# On liftings of modules of finite projective dimension

Nawaj KC  Department of Mathematics, University of Nebraska, Lincoln, NE
68588, U.S.A<EMAIL_ADDRESS>and  Andrew J. Soto Levins  Department of
Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.
<EMAIL_ADDRESS>
###### Abstract.

We introduce and study a notion of dimly liftable modules; these are modules
that are liftable to the right dimension to a regular local ring. We establish
special new cases of Serre’s positivity conjecture over ramified regular local
rings by proving it for dimly liftable modules. Furthermore, we show that the
length of a nonzero finite length dimly liftable module is bounded below by
the Hilbert-Samuel multiplicity of the local ring. This establishes special
cases of the Length Conjecture of Iyengar-Ma-Walker.

## Introduction

Suppose $R$ is a noetherian local ring and $Q$ is a regular local ring that
surjects onto $R$. Classically, a finitely generated $R$-module $M$ is said to
derived lift to $Q$ if there exists a $Q$-module $L$ such that
$L\otimes_{Q}^{\mathsf{L}}R\simeq M$. When this occurs, $L$ is called a lift
of $M$ and the minimal $R$-free resolution of $M$ lifts to the minimal
$Q$-free resolution of $L$. In this situation, many properties of an
$R$-module are dictated by the properties of its lift. Since over a regular
ring, all modules have finite projective dimension, one expects that modules
over an arbitrary local ring rarely lift. In this paper, we introduce a
natural, more geometric notion of lifting $R$-modules that generalizes this
derived lifting.

We begin by showing that if a finitely generated $Q$-module $L$ is a lift of
an $R$-module $M$, then we have the following equality of Krull dimensions;
see Proposition 2.3:

$\dim L+\dim R=\dim Q+\dim M.$

Motivated by this result, we say a $Q$-module $L$ is a dim lift of an
$R$-module $M$ if $L\otimes_{Q}R\cong M$ and the equality of dimension as
above is achieved. When $M$ admits such a lift, we say $M$ is dimly liftable.
While liftable modules must have finite projective dimension, dimly liftable
modules have no such restriction. In fact, dimly liftable modules exist in
abundance. Producing modules of finite projective dimension over a singular
ring is harder but we show there exists a two dimensional hypersurface
singularity that admits a module of finite length and finite projective
dimension that is dimly liftable but unliftable; cf. Example 2.16.

Serre introduced the following fundamental notion of intersection multiplicity
over regular local rings in [33]. Given finitely generated modules $M$ and $N$
over a regular local ring $R$ whose supports intersect at the maximal ideal,
we define

$\chi(M,N)\coloneqq\chi(M\otimes_{R}^{\mathsf{L}}N)=\sum_{i\geq
0}(-1)^{i}\ell_{R}({\operatorname{Tor}}_{i}(M,N)).$

Serre proved that if $R$ is an unramified regular ring and $\dim M+\dim N=\dim
R$, then $\chi(M,N)>0$. He conjectured that this holds in general but it
remains wide open when $R$ is a ramified regular local ring. In fact, the idea
of lifting modules was initially motivated by the fact that Serre’s positivity
conjecture would hold [3] if one of the intersecting modules lifts to an
unramified regular local ring. However, unliftable modules in this setting
were subsequently found in [11]. Nevertheless, there exist unliftable modules
that dimly lift to an unramified regular local ring; see Example 2.8. The
following theorem establishes the positivity conjecture for such modules; see
Theorem 2.9.

######  Theorem A.

Suppose $R$ is a ramified regular local ring and $M$, $N$ finitely generated
$R$-modules such that ${\ell(M\otimes_{R}N)<\infty}$ and ${\dim M+\dim N=\dim
R}$. Fix a surjection $Q\twoheadrightarrow R=Q/f$ where $Q$ is an unramified
regular local ring. If $M$ is dimly liftable to $Q$, then $\chi(M,N)>0$.

Interestingly, we have not been able to produce an example of a module over a
ramified regular local ring that is not dimly liftable to an unramified
regular local ring; cf. 2.11. Our initial interest in liftable and dimly
liftable modules arose in trying to understand lengths of modules of finite
projective dimension over local singularities.

One motivation for studying lengths of modules of finite projective dimension
is to understand the behavior of the Hilbert Samuel multiplicity under local
flat extensions. The long-standing Lech’s conjecture [21] contends that if
$R\to S$ is a faithfully flat extension of local rings, then $e(R)\leq e(S)$.
The intuition is that under a faithfully flat extension, the singularity
cannot get better. However, beyond dimension three [24] or the graded setting
[25], this question remains unanswered. Consider then the following recent
conjecture of Iyengar, Ma, and Walker.

