M=0\\}.$ We are interested in understanding

$\inf\\{\ell\ell_{R}(M)\mid M\neq 0\text{ finitely generated and
}\operatorname{projdim}_{R}(M)<\infty\\}\,,$

which (by the New Intersection Theorem) is finite if and only if $R$ is Cohen-
Macaulay; recall that if $R$ is Cohen-Macaulay then $R/(\bm{x})$ is a finite
length module of finite projective dimension for any system of parameters
$\bm{x}$ on $R$. These observations suggest the following, conjectured by
Corso, Huneke, Polini and Ulrich [10]; we refer to this as the _Loewy Length
Conjecture_.

######  Conjecture A.

For a local ring $R$ and a nonzero $R$-module $M$ with finite projective
dimension, the following inequality holds:

$\ell\ell_{R}(M)\geqslant\min\\{\ell\ell_{R}(R/(\bm{x}))\mid\bm{x}\text{ is a
system of parameters on }R\\}\,.$

The proposed uniform lower bound in the conjecture is called the generalized
Loewy length of $R$, denoted $g\ell\ell_{R}(R)$; this invariant is a Loewy
length analog to the Hilbert-Samuel multiplicity of $R$, and it has been
studied in a number of works; see, for example, [5, 8, 10, 12, 13, 19, 28,
33].

Previously, the most significant progress on the Loewy Length Conjecture was
established by Avramov–Buchweitz–Iyengar–Miller [5, Theorem 1.2]: _If $R$ is a
Gorenstein ring with Cohen-Macaulay associated graded ring $R^{\mathsf{g}}$
and infinite residue field, then A holds_. Without any assumption on the
residue field, they also provide a bound in terms of the (Castelnouvo-Mumford)
regularity of $R^{\mathsf{g}}$, denoted $\operatorname{reg}(R^{\mathsf{g}})$.
The main result of this article extends this result in the following.

######  Theorem B.

If $R$ is a local with Cohen-Macaulay associated graded ring, then any nonzero
finitely generated $R$-module $M$ of finite projective dimension satisfies:

$\ell\ell_{R}(M)\geqslant\operatorname{reg}(R^{\mathsf{g}})+1\,.$

Furthermore, if $R$ has an infinite residue field, then
$\ell\ell_{R}(M)\geqslant g\ell\ell_{R}(R).$

The proof of B can be found at the end of Section 2. An essential ingredient
is Lemma 2.6, where we leverage the hypothesis that $R^{\mathsf{g}}$ is Cohen-
Macaulay (that is, $R$ is _strict_ Cohen-Macaulay) to produce the minimal free
resolution of a certain artinian quotient of $R$; this resolution has the
property that entries in its differentials are in sufficiently high powers of
the maximal ideal. B then follows from the calculation in Lemma 2.7 which
shows that upon tensoring this resolution with modules having ‘small enough’
Loewy length there must be homology in arbitrarily high degrees.

There are several consequences of B to strict Cohen-Macaulay rings. An
immediate one is that the generalized Loewy length of a nonzero Cohen-Macaulay
module with finite projective dimension is bounded below by the generalized
Loewy length of the ring; see Theorem 2.1. Another is that

$g\ell\ell_{R}(R)=\ell\ell_{R}(R/(\bm{x}))$

for all maximal superficial sequences $\bm{x}$ in
$\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$, where $\mathfrak{m}$ denotes the
maximal ideal of $R$; cf. Corollary 2.2. Furthermore, we introduce a version
of the long-standing conjecture of Lech [29], where Hilbert-Samuel
multiplicity is replaced with generalized Loewy length (see 3.1), and we
provide the following evidence:

######  Corollary C.

Suppose $R\to S$ is a map of Cohen-Macaulay local rings with infinite residue
fields. If $\operatorname{flatdim}_{R}(S)<\infty$ and $R$ is strict Cohen-
Macaulay, then

$g\ell\ell_{R}(R)\leqslant g\ell\ell_{S}(S)\,.$

This corollary establishes interesting ring theoretic properties along flat
local extensions whose base is strict Cohen-Macaulay; cf. Corollary 3.3. In
particular, we verify a conjecture of Hanes [18, Conjecture 3.1] on reduction
numbers along flat local extensions when the base is assumed to be strict
Cohen-Macaulay.

In the final part of the paper, Section 4, we turn our attention to lower
bounds for the Loewy lengths of nonzero modules of finite projective dimension
without making assumptions on the structure of $R^{\mathsf{g}}$. Building on
[5], it was shown in [33] that when $R$ is Gorenstein then for such a module
$M$ one has $\ell\ell_{R}(M)\geqslant\operatorname{ord}(R)$; here
$\operatorname{ord}(R)$ denotes the minimal order of a defining relation of
$R$ in one of its minimal Cohen presentations. We improve this bound in a
rather drastic way; below, the complexity of an $R$-module $M$, denoted
$\operatorname{cx}_{R}(M)$, measures the polynomial rate of growth of the
Betti sequence for $M$.

######  Theorem D.

Assume $R$ is a local ring, and fix a minimal Cohen presentation
$\widehat{R}\cong Q/(f_{1},\ldots,f_{c})$ and a nonzero $R$-module $M$.

  1. (1)

If $\ell\ell_{R}(M)<\operatorname{ord}(R),$ then
$\operatorname{cx}_{R}(M)\geqslant c$.

  2. (2)

If $M$ has finite projective dimension, then

$\ell\ell_{R}(M)\geqslant\max\\{\operatorname{ord}(f_{i}):1\leqslant
i\leqslant c\\}\,.$

  3. (3)

If $M$ has finite projective dimension and $R$ is a complete intersection
ring, then

$\ell\ell_{R}(M)\geqslant\sum_{i=1}^{c}\operatorname{ord}(f_{i})-c+1\,.$

The first two parts of the theorem can be found in Theorem 4.3, and their
proofs make use of a dg algebra structure on the minimal free resolution of
the residue field. Part (3) of D, see Theorem 4.6, was first suggested to the
authors by Mark Walker. We present two proofs: the first is similar to the
proof of B, making use of the explicit free resolution of the socle of an
artinian complete intersection, while the second was sketched to the authors
by Walker and makes use of a construction from [21].

Finally, the central problem in this article (A) parallels the _Length
Conjecture_ studied in [26]; cf. Remark 3.4. As Loewy length and length are
subtle invariants in their own way, it is perhaps unsurprising we employ
seemingly different techniques in the present article to establish our uniform
lower bounds for Loewy lengths compared to the methods used by
Iyengar–Ma–Walker for length (with the exception of the second proof of D). It
is worth remarking that the existence of Ulrich modules, which played a
pivotal role in [26], are now known to not exist over rings where the Loewy
Length Conjecture was settled in the positive here; for example, [27] provides
examples of complete intersection rings that are strict Cohen-Macaulay where
the method from [26] cannot be applied.

######  Acknowledgements.

We thank Mark Walker for sharing a proof of Theorem 4.6 with us, and
encouraging us to present that argument here. We also thank Srikanth Iyengar
for several helpful discussions regarding this work, as well as Linquan Ma for
making us aware of the conjecture in [18]. We are grateful to Alberto Corso,
Craig Huneke, Claudia Polini and Bernd Ulrich for discussing their
conjectures, and their upcoming work in [10], with us.

The first author was supported by the National Science Foundation grant
DMS-1928930 and by the Alfred P. Sloan Foundation under grant G-2021-16778,
while he was in residence at the Simons Laufer Mathematical Sciences Institute
(formerly MSRI) in Berkeley, California, during the Spring 2024 semester. The
first author was also partly supported by NSF grants DMS-2200732 and
DMS-2044833. The second author was supported by the NSF grant DMS-2302567.

##  1\. Loewy length and the associated graded ring

This section recounts the necessary background on (generalized) Loewy length.
Throughout, $R$ is a commutative noetherian local ring with maximal ideal
$\mathfrak{m}$.

######  1.1.

The _Loewy length_ of an $R$-module $M$, denoted $\ell\ell_{R}(M)$, is the
least nonnegative integer $i$ such that $\mathfrak{m}^{i}M=0$; if no such
integer exists set $\ell\ell_{R}(M)=\infty$. When $M$ is a finitely generated
$R$-module, its _generalized Loewy length_ is

$g\ell\ell_{R}(M)=\inf\\{\ell\ell_{R}(M/\bm{x}M)\mid\bm{x}\text{ is a system
of parameters on }M\\}.$

The invariant $g\ell\ell_{R}(R)$ can be regarded as a measure of the
singularity of $R$, as it equals one if and only if $R$ is regular.