######  Conjecture B.

[14, Conjecture 1] If $M$ is a nonzero module of finite length and finite
projective dimension over a noetherian local ring $R$, then $\ell_{R}(M)\geq
e(R)$.

Due to Ma, an affirmative answer to this conjecture settles Lech’s conjecture
for Cohen-Macaulay rings; see [23, Chapter V] for details. Unfortunately, B is
also only known in special cases. Iyengar, Ma, and Walker establish a stronger
version of B for standard graded $k$-algebras localized at the homogeneous
maximal ideal where $k$ is a perfect field of characteristic $p>0$; cf.
Theorem 3.10. On the other hand, without the graded structure, B is not known
in full generality even in dimension two. The following theorem is thus of
interest; see Corollary 3.12.

######  Theorem C.

Suppose $M$ is a Cohen-Macaulay module of finite projective dimension over a
noetherian local ring $R$ that is a quotient of a regular local ring $Q$. If
$M$ dimly lifts to a $Q$-module $L$ where $Q$ is an unramified regular local
ring, then

$e(M)\geq e(L)e(R).$

In particular, C establishes B for the class of dimly liftable modules of
finite projective dimension. It is well known that if $I$ is a perfect ideal
of a Cohen-Macaulay local ring $R$ in the linkage class of a complete
intersection, then $R/I$ lifts, and hence dimly lifts, to an unramified
regular ring; see Example 2.1. Hence, for such ideals C establishes that
$e(R/I)\geq e(R)$. However, in general, there are obstructions to lifting
modules of finite projective dimension to a regular local ring and we will
give many such examples in Section 4. Notably, we report a class of perfect
modules of projective dimension two over a Cohen Macaulay ring that cannot
lift to any regular local ring; the main idea is due to Heitmann. We also show
there exist modules of finite length and finite projective dimension over
singular rings that do not dimly lift; see Example 4.2.

Due to the New Intersection Theorem of Roberts [29], the only rings that admit
a module of finite length and finite projective dimension are Cohen-Macaulay.
To generalize B to all local rings, we consider the so called short complexes.
A short complex $F$ over a local ring $R$ is a complex

$F:0\to F_{d}\to\cdots\to F_{0}\to 0$

such that $d=\dim R$, each $F_{i}$ is a finite free $R$-module, and
$\ell_{R}(\operatorname{H}(F))<\infty$. For such a complex $F$ with nonzero
homology, Iyengar, Ma, and Walker also conjecture that $\chi_{\infty}(F)\geq
e(R)$ where $\chi_{\infty}(-)$ is the Dutta multiplicity; see Conjecture 2.
Validity of this conjecture implies Lech’s conjecture in full generality [14,
Proposition 8.3]. Motivated by the module case, we introduce the following
notion of dimly liftable short complexes. A short complex $F$ dimly lifts to a
regular local ring $Q$ if the finite length $R$-module
$\operatorname{H}_{0}(F)$ dimly lifts to $Q$. On this topic, we prove the
following; see Theorem 3.19.

######  Theorem D.

If $F$ is a short complex over a local ring $R$ of prime characteristic $p>0$
with nonzero homology that dimly lifts to an unramified regular local ring,
then

$\chi_{\infty}(F)\geq e(R).$

######  Acknowledgements.

It is a pleasure to thank Mark Walker and Jack Jeffries for providing
innumerable ideas and insights and their comments on preliminary drafts. We
are thankful to Linquan Ma for several discussions, helping us clarify many of
our results, and encouraging us to consider some key generalizations of our
results. We are also grateful to him and Mark Walker for pointing out the
argument in Example 4.2. We thank Ray Heitmann for useful discussions and
Example 4.4. We thank Josh Pollitz for his feedback on a preliminary draft,
many fruitful discussions, and also the ideas behind 2.14, and Ryan Watson for
many enjoyable discussions.

The first author was supported through NSF grants DMS-2200732 and DMS-2044833.
This material is also based upon work supported by the National Science
Foundation under Grant No. DMS-1928930 and by the Alfred P. Sloan Foundation
under grant G-2021-16778, while the authors were in residence at the Simons
Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley,
California, during the Spring 2024 semester.

##  1\. Preliminaries

In this section we establish some notation and give a quick introduction to
the notion of intersection multiplicity over a local ring. Throughout this
paper, all rings are noetherian local rings. We will say a local ring
$(R,\mathfrak{m})$ is unramified if it is either of equal characteristic or it
is of mixed characteristic $(0,p)$ for a prime integer $p$ with
$p\in\mathfrak{m}\setminus\mathfrak{m}^{2}$.

######  1.1.

For basic results on Hilbert-Samuel multiplicities, we refer to [2, 33];
alternatively, see [14, Section 2] as we use similar notation. Given a local
ring $(R,\mathfrak{m})$ and a nonzero finitely generated $R$-module $M$, the
Hilbert-Samuel multiplicity of $M$ is

$e(M):=(\dim M)!\lim_{n\to\infty}\frac{\ell_{R}(M/\mathfrak{m}^{n}M)}{n^{\dim
M}}.$

This limit exists and is strictly positive. When $M=R$ itself, $e(R)$ is the
Hilbert-Samuel multiplicity of the local ring $R$.

######  1.2.

Consider a local ring $R$ and two finitely generated $R$-modules $M$ and $N$
such that their supports intersect at the maximal ideal of $R$. This is
equivalent to the condition ${\ell_{R}(M\otimes_{R}N)<\infty}$. Simple
examples indicate that the integer $\ell_{R}(M\otimes_{R}N)$ fails to
correctly count the multiplicty of this intersection. Serre suggested that
over regular rings one can extract the correct notion by considering the
derived tensor product of $M$ and $N$. Assuming
$\operatorname{projdim}_{R}(M)<\infty$, Serre defined the following
intersection multiplicity pairing

$\chi^{R}(M,N):=\chi(M\otimes_{R}^{\mathsf{L}}N)=\sum_{i\geq
0}(-1)^{i}\ell_{R}({\operatorname{Tor}}_{i}(M,N)).$

When $R$ is an unramified regular local ring, Serre proved several properties
of this pairing that is expected of a good intersection theory. By [33, Ch.V,
Thm 1], we have

  1. (i)

Dimensional inequality: $\dim M+\dim N\leq\dim R$,

  2. (ii)

Non-negativity: $\chi(M,N)\geq 0$,

  3. (iii)

Vanishing: If $\dim M+\dim N<\dim R$, then $\chi(M,N)=0$,

  4. (iv)

Positivity: If $\dim M+\dim N=\dim R$, then $\chi(M,N)>0$.

Dimensional inequality holds for all regular rings [33, Chapter V, Theorem 3].
Although positivity remains an open conjecture for ramified regular local
rings, non-negativity (due to Gabber [32]) and the vanishing property (due to
Gillet-Soulé [8] and Roberts [28] independently) have been proven for all
regular rings. The main property of this invariant that we will utilize is the
following theorem of Serre (in equal characteristic) and Skalit (in the
unramified mixed characteristic).

######  Theorem 1.3.

[34, Theorem A], [33, Ch.V, Thm 1 Complement] Suppose $R$ is an unramified
regular local ring and $M$, $N$ are finitely generated $R$-modules such that
$\ell_{R}(M\otimes_{R}N)<\infty$. If $\chi(M,N)>0$, then $\chi(M,N)\geq
e(M)e(N).$

######  1.4.

In prime characteristic $p>0$ we will also utilize the Frobenius intersection
multiplicities introduced by Dutta [5]. For the rest of this section, we
consider a local ring $R$ of characteristic $p>0$. Let $f\colon R\to R$ denote
the Frobenius endomorphism on $R$. We will assume that our rings of
characteristic $p>0$ are $F$-finite throughout this paper, i.e. $f:R\to R$ is
a finite map of $R$-modules. For $e\in\mathbb{N}$, consider the $e$th iterate
of the Frobenius map $f^{e}(r)=r^{p^{e}}$. Let ${}^{f^{e}}R$ denote $R$-$R$
bimodule with left action via $f^{e}$ and the canonical right action. The
Frobenius functor on the category of $R$-modules is defined as
$F^{e}(-)=-\otimes_{R}\,^{f^{e}}R$ where the $R$-module action is from the
right factor. See [27, Chapter 1] for more details. Suppose $R$ is a local
ring of prime characteristic $p>0$ and $M$, $N$ two finitely generated
$R$-modules such that $\ell_{R}(M\otimes N)<\infty$ and
$\operatorname{projdim}_{R}M<\infty$. Dutta defined the following Frobenius
intersection multiplicity pairing whenever the limit exists.

$\chi_{\infty}(M,N):=\lim_{e\to\infty}\frac{\chi(F^{e}(M),N)}{p^{e\operatorname{codim}M}}.$

If $R$ is a local ring such that the vanishing property as above holds, then
Dutta proved that $\chi_{\infty}(M,N)=\chi(M,N)$ [5, 1.3 Remark]. If $M$ is an
$R$-module of finite length and finite projective dimension, then the Dutta
multiplicity of $M$ is defined as
$\chi_{\infty}(M)\coloneqq\chi_{\infty}(M,R)$. In general,
$\chi_{\infty}(M)\not=\ell_{R}(M)$; see Example 4.3.

##  2\. Dimly liftable modules

Throughout this section, we fix a surjection of local rings $\pi\colon
Q\twoheadrightarrow R$. An $R$-complex is given the structure of $Q$-complex
via restriction of scalars along $\pi$. All the modules considered are
finitely generated.