######  Remark 1.2.

Generalized Loewy length is to Loewy length as Hilbert-Samuel multiplicity is
to length. Whereas there is much known on the Hilbert-Samuel multiplicity of
modules over a local ring, generalized Loewy length is a more subtle and
difficult to understand invariant. For example, the Hilbert-Samuel
multiplicity of a Cohen-Macaulay local ring $R$ is the length of $R/(\bm{x})$
where $\bm{x}$ is any reduction of the maximal ideal. Whether this holds for
generalized Loewy length is a question of DeStefani over Gorenstein rings
having infinite residue fields in [12, Question 4.5]. This is also the content
of a conjecture in [10].

######  Notation 1.3.

The associated graded ring of $R$ is

$R^{\mathsf{g}}\coloneqq\operatorname{gr}_{\mathfrak{m}}(R)=\bigoplus_{j=0}^{\infty}\frac{\mathfrak{m}^{j}}{\mathfrak{m}^{j+1}}\,.$

For a sequence of elements $\bm{x}$ in $R$, write $\bm{x}^{*}$ for its
corresponding sequence of initial forms in $R^{\mathsf{g}}.$ Also, for a
minimal $R$-complex $C$, its linear part is the associated graded complex

$C^{\mathsf{g}}\coloneqq\bigoplus_{j\in\mathbb{Z}}\frac{\mathfrak{m}^{j}C}{\mathfrak{m}^{j+1}C}(-j)\,;$

this is a complex of graded $R^{\mathsf{g}}$-modules where $(-j)$ is shifting
the internal grading (that is, the non-homological one). In particular
$C^{\mathsf{g}}$ is bigraded with

$C^{\mathsf{g}}_{i,j}=\frac{\mathfrak{m}^{j-i}C_{i}}{\mathfrak{m}^{j-i+1}C_{i}}\,.$

######  1.4.

A sequence $\bm{x}$ in $R$ is _superficial_ if its sequence of initial forms
in $R^{\mathsf{g}}$ is part of a system of parameters for $R^{\mathsf{g}}$;
such a sequence is part of a system of parameters for $R$. Moreover, if $R$
has an infinite residue field, then there exists a superficial system of
parameters in $\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$.

######  1.5.

A local ring $R$ is strict Cohen-Macaulay if its associated graded ring
$R^{\mathsf{g}}$ is Cohen-Macaulay. Strict Cohen-Macaulay rings are, in
particular, Cohen-Macaulay; see, for example, [1] (or [16]). Hypersurface
rings are strict Cohen-Macaulay, however complete intersection rings of higher
codimension need not be strict Cohen-Macaulay (see, for example, Remark 4.7).

######  1.6.

For a standard graded algebra $S$ over a field $S_{0}$, write $S_{+}$ for its
homogeneous maximal ideal. For a homogeneous ideal $I$ of $S$, let
$\operatorname{H}_{I}(-)$ denote the local cohomology functor with respect to
$I$. The _regularity of_ $S$ is

$\operatorname{reg}(S)\coloneqq\max\\{i+j\mid\operatorname{H}_{S_{+}}^{i}(S)_{j}\neq
0\text{ for some }i\\}\,.$

Following [22], the _regularity_ of $R$ is
$\operatorname{reg}(R)\coloneqq\operatorname{reg}(R^{\mathsf{g}}).$

######  Lemma 1.7.

If $R$ is a local ring with a superficial system of parameters $\bm{x}$ in
$\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$, and set
$A=\mathrm{Kos}^{R}(\bm{x})$, then

$\ell\ell_{R}(R/(\bm{x}))-1\leqslant\operatorname{reg}(R)=\max\\{j\geqslant
0\mid\operatorname{H}_{i}(A^{\mathsf{g}})_{i+j}\neq 0\text{ for some }i\\}\,.$

###### Proof.

Set $d=\dim(R)$, and fix a Noether normalization
$S=k[t_{1},\ldots,t_{d}]\hookrightarrow R^{\mathsf{g}}$ with each $t_{i}$
mapping to $x_{i}^{*}$, the initial form of $x_{i}$ in $R^{\mathsf{g}}$. By
the assumptions on $\bm{x}$, we have

$\operatorname{H}_{R^{\mathsf{g}}_{+}}^{i}(R^{\mathsf{g}})\cong\operatorname{H}_{(\bm{t})}^{i}(R^{\mathsf{g}})$

and hence it follows from a theorem of Eisenbud–Goto (see, for example, [9,
Theorem 4.3.1]) that

$\operatorname{reg}(R)=\max\\{j\geqslant
0\mid\operatorname{H}_{i}(\mathrm{Kos}^{S}(\bm{t};R^{\mathsf{g}}))_{i+j}\neq
0\text{ for some }i\\}\,.$

Finally, it remains to observe that
$\mathrm{Kos}^{S}(\bm{t};R^{\mathsf{g}})\cong A^{\mathsf{g}}$ to justify the
desired equality.

For the inequality, it follows from [24, Proposition 8.2.4] that

$\ell\ell_{R}(R/(\bm{x}))\leqslant\min\\{j\mid(R^{\mathsf{g}}/(\bm{x}^{*}))_{\geqslant
j}=0\\}\,.$

Now from the already established equality, using that
$\operatorname{H}_{0}(A^{\mathsf{g}})=R^{\mathsf{g}}/(\bm{x}^{*}),$ we obtain
the desired inequality. ∎

######  Lemma 1.8.

If $R$ is strict Cohen-Macaulay, then
$\operatorname{reg}(R)=\ell\ell_{R}(R/(\bm{x}))-1$ for any superficial system
of parameters $\bm{x}$ in $\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$.

###### Proof.

By [35, Lemma 0.1], since $R^{\mathsf{g}}$ is Cohen-Macaulay, for any
superficial sequence $\bm{x}$ we have

$(R/(\bm{x}))^{\mathsf{g}}\cong R^{\mathsf{g}}/(\bm{x}^{*})\,.$

Also, when $\bm{x}$ is a system of parameters for $R$ in
$\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$ it follows from [19, Lemma 6.2]
that

$\ell\ell_{R}(R/(\bm{x}))=\min\\{j\mid(R^{\mathsf{g}}/(\bm{x}^{*}))_{\geqslant
j}=0\\}\,.$

Now since $R^{\mathsf{g}}$ is Cohen-Macaulay, applying Lemma 1.7 yields

$\ell\ell_{R}(R/(\bm{x}))=\operatorname{reg}(R)+1\,.\qed$

##  2\. Main result

This section is devoted to proving B from the introduction. In fact, we prove
the following stronger statement, affirming a conjecture of
Corso–Huneke–Polini–Ulrich [10] for strict Cohen-Macaulay rings.

######  Theorem 2.1.

Suppose a local ring $R$ is strict Cohen-Macaulay. If $M$ is a nonzero Cohen-
Macaulay $R$-module of finite projective dimension, then

$g\ell\ell_{R}(M)\geqslant\operatorname{reg}(R)+1\,.$

Furthermore, if $R$ has an infinite residue field, then
$g\ell\ell_{R}(M)\geqslant g\ell\ell_{R}(R).$

Before starting on the proof, we present the following corollary. This
establishes another conjecture from [10] (again, for strict Cohen-Macaulay
rings).

######  Corollary 2.2.

For a strict Cohen-Macaulay ring $(R,\mathfrak{m})$ with infinite residue
field,

$g\ell\ell_{R}(R)=\ell\ell_{R}(R/(\bm{x}))=\operatorname{reg}(R)+1$

where $\bm{x}$ is a sufficiently general system of parameters for $R$ in
$\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$.

###### Proof.

As the residue field is infinite, a sufficiently general system of parameters
$\bm{x}$ in $\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$ is, in particular,
superficial; see [24, Theorem 8.6.6]. Since $R$ is strict Cohen-Macaulay, from
Lemma 1.8 we have $\ell\ell_{R}(R/(\bm{x}))=\operatorname{reg}(R)+1$ and from
the inequality in Theorem 2.1 (applied for $M=R$), we obtain

$g\ell\ell_{R}(R)=\ell\ell_{R}(R/(\bm{x}))=\operatorname{reg}(R)+1\,.\qed$

The proof of Theorem 2.1 is given at the end of the section; it takes a bit of
setup and is an immediate consequence of the more technical result in Lemma
2.7.

######  Notation 2.3.

For the remainder of the section we assume $R$ is a commutative noetherian
local ring with maximal ideal $\mathfrak{m}$ and infinite residue field $k$;
the fact that Loewy length decreases and regularity is unchanged among passage
to an infinite residue field is due to [19, Proposition 6.3] and that local
cohomology is invariant under flat base change, respectively. We can also
assume that $R$ is Cohen-Macaulay because of the New Intersection Theorem
[34].

The next lemma abstracts a discussion in [11, Section 2], building on work in
[22]. It is also related to the main theorem of [38], as well as [7, Section
3].

######  Lemma 2.4.

Suppose $(R,\mathfrak{m})$ is a local ring, $\bm{x}$ is a superficial system
of parameters in $\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$, and set
$A=\mathrm{Kos}^{R}(\bm{x})$. Then for $i\geqslant 0$ and
$j\geqslant\operatorname{reg}(R)$ we have

$\partial(A_{i+1})\cap\mathfrak{m}^{j+1}A_{i}=\partial(\mathfrak{m}^{j}A_{i+1})\,.$

###### Proof.

There is no harm in completing $R$ to show the desired equality. As $A$ is a
minimal complex, it is clear that
$\partial(\mathfrak{m}^{j}A_{i+1})\subseteq\partial(A_{i+1})\cap\mathfrak{m}^{j+1}A_{i}.$

For the reverse containment, consider $\partial
x\in\partial(A_{i+1})\cap\mathfrak{m}^{j+1}A_{i}.$ Its image in
$A^{\mathsf{g}}$ is a cycle in $A^{\mathsf{g}}_{i,i+j+1}$, and so it defines
the following homology class

$[\partial x]\in\operatorname{H}_{i}(A^{\mathsf{g}})_{i+j+1}\,.$

Since $j\geqslant\operatorname{reg}(R)$, by Lemma 1.7, the homology class
$[\partial x]$ is zero. That is to say, there exists
$u_{0}\in\mathfrak{m}^{j}A_{i+1}$ such that

$\partial^{A}(x)-\partial^{A}(u_{0})=\partial^{A}(x-u_{0})\in\mathfrak{m}^{j+2}A_{i}\,.$

Repeating the argument above produces $u_{n}\in\mathfrak{m}^{j+n}A_{i+1}$ with

$\partial^{A}(x)-\partial^{A}\left(\sum_{\ell=0}^{n}u_{\ell}\right)=\partial^{A}\left(x-\sum_{\ell=0}^{n}u_{\ell}\right)\in\mathfrak{m}^{j+n+2}A_{i}\,.$

Since $R$ is complete, we can let $u=\sum_{\ell=0}^{\infty}u_{\ell}$, and by
construction $\partial^{A}(x)=\partial^{A}(u)$ with
$u\in\mathfrak{m}^{j}A_{i+1}$. ∎

######  2.5.

Assuming the setup in 2.3, there exists a superficial system of parameters
$\bm{x}$ in $\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$; moreover, as $R$ is
Cohen-Macaulay, $\bm{x}$ is a maximal regular sequence of $R$. Set
$\overline{R}=R/(\bm{x})$ and let $A$ be the Koszul complex on $\bm{x}$ over
$R$. Note that $A$ is the minimal free resolution of $\overline{R}$ over $R$.
Also, fix the minimal free resolution $F\xrightarrow{\simeq}k$ over $R$.

Let $\bar{s}$ denote the element of highest $\mathfrak{m}\overline{R}$-adic
order in the socle of $\overline{R}$. That is to say, the
$\mathfrak{m}\overline{R}$-adic order of $\bar{s}$, denoted $n$, is exactly
one less than the Loewy length of $\overline{R}$. Moreover, a standard lifting
property guarantees the existence of a lift between the complexes of
$R$-modules below:

${F}$${A}$${k}$${{\overline{R}}}$$\scriptstyle{\simeq}$$\scriptstyle{1\mapsto{\bar{s}}}$$\scriptstyle{\exists}$$\scriptstyle{\simeq}$

.

In what follows we construct a particular lift $\sigma\colon F\to A$ with
$\sigma(F)\subseteq\mathfrak{m}^{n}A$ provided that
$\operatorname{reg}(R)=\ell\ell_{R}(R/(\bm{x}))-1$.

######  Lemma 2.6.

Assume $(R,\mathfrak{m},k)$ is a local ring with
$\operatorname{reg}(R)=\ell\ell_{R}(R/(\bm{x}))-1$ for some superficial system
of parameters $\bm{x}$ in $\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$. Set
$\overline{R}=R/(\bm{x})$ and $n=\operatorname{reg}(R)$, and let $F$ denote
the minimal $R$-free resolution of $k$ and $A=\mathrm{Kos}^{R}(\bm{x})$ be the
minimal $R$-free resolution $\overline{R}$.

For any nonzero $\overline{s}\in\mathfrak{m}^{n}\overline{R}$, the $R$-module
map $k\mapsto\overline{R}$ given by $1\mapsto\overline{s}$ admits a lift
$\sigma\colon F\to A$ satisfying:

  1. (1)

$\sigma\colon F\to A$ is map of complexes with
$\sigma(F)\subseteq\mathfrak{m}^{n}A$;

  2. (2)

$\operatorname{cone}(\sigma)$ is the minimal free resolution of
$\overline{R}/(\bar{s})$ over $R$.

###### Proof.

First, we inductively construct $\sigma$. In degree zero, let
$\sigma_{0}\colon F_{0}=R\to R=A_{0}$ be given by multiplication by $s$ where
the image of $s$ in $\overline{R}$ is $\bar{s}$ and $s\in\mathfrak{m}^{n}.$
Now assume we have constructed $\sigma_{0},\ldots,\sigma_{i}$ with

$\sigma_{j}(F_{j})\subseteq\mathfrak{m}^{n}A_{j}\quad\text{and}\quad\sigma_{j-1}\partial^{F}_{j}=\partial^{A}_{j-1}\sigma_{j}$

for each $j=0,\ldots,i.$ Observe that

$\sigma_{i}\partial_{i+1}^{F}(F_{i+1})\subseteq\mathfrak{m}^{n+1}A_{i}\cap\ker\partial^{A}_{i}=\mathfrak{m}^{n+1}A_{i}\cap\partial^{A}(A_{i+1})$

where the first containment holds by the inductive hypothesis and the equality
holds since $A$ is exact. As a consequence, combined with Lemma 2.4, we obtain

$\sigma_{i}\partial_{i+1}^{F}(F_{i+1})\subseteq\partial^{A}(\mathfrak{m}^{n}A_{i+1})\,;$

this is where the assumption that
$\operatorname{reg}(R)=\ell\ell_{R}(R/(\bm{x}))-1$ is used. Hence by the
lifting property of a free $R$-module, there exists an $R$-lienar map
$\sigma_{i+1}\colon F_{i+1}\to A_{i+1}$ such that

$\sigma_{i}\partial^{F}_{i+1}=\partial^{A}_{i+1}\sigma_{i+1}\quad\text{and}\quad\sigma_{i+1}(F_{i+1})\subseteq\mathfrak{m}^{n}A_{i+1}\,.$

This completes the construction of the lift $\sigma\colon F\to A,$ and
verifies Item 1.

For Item 2, set $C=\operatorname{cone}(\sigma)$ and observe that the cone
exact sequence

$0\to A\to C\to{\scriptstyle\mathsf{\Sigma}}F\to 0$

induces the exact sequence

$\cdots\to 0\to 0\to\operatorname{H}_{2}(C)\to 0\to
0\to\operatorname{H}_{1}(C)\to\operatorname{H}_{0}(F)\xrightarrow{\operatorname{H}_{0}(\sigma)}\operatorname{H}_{0}(A)\to\operatorname{H}_{0}(C)\to
0\,.$

In particular, $\operatorname{H}_{i}(C)=0$ for $i>1$. Furthermore, note that
$\operatorname{H}_{0}(\sigma)$ is the inclusion $k\hookrightarrow\overline{R}$
sending $1$ to $\bar{s}$, and so it follows from the exact sequence above that

$\operatorname{H}_{i}(C)=\begin{cases}\overline{R}/(\bar{s})&i=0\\\
0&\text{otherwise}\end{cases}\,.$

Also, as $\sigma$ is a map between free resolutions of $R$-modules, $C$ is a
nonnegatively graded complex of free $R$-modules and hence it is a free
resolution of $\overline{R}/(\bar{s})$ over $R$. Finally, by definition of the
differential of $C$, it is in fact minimal since both $F$ and $A$ are minimal
$R$-complexes and $\sigma(F)\subseteq\mathfrak{m}^{n}A$; cf. Item 1. ∎

######  Lemma 2.7.

Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay ring with a superficial system of
parameters $\bm{x}$ in $\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$ such that
$\operatorname{reg}(R)=\ell\ell_{R}(R/(\bm{x}))-1$. Set
$\overline{R}=R/(\bm{x})$ and let $\bar{s}$ be a socle element of highest
$\mathfrak{m}\overline{R}$-adic order.