###  2.1. Liftability.

A finitely generated $R$-module $M$ (derived) lifts to $Q$ if there exists a
finitely generated $Q$-module $L$ such that $L\otimes^{\mathsf{L}}_{Q}R\simeq
M$. We say $L$ is a (derived) lift of $M$. This means that $L\otimes_{Q}R\cong
M$ as modules and ${\operatorname{Tor}}^{Q}_{i}(L,R)=0$ for $i>0$. In other
words, the minimal $Q$-free resolution of $L$ tensors down to the minimal
$R$-free resolution of $M$. If $\operatorname{projdim}_{Q}(L)<\infty$, we have
$\operatorname{projdim}_{R}(M)<\infty$. This is a classical definition and is
related to “Grothendieck’s lifting problem”; see, for example, [3, 4, 11, 16,
17, 27]. It is now well-known that modules, even over a regular local rings,
do not lift in general; see Example 2.8 for a particular one. In Section 4, we
will report several examples and obstructions to liftability of modules over
singular rings to regular rings. The following is one class of modules of
finite projective dimension that lift to a regular ring.

######  Example 2.1.

(Licci ideals) Suppose $R$ is a Cohen-Macaulay local ring and fix a surjection
$Q\twoheadrightarrow R$ where $Q$ is an unramified regular local ring. Two
perfect ideals $I$ and $J$ of $R$ are said to be linked by a complete
intersection, denoted $I\sim J$, if there exists a regular sequence
$\bm{x}=x_{1},\ldots,x_{c}$ such that $J=(\bm{x}):I$ and $I=(\bm{x}):J$.
Jorgensen has shown that if $I\sim J$, $R/I$ lifts to $Q$ if and only if $R/J$
also lifts to $Q$ [17, Proposition 2.3]. Jorgensen assumes $R=Q/(x)$ and $R$
is Gorenstein, but given our definition of linkage the proof applies at the
level of generality stated.

We say $I$ is in the linkage class of a complete intersection, or is licci, if
there exists a finite sequence of links $I=I_{0}\sim I_{1}\sim\cdots\sim J$
where $J=(\bm{x})$ is an ideal generated by a regular sequence. Since the
resolution of $R/(\bm{x})$ is the Koszul complex, it lifts to $Q$, whence any
licci ideal lifts. It is well known that any perfect ideal of codimension two
and any perfect, Gorenstein ideal of codimension three are licci. More
recently, in equal characteristic zero, Gurrieri, Ni, and Weyman have shown
that many classes of perfect ideals of codimension three are licci [9, Theorem
5.2].

###  2.2. Dim liftability.

We now introduce the main objects of interest in this paper. We begin by
recording a preliminary lemma.

######  Lemma 2.2.

Suppose $R$ is a local ring, $M$ is an $R$-module of finite projective
dimension and $N$ is an $R$-complex with bounded homology. If
$\ell(M\otimes_{R}\operatorname{H}_{i}(N))<\infty$ for all $i$,

$\chi(M\otimes_{R}^{\mathsf{L}}N)=\sum_{i}(-1)^{i}\chi(M\otimes_{R}^{\mathsf{L}}\operatorname{H}_{i}(N)).$

###### Proof.

We induct on the number of nonzero homology modules of $N$. The base case is
clear. Suppose $n=\max\\{i:\operatorname{H}_{i}(N)\not=0\\}$. We have an exact
triangle in $\operatorname{\mathsf{D}}(R)$, the derived category of
$R$-modules.

${\scriptstyle\mathsf{\Sigma}}^{n}\operatorname{H}_{n}(N)\xrightarrow{\iota}N\to
C\to$

Hence, by additivity of $\chi(M\otimes_{R}^{\mathsf{L}}-)$ on triangles in
$\operatorname{\mathsf{D}}(R)$, we get

$\chi(M\otimes_{R}^{\mathsf{L}}N)=\chi(M\otimes_{R}^{\mathsf{L}}C)+\chi(M\otimes_{R}^{\mathsf{L}}{\scriptstyle\mathsf{\Sigma}}^{n}\operatorname{H}_{n}(N)).$

Now by inductive hypothesis, we get

$\chi(M\otimes_{R}^{\mathsf{L}}N)=\sum_{i\geq
0}\chi(M\otimes_{R}^{\mathsf{L}}{\scriptstyle\mathsf{\Sigma}}^{i}\operatorname{H}_{i}(N)).$

since $\operatorname{H}_{n}(C)=0$ and
$\operatorname{H}_{j}(C)=\operatorname{H}_{j}(N)$ for $j\not=n$. The claimed
equality follows as
$\chi(M\otimes_{R}^{\mathsf{L}}{\scriptstyle\mathsf{\Sigma}}^{i}\operatorname{H}_{i}(N))=(-1)^{i}\chi(M\otimes_{R}^{\mathsf{L}}\operatorname{H}_{i}(N))$.
∎