For an $R$-module $M$ with $\ell\ell_{R}(M)<\ell\ell_{R}(\overline{R})$, we
have

${\operatorname{Tor}}^{R}(M,\overline{R}/(\bar{s}))\cong{\operatorname{Tor}}^{R}(M,\overline{R})\oplus{\scriptstyle\mathsf{\Sigma}}{\operatorname{Tor}}^{R}(M,k)\,.$

In particular, if $M$ is a nonzero, finite length $R$-module with
$\ell\ell_{R}(M)<\ell\ell_{R}(\overline{R})$ then
$\operatorname{projdim}_{R}M=\infty$.

###### Proof.

We are exactly in the setting of Lemma 2.6. By Lemma 2.6(2),
$\operatorname{cone}(\sigma)$ is the minimal free resolution of
$\overline{R}/(\bar{s})$ over $R$. Hence we have the first isomorphism below:

$\displaystyle{\operatorname{Tor}}^{R}(M,\overline{R}/(\bar{s}))$
$\displaystyle\cong\operatorname{H}(M\otimes_{R}\operatorname{cone}(\sigma))$
$\displaystyle\cong\operatorname{H}(\operatorname{cone}(M\otimes_{R}\sigma))$
$\displaystyle\cong\operatorname{H}(\operatorname{cone}(M\otimes_{R}F\xrightarrow{0}M\otimes_{R}A))$
$\displaystyle=\operatorname{H}(M\otimes_{R}A\oplus{\scriptstyle\mathsf{\Sigma}}(M\otimes_{R}F))$
$\displaystyle\cong\operatorname{H}(M\otimes_{R}A)\oplus{\scriptstyle\mathsf{\Sigma}}\operatorname{H}(M\otimes_{R}F)$
$\displaystyle\cong{\operatorname{Tor}}^{R}(M,\overline{R})\oplus{\scriptstyle\mathsf{\Sigma}}{\operatorname{Tor}}^{R}(M,k)\,;$

the remaining isomorphisms are all immediate except for the third one. This
isomorphism is where we use the hypothesis $\ell\ell_{R}(M)\leqslant
n=\ell\ell_{R}(\overline{R})-1$ and Lemma 2.6Item 1 to establish the
isomorphism of the underlying complexes.

Now by way of contradiction, assume that $M$ is a nonzero finite length
$R$-module of finite projective dimension. By the Auslander-Buchsbaum formula
(and Nakayama’s lemma), we have that ${\operatorname{Tor}}^{R}_{d}(M,k)\neq 0$
and ${\operatorname{Tor}}_{i}^{R}(M,-)=0$ whenever $i>d$, where $d=\dim(R)$.
However, observe that the already established isomorphism yields

${\operatorname{Tor}}_{d+1}^{R}(M,\overline{R}/(\bar{s}))\cong{\operatorname{Tor}}^{R}_{d+1}(M,\overline{R})\oplus{\operatorname{Tor}}_{d}^{R}(M,k)\,,$

giving us a contradiction. Hence, $M$ must have infinite projective dimension.
∎

###### Proof of Theorem 2.1.

As discussed in 2.3, we can assume $R$ is a Cohen-Macaulay local ring with
infinite residue field, and by our hypothesis, there exists a superficial
system of parameters $\bm{x}$ in $\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$
such that $\operatorname{reg}(R)=\ell\ell_{R}(R/(\bm{x}))-1$. Hence we are in
the context of Lemma 2.7, and so we obtain

$\ell\ell_{R}(M)\geqslant g\ell\ell_{R}(R)\,,$

whenever $M$ has nonzero finite length and finite projective dimension.

Next assume that $M$ is a nonzero Cohen-Macaulay module having finite
projective dimension over $R$. For any system of parameters $\bm{x}$ on $M$,
the $R$-module $M/(\bm{x})M$ is a nonzero finite length module with finite
projective dimension. Now one need only apply Theorem 2.1 to $M/(\bm{x})M$ to
deduce

$\ell\ell_{R}(M/(\bm{x})M)\geqslant\operatorname{reg}(R)+1\,,$

and when $R$ has infinite residue field $\ell\ell_{R}(M/(\bm{x})M)\geqslant
g\ell\ell_{R}(R)$. Since these inequalities hold for each system of parameters
$\bm{x}$ on $M$, we obtain the desired inequalities. ∎

######  Remark 2.8.

For a local ring $(R,\mathfrak{m})$, write
$(R^{\prime},\mathfrak{m}^{\prime})$ for the inflation
$R[t]_{\mathfrak{m}[t]}$ of $R$. We say $R$ has _minimal regularity_ if

$\operatorname{reg}(R^{\prime})=\ell\ell_{R^{\prime}}(R^{\prime}/(\bm{x}))-1$

for some superficial system of parameters $\bm{x}$ in
$\mathfrak{m}^{\prime}\smallsetminus(\mathfrak{m}^{\prime})^{2}$. When the
ring has an infinite residue field, having minimal regularity is an intrinsic
property. That is to say, a local ring $(R,\mathfrak{m})$ with infinite
residue field has minimal regularity if and only if
$\operatorname{reg}(R)=\ell\ell_{R}(R/(\bm{x}))-1$ for some superficial system
of parameters $\bm{x}$ in $\mathfrak{m}\smallsetminus\mathfrak{m}^{2}$.
Indeed, the backwards direction is clear, while the forward direction is
actually a consequence of Theorem 2.1. Tracking through the proof of Theorem
2.1, one sees that the theorem holds for rings having minimal regularity. It
is easy to write down examples of such rings, however we do not know examples
of _Cohen-Macaulay_ local rings having minimal regularity that are not strict
Cohen-Macaulay.

##  3\. Generalized Loewy length along flat extensions

A tremendous amount of research (see, for example, [23, 30, 31, 32] and the
references therein) in commutative algebra has been motivated by the
longstanding conjecture of Lech [29]: For a flat local extension $R\to S$, one
has the following inequality on Hilbert-Samuel multiplicities:

$e(R)\leqslant e(S)\,.$

Ma has established the conjecture for equicharacterisitc rings of dimension at
most three [31], and for all standard graded algebras localized at their
homogeneous maximal ideal [32]. In light of Remark 1.2 and Lech’s conjecture,
we are led to (perhaps optimistically) conjecture the following.

######  Conjecture 3.1.

If $R\to S$ is a flat local extension between Cohen-Macaulay rings with
infinite residue fields, then $g\ell\ell_{R}(R)\leqslant g\ell\ell_{S}(S).$

By combining [13, Theorem 2.1] and [19, Corollary 5.2], the conjecture is
already known to hold when $R\to S$ is a flat local extension of Gorenstein,
strict Cohen-Macaulay rings having infinite residue fields and a regular
fiber. We give a substantial improvement below.

######  Theorem 3.2.

Suppose $R\to S$ is a local extension of Cohen-Macaulay rings with infinite
residue fields, and assume $\operatorname{flatdim}_{R}(S)<\infty$. If $R$ is
strict Cohen-Macaulay, then $g\ell\ell_{R}(R)\leqslant g\ell\ell_{S}(S).$ In
particular, 3.1 holds in this setting.

###### Proof.

By [19, Lemma 3.3] generalized Loewy length is invariant upon completion, so
we can assume both $R$ and $S$ are complete. Now by [6, Theorem 1.1], there
exists a Cohen factorization of $\varphi$:

$R\xrightarrow{\iota}R^{\prime}\xrightarrow{\varphi^{\prime}}S$

where $\iota$ weakly regular (that is to say, $\iota$ is flat with regular
fiber) and $\varphi^{\prime}$ is surjective.

As $\iota$ is weakly regular, the map $\iota^{\mathsf{g}}\colon
R^{\mathsf{g}}\to(R^{\prime})^{\mathsf{g}}$ is tangentially flat, in the sense
that it is a flat extension of standard graded algebras with a symmetric
algebra fiber; see [20, Theorem 1.2]. In particular, $R^{\prime}$ is strict
Cohen-Macaulay. As a consequence, the second equality below holds:

(1)
$g\ell\ell_{R}(R)=\operatorname{reg}(R)+1=\operatorname{reg}(R^{\prime})+1=g\ell\ell_{R^{\prime}}(R^{\prime})\,;$

the outside equalities are from Corollary 2.2. Again, using that $\iota$ is
weakly regular and $\operatorname{flatdim}_{R}(S)<\infty$ we have that
$\operatorname{projdim}_{R^{\prime}}(S)<\infty$; see [6, Lemma 3.2]. Also,
since $S$ is Cohen-Macaulay it follows that it is a Cohen-Macaulay
$R^{\prime}$-module. Therefore, we have

$g\ell\ell_{S}(S)\geqslant
g\ell\ell_{R^{\prime}}(R^{\prime})=g\ell\ell_{R}(R)\,,$

where inequality uses Theorem 2.1, and the equality comes from Eq. 1. ∎

With Lemma 1.7, we obtain an immediate corollary in the following, where we
refer the reader to the nice introduction on reductions and reduction numbers
in [24, Chapter 8]. Furthermore, the next corollary verifies a conjecture of
Hanes [18, Conjecture 3.1] when $R$ is strict Cohen-Macaulay: _If
$(R,\mathfrak{m})\to(S,\mathfrak{n})$ is a local extension of Cohen-Macaulay
rings with infinite residue fields, then $r(\mathfrak{m})\leqslant
r(\mathfrak{n})$._ Hanes had previously established the conjecture in the
standard graded setting.

######  Corollary 3.3.

Suppose $(R,\mathfrak{m})\to(S,\mathfrak{n})$ is a local extension of Cohen-
Macaulay rings with infinite residue fields, and assume
$\operatorname{flatdim}_{R}(S)<\infty$. If $R$ is strict Cohen-Macaulay, then
$r(\mathfrak{m})\leqslant r(\mathfrak{n})$ and
$\operatorname{reg}(R)\leqslant\operatorname{reg}(S)$.

######  Remark 3.4.

The _Length Conjecture_ of Iyengar–Ma–Walker [26, Conjecture 1] implies Lech’s
conjecture in full generality for Cohen-Macaulay rings; see also [30, Chapter
V]. The former posits that over a local ring $R$, any nonzero module of finite
projective dimension has $\ell_{R}(M)\geqslant e(R)$. This is known to hold
when $R$ is a strict complete intersection (that is, the associated graded is
a complete intersection) or when $R$ is a localization of a standard graded
algebra, at its homogeneous maximal ideal, over a perfect field of positive
characteristic [26]. The Loewy Length Conjecture (A) is the analog of the
Length Conjecture; cf. Remark 1.2. Hence, it seems appropriate to ask the
following.

######  Question 3.5.

If the Loewy Length Conjecture holds for all Cohen-Macaulay rings having
infinite residue field, does this imply 3.1 holds?

The only thing to determine to answer the question in the positive is whether
the generalized Loewy length remains the same along a weakly regular map
between Cohen-Macaulay rings having infinite residue field; without infinite
residue fields this is false because of the example of Hashimoto–Shida [19].

##  4\. General bounds

In this section, we show that there are lower bounds for the Loewy lengths of
nonzero modules of finite projective dimension without any assumption on the
associated graded ring. We give a significant strengthening of known results
in Theorem 4.3, and then present an even stronger bound for local complete
intersection rings in Theorem 4.6.

######  4.1.

For a Gorenstein local ring $R$, if $M$ is a nonzero finite length module
having finite projective dimension, then

$\ell\ell_{R}(M)\geqslant\operatorname{ord}(R)\,;$

here $\operatorname{ord}(R)$, the order of $R$, is minimal
$\mathfrak{m}_{Q}$-adic order of the kernel of a minimal Cohen presentation
$(Q,\mathfrak{m}_{Q})\to\widehat{R}$. This was first established in the case
that $R$ is also assumed to be strict Cohen-Macaulay with infinite residue
field in [5, Theorem 1.1], and for general Gorenstein rings in [33, Theorem
1.1].

######  4.2.

For an element $f$ in a local ring $(R,\mathfrak{m})$ we write
$\operatorname{ord}(f)$ for its $\mathfrak{m}$-adic order. Define the _max-
order of_ $R$, denoted $\max\operatorname{ord}(R)$, to be the supremum over
all nonnegative integers $n$ such that $R$ admits a minimal Cohen presentation
$\widehat{R}\cong Q/I$ where $I$ has a minimal generator $f$ of order $n$. We
always have inequalities

$\operatorname{ord}(R)\leqslant\max\operatorname{ord}(R)\leqslant\operatorname{reg}(R)+1\,,$

where the first is by definition. The second inequality holds since
$\max\operatorname{ord}(f)$ is the degree of a minimal homogeneous generator
for the kernel of a surjective map from a standard graded polynomial ring to
$R^{\mathsf{g}}.$

Next is one of the main result of the section; this removes the Gorenstein
hypothesis and improves the bound in 4.1 in a rather drastic way. The proof is
similar to that of [28, Proposition 1.2] which identifies homology classes in
Tor-modules depending on the Loewy length of a module.

######  Theorem 4.3.

Assume $R$ is a local ring, and fix a minimal Cohen presentation
$\widehat{R}\cong Q/(f_{1},\ldots,f_{t})$ and a nonzero finite length
$R$-module $M$.

  1. (1)

If $\ell\ell_{R}(M)<\operatorname{ord}(R),$ then
$\operatorname{cx}_{R}(M)\geqslant t$.

  2. (2)

If $M$ has finite projective dimension, then
$\ell\ell_{R}(M)\geqslant\max\operatorname{ord}(R).$

Above, recall the complexity of an $R$-module $M$ is

$\operatorname{cx}_{R}(M)=\inf\\{d\in\mathbb{N}\mid\beta_{n}^{R}(M)\leqslant
an^{d-1}\text{ for some $a>0$ and all $n$}\\}\,;$

this is a measure of the polynomial growth of the Betti numbers of $M$ over
$R$. In particular $\operatorname{cx}_{R}(M)=0$ if and only if
$\operatorname{projdim}_{R}(M)<\infty$. See [3, Chapter 5] for more on this
and other asymptotic homological invariants defined over local rings; see also
[7].

###### Proof of Theorem 4.3.

We make use of a well-known dg algebra structure on the minimal free
resolution of the residue field $k$ over $R$; see [3, Chapter 6], or [17], for
more details and any unexplained notation or terminology.

The essential point is that the minimal free resolution of $k$ over $R$ has
the form $R\langle X\rangle$ where as a graded $R$-algebra it is the free
divided power algebra with $X=X_{1},X_{2},X_{3},\ldots$, and $X_{i}$ consists
of variables of degree $i$. The differential is determined by its values on
the variables and extended via the Leibniz rule and respecting divided powers.
We are particularly interested in the differential on the divided power
subalgebra on $X_{2}$, which can be described explicitly as follows.