######  Proposition 2.3.

Suppose a finitely generated $Q$-module $L$ is a lift of a finitely generated
$R$-module $M$. If $Q$ is regular, then

$\dim L+\dim R=\dim Q+\dim M.$

###### Proof.

Suppose $\dim M=r$. Choose $\bm{x}=x_{1},\ldots,x_{r}$ in the maximal ideal of
$R$ which is a parameter sequence on both $M$ and $R$. Since
$L\otimes_{Q}R/(\bm{x})\cong M/(\bm{x})M$ is of finite length, we have $\dim
L+\dim R\leq\dim Q+\dim M$ by Serre’s dimensional inequality; see 1.2. Let
$K_{R}=K(\bm{x})$ be the Koszul complex on $\bm{x}$ over $R$. As $L$ is a lift
of $M$, we have the following quasi-isomorphism of complexes

$L\otimes^{\mathsf{L}}_{Q}K_{R}\simeq M\otimes^{\mathsf{L}}_{R}K_{R}.$

As $\operatorname{H}_{i}(K_{R})$ is annihilated by $(\bm{x})$ as an
$R$-module,
${\operatorname{Supp}}(\operatorname{H}_{i}(K_{R}))\subseteq{\operatorname{Supp}}(R/(\bm{x}))$,
whence $\dim\operatorname{H}_{i}(K_{R})\leq\dim R/(\bm{x})$ and
$L\otimes_{Q}\operatorname{H}_{i}(K_{R})$ has finite length for all $i$. By
Lemma 2.2, we have

$\chi(L\otimes_{R}^{\mathsf{L}}K_{R})=\sum_{i\geq
0}(-1)^{i}\chi(L\otimes_{R}^{\mathsf{L}}\operatorname{H}_{i}(K_{R})).$

By way of contradiction, assume ${\dim L+\dim R<\dim Q+\dim M}$. Equivalently,
$\dim L+\dim R/(\bm{x})<\dim Q$, whence $\dim
L+\dim\operatorname{H}_{i}(K_{R})<\dim Q$ for all $i$. The vanishing theorem
of Roberts and Gillet-Soulé (see 1.2) implies
$\chi(L\otimes^{\mathsf{L}}_{Q}K_{R})=0$. This is a contradiction as
$\chi(L\otimes^{\mathsf{L}}_{Q}K_{R})=\chi(M\otimes^{\mathsf{L}}_{R}K_{R})=\chi(\bm{x};M)=e(\bm{x};M)>0$
by Serre’s theorem [33, Chapter IV, Theorem 1]. This completes our proof. ∎

######  Remark 2.4.

The proof above actually generalizes the conclusion of Proposition 2.3 to the
case when $Q$ is a hypersurface and $R$ is Cohen-Macaulay. Indeed, the
dimensional inequality of Serre is known to hold over hypersurfaces by the
work of Hochster [12]. Furthermore, since $R$ is Cohen-Macaulay, $K_{R}\simeq
R/(\bm{x})$ and as a $Q$-module $R/(\bm{x})$ has finite projective dimension.
In fact, assuming $R$ is Cohen-Macaulay, if $Q$ is a local ring such that the
dimensional inequality and vanishing conjectures hold, then the proof of
Proposition 2.3 establishes the analogue of the derived depth formula for
Krull dimensions. This raises the following question.

######  Question 2.5.

Suppose $Q\twoheadrightarrow R$ is a surjection of noetherian local Cohen-
Macaulay rings. Suppose a finitely generated $R$-module of finite projective
dimension lifts to a $Q$-module $L$, i.e. $L\otimes^{\sf L}_{Q}R\simeq M$. Do
we have $\dim L+\dim R=\dim Q+\dim M$ ?

######  2.6.

Suppose a $Q$-module $L$ is a lift of an $R$-module $M$, i.e.
$L\otimes^{\mathsf{L}}_{Q}R\simeq M$. If
$\operatorname{projdim}_{R}(M)<\infty$, by the derived depth formula [13,
Corollary 2.2],

$\operatorname{depth}L+\operatorname{depth}R=\operatorname{depth}Q+\operatorname{depth}M.$

If $Q$ is regular, by Proposition 2.3, $\dim L+\dim R=\dim Q+\dim M$ whence

$(\dim L-\operatorname{depth}L)+(\dim R-\operatorname{depth}R)=(\dim
M-\operatorname{depth}M).$

Therefore, when $Q$ is regular and $R$ is Cohen-Macaulay, if $L$ is a lift of
$M$,

$\dim L-\operatorname{depth}L=\dim M-\operatorname{depth}M.$

In other words, $M$ is Cohen-Macaulay if, and only if, so is its lift $L$.