There is no harm in assuming $R$ is complete and so using the minimal Cohen
presentation $R=Q/(f_{1},\ldots,f_{t})$, fix a minimal set of generators
$x_{1},\ldots,x_{d}$ of $\mathfrak{m}_{Q}$, and write

$f_{i}=\sum\tilde{a}_{ij}x_{j}\quad\text{with
}\tilde{a}_{ij}\in\mathfrak{m}_{Q}\,.$

Then $X_{1}=\\{e_{1},\ldots,e_{d}\\}$, $X_{2}=\\{y_{1},\ldots,y_{t}\\}$ and we
have

$\partial(y_{i})=\sum a_{ij}e_{j}\quad\text{ for each }i=1,\ldots,t$

where $a_{ij}$ is the image of $\tilde{a}_{ij}$ in $\mathfrak{m}$, the maximal
ideal of $R$; this calculation is classical and due to Tate [37, Theorem 4],
but it is also explained in the references above. As a consequence, the
differential on the divided power monomial $y_{1}^{(c_{1})}\cdots
y_{t}^{(c_{t})}$ is

(2) $\partial(y_{1}^{(c_{1})}\cdot\cdots\cdot
y_{t}^{(c_{t})})=\sum_{i=1}^{t}\sum_{j=1}^{d}a_{ij}e_{j}y_{1}^{(c_{1})}\cdots
y_{i}^{(c_{i}-1)}\cdots y_{t}^{(c_{t})}\,.$

Now assume $\ell\ell_{R}(M)<\operatorname{ord}(R)=n$, then in the Cohen
presentation above each $f_{i}\in\mathfrak{m}_{Q}^{n}$ and so we can assume
$a_{ij}\in\mathfrak{m}^{n-1}$. In particular, from Eq. 2 it follows that

$\partial(y_{1}^{(c_{1})}\cdot\cdots\cdot y_{t}^{(c_{t})}\otimes
m)=\sum_{i=1}^{t}\sum_{j=1}^{d}a_{ij}e_{j}y_{1}^{(c_{1})}\cdots
y_{i}^{(c_{i}-1)}\cdots y_{t}^{(c_{t})}\otimes m=0$

as each $a_{ij}\in\mathfrak{m}^{n-1}\subseteq{\operatorname{ann}}_{R}(M).$
Hence, for any $m\in M\smallsetminus\mathfrak{m}M$, we have a cycle

$y_{1}^{(c_{1})}\cdots y_{t}^{(c_{t})}\otimes m$

that cannot be a boundary as $\partial(R\langle
X\rangle\otimes_{R}M)\subseteq\mathfrak{m}R\langle X\rangle\otimes_{R}M$.
Therefore, we have obtained the inequality below:

$\beta^{R}_{i}(M)=\operatorname{rank}_{k}{\operatorname{Tor}}^{R}_{i}(k,M)=\operatorname{rank}_{k}\operatorname{H}_{i}(R\langle
X\rangle\otimes_{R}M)\geqslant\operatorname{rank}_{k}(\Gamma_{i}\otimes_{k}M/\mathfrak{m}M)$

where $\Gamma=k\langle X_{2}\rangle$, the free divided power algebra on the
degree two variables $X_{2}$. The Hilbert series for $\Gamma$ (over $k$) is
$(1-z^{2})^{-t}$ and so the desired result on complexity has been established.

Next, assume $\ell\ell_{R}(M)<\max\operatorname{ord}(R)=n$, then in the Cohen
presentation above at least one $f_{i}$ belongs to $\mathfrak{m}^{n}_{Q}$ and
so for that $i$, we can assume $a_{ij}\in\mathfrak{m}^{n-1}$ for each $j$. In
this case, from Eq. 2 it follows that

$\partial(y_{i}^{(c)}\otimes m)=\sum_{j=1}^{t}a_{ij}e_{j}y_{i}^{(c)}\otimes
m=0$

as each $a_{ij}\in\mathfrak{m}^{n-1}\subseteq{\operatorname{ann}}_{R}(M).$
Again, the minimality of $R\langle X\rangle\otimes_{R}M$ gives us cycles that
are not boundaries in each even degree:

$\\{y_{i}^{(c)}\otimes m\mid c\geqslant 0\text{ and }m\in
M\smallsetminus\mathfrak{m}M\\}\,.$

Finally, recalling the homology of $R\langle X\rangle\otimes_{R}M$ is
${\operatorname{Tor}}^{R}(k,M)$ we have shown that
$\operatorname{projdim}_{R}(M)=\infty$, as claimed. ∎

######  Remark 4.4.

Theorem 4.3 shows that when $R$ is a local complete intersection ring and $M$
is a nonzero $R$-module with $\ell\ell_{R}(M)<\operatorname{ord}(R)$, then $M$
is extremal in the sense of Avramov [2]. Roughly speaking, this means the
Betti sequence of $M$ grows at the same rate as the Betti sequence of the
residue field (the latter has maximal growth, in a precise sense, among all
finitely generated $R$-modules). In light of this and Theorem 4.3, we ask the
following.

######  Question 4.5.

For a local ring $R$, if a nonzero $R$-module $M$ has
$\ell\ell_{R}(M)<\operatorname{ord}(R)$, is $M$ extremal?

We end the article with a lower bound for Loewy lengths over complete
intersection rings. We give two proofs: the first proof is closer to the proof
of Theorem 2.1, while the second proof, suggested to us by Mark Walker, uses a
construction also used to establish the Length Conjecture for strict complete
intersection rings in [30, Corollary V.29].

######  Theorem 4.6.

Assume $R$ is a complete intersection ring of codimension $c$. If $M$ is a
nonzero Cohen-Macaulay module of finite projective dimension, then

$g\ell\ell_{R}(M)\geqslant\sum_{i=1}^{c}\operatorname{ord}(f_{i})-c+1$

where $\widehat{R}\cong Q/(f_{1},\ldots,f_{c})$ is any minimal Cohen
presentation of $R$.

######  Remark 4.7.

The bound in Theorem 4.6 is usually lower than $g\ell\ell_{R}(R)$ of a local
complete intersection ring that is not strictly Cohen-Macaulay. For example,
in [12], DeStefani showed that the following one dimensional complete
intersection $k$-algebra, with $k$ a field,

$R=\frac{k\llbracket x,y,z\rrbracket}{(x^{2}-y^{5},xy^{2}+yz^{3}-z^{5})}$

has $g\ell\ell_{R}(R)=6$ and a calculation, using Macaulay2, shows
$\operatorname{reg}(R)=6$. In particular, $R$ cannot be strict Cohen-Macaulay;
one can also see this by calculating $R^{\mathsf{g}}$. In fact, by Theorem
2.1, it follows that $R$ does not have minimal regularity (in the sense
defined in Remark 2.8). Here, the bound from Theorem 4.6 is

$\operatorname{ord}(x^{2}-y^{5})+\operatorname{ord}(xy^{2}+yz^{3}-z^{5})-1=4\,.$

######  Remark 4.8.

In Theorem 4.6, each $\operatorname{ord}(f_{i})$ is at least two and so
Theorem 4.6 implies $\ell\ell_{R}(M)\geqslant c+1$. Hence the bound from
Theorem 4.6 strengthens the already known bounds for _modules_ over complete
intersection rings in [5]; the bounds from the latter are known to hold for
the sum of the Loewy lengths of the homology modules of perfect complexes, and
is tight as the Koszul complex $K$ on the maximal ideal always has
$\sum\ell\ell_{R}\operatorname{H}_{i}(K)=c+1.$

###### First proof of Theorem 4.6.

First, we need a bit of notation; see [3] for background on dg modules in
commutative algebra.

We can assume $R$ is complete and hence $R\cong Q/(f_{1},\ldots,f_{c})$ with
$f_{1},\ldots,f_{c}$ a regular sequence in the regular ring
$(Q,\mathfrak{m}_{Q})$. Let
$\mathfrak{m}_{Q}=(t_{1},\ldots,t_{c},x_{1},\ldots,x_{d})$ where the image of
$x_{1},\ldots,x_{d}$ is a maximal regular sequence in
$\mathfrak{m}_{R}\smallsetminus\mathfrak{m}_{R}^{2}$, and write

$f_{i}=\sum_{j=i}^{c}a_{ij}t_{j}+\sum_{j=1}^{d}b_{ij}x_{j}$

with each $a_{ij}$ and $b_{ij}$ belonging to
$\mathfrak{m}_{Q}^{\operatorname{ord}(f_{i})-1}.$ Set

$\displaystyle E$ $\displaystyle=\mathrm{Kos}^{Q}(\bm{f})=Q\langle
e_{1},\ldots,e_{c}\mid\partial e_{i}=f_{i}\rangle$ $\displaystyle\tilde{A}$
$\displaystyle=E\otimes_{Q}\mathrm{Kos}^{Q}(\bm{x})=Q\langle
e_{1},\ldots,e_{c},e^{\prime}_{1},\ldots,e^{\prime}_{d}\mid\partial
e_{i}=f_{i}\,,\partial e_{i}^{\prime}=x_{i}\rangle$ $\displaystyle K$
$\displaystyle=\mathrm{Kos}^{Q}(\bm{t},\bm{x})=Q\langle
e_{1}^{\prime\prime},\ldots,e_{c}^{\prime\prime},e^{\prime}_{1},\ldots,e^{\prime}_{d}\mid\partial
e_{i}^{\prime\prime}=t_{i}\,,\partial e_{i}^{\prime}=x_{i}\rangle\,,$

which are dg $E$-modules; the $E$-actions on $E,\tilde{A}$ are the obvious
ones and the $E$ action on $K$ is given by the map on dg $Q$-algebras $E\to K$
determined by

$e_{i}\mapsto\sum_{j=i}^{c}a_{ij}e_{j}^{\prime\prime}+\sum_{j=1}^{d}b_{ij}e_{j}^{\prime}\,.$

Finally, consider the map $\alpha\colon\tilde{A}\to K$ of dg $E$-modules
extending the map above and sending each $e_{j}^{\prime}$ in $\tilde{A}$ to
$e_{j}^{\prime}$ in $K$.

Note that $\alpha_{i}=\bigwedge^{i}\alpha_{1}\colon\tilde{A}_{i}\to K_{i}$ and
now using that each $a_{ij}$, $b_{ij}$ is in
$\mathfrak{m}_{Q}^{\operatorname{ord}(f_{i})-1}$, a direct calculation shows

(3)
$\alpha(E_{c}\otimes_{Q}\mathrm{Kos}^{Q}(\bm{x}))\subseteq\mathfrak{m}_{Q}^{n}K\quad\text{where}\quad
n=\sum_{i=1}^{c}\operatorname{ord}(f_{i})-c\,.$

Also, it is well known (see, for instance, [15, Exercise 21.23]) that the cone
of

${\scriptstyle\mathsf{\Sigma}}^{c+d}\alpha^{\vee}\colon{\scriptstyle\mathsf{\Sigma}}^{c+d}K^{\vee}\to{\scriptstyle\mathsf{\Sigma}}^{c+d}\tilde{A}^{\vee}\,,$

is a $Q$-free resolution of $R/(\bm{x},\det(a_{ij}))$ where
$(-)^{\vee}={\operatorname{Hom}}_{Q}(-,Q)$. Using the self-duality of Koszul
complexes, ${\scriptstyle\mathsf{\Sigma}}^{c+d}\alpha^{\vee}$ can regarded as
a dg $E$-module map from $K$ to $\tilde{A}$; write $\tilde{\sigma}\colon
K\to\tilde{A}$ for this map using these identifications of each Koszul complex
with its dual. By Eq. 3, composing $\tilde{\sigma}$ with the projection of
$\tilde{A}\to E_{0}\otimes_{Q}\mathrm{Kos}^{Q}(\bm{x})$ factors as the map of
graded $Q$-modules:

(4)
${K}$${\tilde{A}}$${\mathfrak{m}_{Q}^{n}E_{0}\otimes_{Q}\mathrm{Kos}^{Q}(\bm{x})}$${E_{0}\otimes_{Q}\mathrm{Kos}^{Q}(\bm{x})\,;}$$\scriptstyle{\tilde{\sigma}}$

here we used that the identification of
${\scriptstyle\mathsf{\Sigma}}^{c+d}\tilde{A}$ with $\tilde{A}$, restricts to
an isomorphism

${\scriptstyle\mathsf{\Sigma}}^{c+d}(E_{c}\otimes_{Q}\mathrm{Kos}^{Q}(\bm{x}))^{\vee}\cong{\scriptstyle\mathsf{\Sigma}}^{c}E_{c}^{\vee}\otimes_{Q}{\scriptstyle\mathsf{\Sigma}}^{d}(\mathrm{Kos}^{Q}(\bm{x}))^{\vee}\cong
E_{0}\otimes_{Q}\mathrm{Kos}^{Q}(\bm{x})\,.$

Now letting $\Gamma\coloneqq R\langle y_{1},\ldots,y_{c}\rangle$ where each
$y_{i}$ is a degree two divided power variable,
$1\otimes\tilde{\sigma}\colon\Gamma\otimes_{Q}^{\tau}K\to\Gamma\otimes_{Q}^{\tau}\tilde{A}$
is a map of $R$-complexes; here, for a dg $E$-module $N$, the $R$-complex
$\Gamma\otimes_{Q}^{\tau}N$ is the construction of Eisenbud [14] and Shamash
[36] (see also [4, Section 2]). Explicitly, $\Gamma\otimes_{Q}^{\tau}N$ is the
free graded $R$-module $\Gamma\otimes_{Q}N$ with differential

$y_{1}^{(h_{1})}\cdots y_{c}^{(h_{c})}\otimes n\mapsto y_{1}^{(h_{1})}\cdots
y_{c}^{(h_{c})}\otimes\partial^{N}(n)+\sum_{i=1}^{c}y_{1}^{(h_{1})}\cdots
y_{i}^{(h_{i}-1)}\cdots y_{c}^{(h_{c})}\otimes e_{i}n\,.$

Set $F=\Gamma\otimes_{Q}^{\tau}K$, which by [4, Theorem 2.4] (see also [37]),
is the minimal free resolution of $k$ over $R$. Also, there is an isomorphism
of complexes

$\Gamma\otimes_{Q}^{\tau}\tilde{A}\cong R\langle
e_{1},\ldots,e_{c},e_{1}^{\prime},\ldots,e_{d}^{\prime},y_{1},\ldots,y_{c}\mid\partial
e_{i}=0,\,\partial e^{\prime}_{i}=x_{i},\,\partial y_{i}=e_{i}\rangle$

and so we have quasi-isomorphisms

$\Gamma\otimes_{Q}^{\tau}\tilde{A}\xrightarrow{\simeq}R\otimes_{E}\tilde{A}\cong\mathrm{Kos}^{R}(\bm{x})\xrightarrow{\simeq}R/(\bm{x})\,.$

Set $A=R\otimes_{E}\tilde{A}$ and let $\sigma$ be the composition of chain
maps

$F=\Gamma\otimes_{Q}^{\tau}K\xrightarrow{1\otimes\tilde{\sigma}}\Gamma\otimes_{Q}^{\tau}\tilde{A}\xrightarrow{\simeq}A\,.$

Forgetting differentials, $\sigma$ factors as the following map of graded
$Q$-graded modules:

${F}$${\Gamma\otimes_{Q}^{\tau}\tilde{A}}$${R\otimes_{Q}K}$${R\otimes_{Q}\tilde{A}}$${R\otimes_{E}\tilde{A}=A}$$\scriptstyle{1\otimes\tilde{\sigma}}$$\scriptstyle{\sigma}$$\scriptstyle{1\otimes\sigma}$

and so by Eq. 4 we have that $\sigma(F)\subseteq\mathfrak{m}^{n}_{R}A.$ Also,
observe that $\operatorname{H}_{0}(\sigma)\colon k\hookrightarrow R/(\bm{x})$,
is given by $1\mapsto\det(a_{ij})$, and so $\operatorname{cone}(\sigma)$ is a
free resolution of $N\coloneqq R/(\bm{x},\det(a_{ij}))$; cf. the proof of
Lemma 2.6Item 1. Therefore, the same argument used to establish Lemma 2.7
shows that for each $i$ we have isomorphisms

${\operatorname{Tor}}^{R}_{i}(M,N)\cong{\operatorname{Tor}}^{R}_{i}(M,R/(\bm{x}))\oplus{\operatorname{Tor}}^{R}_{i-1}(M,k)$

whenever $M$ is a finite length module with $\ell\ell_{R}(M)\leqslant n$;
hence, if $M$ is also assumed to be nonzero and of finite projective dimension
we reach a contradiction through applications of the Auslander–Buchsbaum
formula (again, as argued in Lemma 2.7). It remains to repeat the argument
from Corollary 2.2 to establish the desired lower bound for perfect modules of
positive dimension. ∎

Before presenting the second proof of Theorem 4.6, we need the following
construction from [21]; see also [30, Theorem V.27].

######  4.9.

Let $(Q,\mathfrak{m})$ be a Cohen-Macaulay ring and $f\in\mathfrak{m}^{d}$ a
regular element. Then for some integer $s\geqslant 1$, there exists a
filtration of $R=Q/(f)$-modules

$0=U_{0}\subseteq U_{1}\subseteq\ldots\subseteq U_{d}=R^{s}$

such that $U_{i-1}\subseteq\mathfrak{m}U_{i}$ and each $U_{i}/U_{i-1}$ is a
maximal Cohen-Macaulay (abbreviated, as usual, to MCM) $R$-module having
finite projective dimension over $Q$. The condition on $U_{i}/U_{i-1}$ is
equivalent to $\operatorname{projdim}_{Q}(U_{i}/U_{i-1})=1$.