Proposition 2.3 motivates the following weaker, more geometric notion of
lifting.

######  Definition 2.7.

A finitely generated $Q$-module $L$ is a dim lift of a finitely generated
$R$-module $M$ if $L\otimes_{Q}R\cong M$ as modules and $\dim L+\dim R=\dim
Q+\dim M$. If $M$ admits such a lift, we say $M$ is dimly liftable to $Q$.

###  2.3. Dimly liftable modules over regular rings.

The motivation behind lifting modules over a regular ring comes from the
“Grothendieck’s lifting problem”. Suppose $Q$ is an unramified regular local
ring and $R$ is a ramified regular local ring. If every $R$-module $M$ lifts
to $Q$, then the still open Serre’s positivity conjecture (see 1.2) would be
resolved; see [3] for an argument. However, counterexamples were subsequently
found in [11] (see also [4, 16]).

######  Example 2.8.

(Hochster) [11, Example 2] Let $Q=\mathbb{Z}_{(2)}[[x,y,z,a,b,c]]$ and let
$f=2+u$ where $u=x^{2}+x^{4}+y^{4}$. Consider the ramified regular local ring
$R=Q/f$ and the surjection $Q\twoheadrightarrow R$. Let
${I=(x^{2},y^{2},z^{2},a^{2},b^{2},c^{2},xa+yb+zc)}$. Note that $M=R/(I)$ is
an $R$-module of finite length. Hochster proved that $M$ does not lift to $Q$.
It is dimly liftable however; $Q/I$ is a dim lift of $M$. Indeed,
$I\subseteq\mathfrak{b}$ where $\mathfrak{b}=(x,y,z,a,b,c)$, so $V(I)\supseteq
V(\mathfrak{b})$. Clearly, $V(\mathfrak{b})$ is $1$-dimensional so, $\dim
Q/I=\dim V(I)\geq\dim V(\mathfrak{b})=1$. Note that the dimension is also at
most one, and therefore $\dim Q/I+\dim R=\dim Q$.

######  Theorem 2.9.

Suppose $R$ is a ramified regular local ring and $M$, $N$ two finitely
generated $R$-modules such that ${\ell(M\otimes_{R}N)<\infty}$ and ${\dim
M+\dim N=\dim R}$. Fix a surjection $\pi\colon Q\twoheadrightarrow R=Q/f$
where $Q$ is an unramified regular local ring. Suppose $M$ is dimly liftable
to $Q$. Then, $\chi^{R}(M,N)>0$.

###### Proof.

Suppose a $Q$-module $L$ is a dim lift of $M$. Let $G\xrightarrow{\simeq}L$
denote the minimal $Q$-free resolution of $L$ and set $\overline{G}\coloneqq
G\otimes_{Q}R$. As
$\operatorname{H}_{i}(\overline{G})={\operatorname{Tor}}^{Q}_{i}(L,R)$, we
have that $\overline{G}$ is a perfect complex over $R$ with
$\operatorname{H}_{i}(\overline{G})=0$ for $i\geq 2$ and
$\operatorname{H}_{0}(\overline{G})\cong M$. Furthermore,
${\operatorname{Supp}}(\operatorname{H}_{1}(\overline{G}))\cap{\operatorname{Supp}}(N)\subseteq{\operatorname{Supp}}(M)\cap{\operatorname{Supp}}(N)=\\{\mathfrak{m}\\}$
where $\mathfrak{m}$ is the maximal ideal of $R$. Thus we may apply Lemma 2.2
to get

$\chi(\overline{G}\otimes_{R}^{\mathsf{L}}N)=\chi(M\otimes_{R}^{\mathsf{L}}N)-\chi(\operatorname{H}_{1}(\overline{G})\otimes_{R}^{\mathsf{L}}N).$

Since $L$ is a dim lift of $M$, $\dim L+\dim N=\dim Q$ so by Serre’s
positivity theorem over unramified regular local rings (see 1.2),

$\chi^{Q}(L,N)=\chi(\overline{G}\otimes_{R}^{\mathsf{L}}N)>0.$

On the other hand, by Gabber’s nonnegativity theorem [32] over ramified
regular local rings,
$\chi(\operatorname{H}_{1}(\overline{G})\otimes_{R}^{\mathsf{L}}N)\geq 0$.
Therefore, $\chi^{R}(M,N)>0$ as claimed. ∎

######  Remark 2.10.

As in Example 2.8, there exist unliftable modules over ramified regular local
rings that dimly lift to an unramified regular local ring. Hence Theorem 2.9
establishes Serre’s positivity conjecture in new cases. The obstructions to
dim liftability we have discovered so far are only applicable for modules over
singular rings, and so we have not been able to find an example of a module
over a ramified regular local ring that does not dimly lift in the setting of
Theorem 2.9. This raises the following question.