######  Lemma 4.10.

Suppose $(Q,\mathfrak{m})$ is a Cohen-Macaulay ring and
$R=Q/(f_{1},\ldots,f_{c})$ where $f_{1},\ldots,f_{c}$ is a $Q$-regular
sequence. Set

$n=\sum_{i=1}^{c}\operatorname{ord}(f_{i})-c+1\,.$

Then for some integer $t\geqslant 1$, there exists a filtration of $R$-modules

$0=U_{0}\subseteq U_{1}\subseteq\ldots\subseteq U_{n}=R^{t}$

such that $U_{i-1}\subseteq\mathfrak{m}U_{i}$ and $U_{i}/U_{i-1}$ is an MCM
$R$-module for $1\leqslant i\leqslant n.$

###### Proof.

We induct on $c$. The base case follows by 4.9. Let
$R^{\prime}=Q/(f_{1},\ldots,f_{c-1})$ and

$n^{\prime}=\sum_{i=1}^{c-1}\operatorname{ord}(f_{i})-(c-1)+1.$

By the induction hypothesis, for some $t^{\prime}\geqslant 1$, we have a
filtration of $R^{\prime}$-modules

$0=U^{\prime}_{0}\subseteq U^{\prime}_{1}\subseteq\ldots\subseteq
U^{\prime}_{n^{\prime}}\cong(R^{\prime})^{t^{\prime}}$

where $U^{\prime}_{i-1}\subseteq\mathfrak{m}U^{\prime}_{i}$ and
$U^{\prime}_{i}/U^{\prime}_{i-1}$ is an MCM $R^{\prime}$-module. Since
$R^{\prime}$ is Cohen-Macaulay and $f=f_{c}$ regular on $R^{\prime}$ in
$\mathfrak{m}^{\operatorname{ord}(f)}R^{\prime}$, we can apply 4.9 to obtain a
filtration of $R=R^{\prime}/(f)$-modules

$0=V_{0}\subseteq V_{1}\subseteq\cdots\subseteq V_{\operatorname{ord}(f)}\cong
R^{t}$

for some $t\geqslant 1$ where each $V_{i-1}\subseteq\mathfrak{m}V_{i}$ and
$\operatorname{projdim}_{R^{\prime}}(V_{i}/V_{i-1})=1$; in particular,
$\operatorname{projdim}_{R^{\prime}}V_{1}=1$.

We claim that the following filtration of $R$-modules of length $n$ has all
the desired properties:

$U^{\prime}_{0}\otimes V_{1}\subseteq U^{\prime}_{1}\otimes
V_{1}\subseteq\cdots\subseteq U^{\prime}_{n^{\prime}}\otimes V_{1}\subseteq
U^{\prime}_{n^{\prime}}\otimes V_{2}\subseteq\cdots\subseteq
U^{\prime}_{n^{\prime}}\otimes V_{\operatorname{ord}(f)}\,.$

Indeed, $U_{0}^{\prime}\otimes V_{1}=0$ and $U^{\prime}_{n^{\prime}}\otimes
V_{\operatorname{ord}(f)}\cong R^{tt^{\prime}}$, and for $i\geqslant 1$ every
term in the filtration is of the form

$U_{n^{\prime}}^{\prime}\otimes V_{i}\cong(R^{\prime})^{t^{\prime}}\otimes
V_{i}\cong V_{i}^{t^{\prime}}\,;$

as a consequence, there are the desired containments
$U_{n^{\prime}}^{\prime}\otimes
V_{i-1}\subseteq\mathfrak{m}U_{n^{\prime}}\otimes V_{i}$ and

$\operatorname{projdim}_{Q}(U_{n^{\prime}}^{\prime}\otimes
V_{i}/U_{n^{\prime}}^{\prime}\otimes
V_{i-1})=\operatorname{projdim}_{Q}(V_{i}^{t^{\prime}}/V_{i-1}^{t^{\prime}})=c$

where the last equality used
$\operatorname{projdim}_{R^{\prime}}(V_{i}/V_{i-1})=1;$ thus each of these
subquotients are MCM $R$-modules. Therefore, it remains to verify the first
$n^{\prime}$-steps satisfy the desired properties as well.

To this end, for each $i$, as $U^{\prime}_{i}/U^{\prime}_{i-1}$ is MCM over
$R^{\prime}$ and $\operatorname{projdim}_{R^{\prime}}V_{1}<\infty$ it follows
that
${\operatorname{Tor}}^{R^{\prime}}_{>0}(U^{\prime}_{i}/U^{\prime}_{i-1},V_{1})=0$
and hence the maps along the bottom are injective:

${{\mathfrak{m}U^{\prime}_{i}\otimes_{R^{\prime}}V_{1}}}$${{U^{\prime}_{i-1}\otimes_{R^{\prime}}V_{1}}}$${{U^{\prime}_{i}\otimes_{R^{\prime}}V_{1}}}$

and so $U^{\prime}_{i-1}\otimes
V_{1}\subseteq\mathfrak{m}U^{\prime}_{i}\otimes
V_{1}=\mathfrak{m}(U^{\prime}_{i}\otimes V_{1})$. It only remains to observe

$\displaystyle\operatorname{depth}(U^{\prime}_{i}/U^{\prime}_{i-1})$
$\displaystyle=\operatorname{depth}(U^{\prime}_{i}/U^{\prime}_{i-1}\otimes^{\mathsf{L}}_{R^{\prime}}V_{1})+\operatorname{projdim}_{R^{\prime}}(V_{1})$
$\displaystyle=\operatorname{depth}(U^{\prime}_{i}/U^{\prime}_{i-1}\otimes_{R^{\prime}}V_{1})+1$
$\displaystyle=\operatorname{depth}(U^{\prime}_{i}\otimes
V_{1}/U^{\prime}_{i-1}\otimes V_{1})+1$

where the first equality uses the derived depth formula [25, Corollary 2.2],
and second equality uses that $\operatorname{projdim}_{R^{\prime}}(V_{1})=1$
and
${\operatorname{Tor}}^{R^{\prime}}_{>0}(U^{\prime}_{i}/U^{\prime}_{i-1},V_{1})=0$.
We need only note that $U_{i}^{\prime}/U_{i-1}^{\prime}$ is an MCM
$R^{\prime}$-module to deduce that $U^{\prime}_{i}\otimes
V_{1}/U^{\prime}_{i-1}\otimes V_{1}$ is MCM over $R$. ∎

###### Second proof of Theorem 4.6.

We can assume $R\cong Q/(f_{1},\ldots,f_{c})$ with $f_{1},\ldots,f_{c}$ a
regular sequence in the regular ring $Q$. By Lemma 4.10, there exists a
filtration by $R$-submodules

$0=U_{0}\subseteq U_{1}\subseteq\ldots\subseteq
U_{n}=R^{t}\,,\quad\text{with}\quad
n=\sum_{i=1}^{c}\operatorname{ord}(f_{i})-c+1\,,$

such that each $U_{i-1}\subseteq\mathfrak{m}U_{i}$ and $U_{i}/U_{i-1}$ is an
MCM $R$-module, and $t\geqslant 1$. Arguing as in the proof of Lemma 4.10,
since $\operatorname{projdim}_{R}M<\infty$ and each subquotient of the
filtration is an MCM $R$-module, we obtain inclusions $U_{i-1}\otimes
M\subseteq\mathfrak{m}(U_{i}\otimes M)$ for each $i$. In particular,

$0\neq U_{1}\otimes M\subseteq\mathfrak{m}U_{2}\otimes
M\subseteq\ldots\subseteq\mathfrak{m}^{n-1}U_{n}\otimes
M=\mathfrak{m}^{n-1}M^{t}\,,$

and thus, $\ell\ell_{R}(M)\geqslant n$. Again, it remains to repeat the
argument from Corollary 2.2 to establish the desired lower bound for perfect
modules of positive dimension. ∎

In fact, the second proof establishes a bound in the relative setting. That is
to say, when a Cohen-Macaulay local ring $R$ admits a deformation, there is
the following uniform lower bound on the Loewy length of nonzero modules of
finite projective dimension.

######  Theorem 4.11.

Assume a Cohen-Macaulay local ring $R$ is a deformation:

$R\cong Q/(f_{1},\ldots,f_{c})\quad\text{with}\quad f_{1},\ldots,f_{c}\text{ a
regular sequence.}$

For any nonzero Cohen-Macaulay module $M$ of finite projective dimension, we
have

$g\ell\ell_{R}(M)\geqslant\sum_{i=1}^{c}\operatorname{ord}(f_{i})-c+1\,.\qed$

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