######  Question 2.11.

Fix a surjection $Q\twoheadrightarrow Q/f=R$ where $Q$ and $R$ are unramified
and ramified regular local rings respectively. Does there exist a prime ideal
$\mathfrak{p}$ of $R$ such that $R/\mathfrak{p}$ does not dimly lift to $Q$?

###  2.4. Dimly liftable modules over local singularities.

We now focus our attention on dimly liftable modules over local singularities.
As we alluded to earlier, an $R$-module $M$ that lifts to a regular ring $Q$
must have finite projective dimension and this is a strong obstruction when
$R$ is not regular. However, dimly liftable modules do not have this
restriction, so they exist in abundance.

######  Example 2.12.

In fact, it is easy to produce examples of dimly liftable modules. Suppose
$(Q,\mathfrak{n})$ is a regular ring and $L$ is some $Q$-module of dimension
$r$. Choose a sequence of elements $\bm{x}=x_{1},\ldots,x_{r}\in\mathfrak{n}$
that is a parameter sequence on $L$. Set $M\coloneqq L/(\bm{x})L$. Then $M$ is
a module of finite length over $R=Q/(\bm{x})$ that is dimly liftable to $Q$.

######  Example 2.13.

There exist dimly liftable but not liftable finite length $R$-modules. Suppose
$R$ is a non-Cohen-Macaulay ring such that $\dim R=d$. Given some system of
parameters $\bm{x}=x_{1},\ldots,x_{d}$ on $R$, we have that
$\operatorname{projdim}_{R}(R/(\bm{x}))=\infty$ by the New Intersection
theorem; see 3.2. Therefore, $M=R/(\bm{x})$ is an $R$-module of finite length
that cannot lift to any regular ring. However, it dimly lifts to any regular
ring $Q$ such that $\pi\colon Q\twoheadrightarrow R$. Indeed, we may choose
arbitrary $\widetilde{x}_{i}\in Q$ such that $\pi(\widetilde{x}_{i})=x_{i}$.
Let $\widetilde{\bm{x}}=\widetilde{x}_{1},\ldots,\widetilde{x}_{d}$ and let
$L=Q/(\widetilde{\bm{x}})$. As $L\otimes_{Q}R=M$, we have by Serre’s
dimensional inequality, $\dim L+\dim R\leq\dim Q$. We also have $\dim
L\geq\dim Q-d$. That is, $\dim L+\dim R=\dim Q$. Hence $L$ is a dim lift of
$M$.

We now show that there exist a dimly liftable but unliftable module of finite
length and finite projective dimension over a local singularity.

######  2.14.

We begin with outlining an obstruction to liftability of modules over complete
intersection rings via the notion of cohomological operators due to Gulliksen
[10] and Eisenbud [7]. Suppose $R$ is a local complete intersection ring and
assume $R$ is complete. Then $R\cong Q/(f_{1},\ldots,f_{c})$ where $Q$ is a
regular ring and $f_{1},\ldots,f_{c}$ a regular sequence on $Q$. Consider a
finitely generated $R$-module $M$ and its minimal free resolution
$F\xrightarrow{\simeq}M$ over $R$.

Following [7], we naively lift the complex $F$ to $Q$, denoted
$\widetilde{F}$. The lift of differentials $\widetilde{d}$ square to zero
modulo $(f_{1},\ldots,f_{c})$. That is,
$\widetilde{d}^{2}=\sum_{i=1}^{c}t_{i}f_{i}$ where $t_{i}$ denote the maps
$t_{i}:\widetilde{F}\to{\scriptstyle\mathsf{\Sigma}}^{2}\widetilde{F}$. The
maps $\chi_{i}=t_{i}\otimes_{Q}R$ induce a well-defined action on
${\operatorname{Ext}}^{*}_{R}(M,N)$ for $N$ some finitely generated $R$-module
and are independent of the choice of the lift $\widetilde{F}$ to $Q$ [7,
Section 1]. The ring of cohomological operators
$E=R[\chi_{1},\ldots,\chi_{c}]$ acts on ${\operatorname{Ext}}^{*}_{R}(M,N)$.

######  Proposition 2.15.

With setup as above, if $M$ is an $R$-module that lifts to $Q$, then
$(\chi_{1},\ldots,\chi_{c}){\operatorname{Ext}}^{*}_{R}(M,N)=0$ for any
finitely generated $R$-module $N$.

###### Proof.

Since $M$ lifts, its minimal $R$-free resolution $F\simeq M$ lifts to the
minimal $Q$-free resolution $G\simeq L$ where $L$ is the lift of $M$. That is,
$E$ acts on ${\operatorname{Ext}}^{*}_{R}(M,N)$ trivially, whence the
assertion follows. ∎

######  Example 2.16.

Let $(R,\mathfrak{m})$ be the two dimensional local hypersurface

$k[[x,y,z]]/(z^{2}-xy)$

defined over some field $k$. Let $M$ be the module with the following
presentation:

$d_{1}\colon R^{4}\xrightarrow[]{\left(\\!\begin{array}[]{cccc}x&0&z&y\\\
y&x&y&0\end{array}\\!\right)}R^{2}.$

The $0$th Fitting ideal of $M$ is
$F_{0}(M)=I_{2}(d_{1})=\left(z^{2},\,y\,z,\,x\,z,\,y^{2},\,x^{2}\right)$ and
hence, ${\operatorname{Supp}}(M)=\\{\mathfrak{m}\\}$. This module also has
finite projective dimension. Indeed,

$0\xrightarrow{}R^{2}\,\xrightarrow{\left(\\!\begin{array}[]{cc}-y-z&-y\\\
-y&0\\\ x+y+z&y\\\
-z&x-z\end{array}\\!\right)}R^{4}\,\xrightarrow{\left(\\!\begin{array}[]{cccc}x&0&z&y\\\
y&x&y&0\end{array}\right)}R^{2}.$

is an acyclic complex: it is easy to check that this is indeed an $R$-complex
and, using the Buchsbaum-Eisenbud acyclicity criterion, one can show that this
is acyclic. Therefore $\operatorname{projdim}_{R}(M)=2$.

######  Proposition 2.17.

In the notation of Example 2.16, $M$ is dimly liftable to $S=k[[x,y,z]]$ but
it does not lift to any regular ring.

###### Proof.

We claim that $L$ with the following presentation is a dim lift of $M$:

$\widetilde{d}_{1}\colon
S^{4}\xrightarrow{\left(\begin{array}[]{cccc}x&0&z&y\\\
y&x&y&0\end{array}\right)}S^{2}.$

Clearly $L\otimes_{S}R\cong M$. Computing the support via the $0$th Fitting
ideal, we get $\dim
L=\dim{\rm{V}}(\left(y\,z,\,x\,z,\,y^{2},\,x\,y,\,x^{2}\right))=\dim{\rm{V}}(x,y)=1$.
So $\dim L=1$. Hence $\dim L+\dim R=1+2=3=\dim S+\dim M$.

Now we show $M$ does not lift to any regular ring. If $M$ is liftable to some
regular ring $Q$ along a surjection $\pi:Q\twoheadrightarrow R$, then it also
lifts to the completion of $Q$ at its maximal ideal, denoted $\widehat{Q}$.
Furthermore, since we can factor the map ${\widehat{Q}\twoheadrightarrow R}$
to ${\widehat{Q}\twoheadrightarrow k[[x,y,z]]\twoheadrightarrow R}$, we may
assume $M$ lifts to the regular local ring $S=k[[x,y,z]]$. Let $E=k[\chi]$ be
the ring of Eisenbud operator that acts on
${\operatorname{Ext}}^{*}_{R}(M,k)$. The action of $\chi$ on
${\operatorname{Ext}}^{*}_{R}(M,k)$ is independent of the choice of lift of
the minimal free resolution $F$ of $M$ to $S$ [7, Proposition 1.2]. Naively
lifting the $R$-free resolution in Example 2.16 above to $S$, we can compute
that

$\widetilde{d}^{2}=\left(\begin{array}[]{cc}z^{2}-xy&0\\\
0&0\end{array}\\!\right)=(z^{2}-xy)\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\\!\right)$

Hence, $\chi$ acts on ${\operatorname{Ext}}_{R}^{*}(M,k)$ via the matrix
$\left(\begin{array}[]{cc}1&0\\\ 0&0\end{array}\right)$, therefore does not
annihilate it. Therefore, by Proposition 2.15, $M$ cannot lift to any regular
ring. ∎

##  3\. Lengths of dimly liftable modules.

In this section, we will show that lengths of dimly liftable modules over a
local ring $R$ is bounded below by the Hilbert-Samuel multiplicity of $R$.
This is closely related to two recent conjectures of Iyengar, Ma, and Walker.
We begin by introducing these conjectures and surveying some known results.

###  3.1. Length Conjectures.

######  Conjecture 1.

[14, Conjecture 1] (Length Conjecture) If $M$ is a nonzero module of finite
length and finite projective dimension over a local noetherian ring $R$, then
$\ell_{R}(M)\geq e(R)$.

######  Remark 3.1.

This question has been around for a long time although not formally stated as
a conjecture until recently. In their 1993 paper [1], Avramov and Buchweitz
remarked that it is “not known whether over a local or graded ring $R$ there
can be a non-zero module $M$ of finite projective dimension whose multiplicity
is smaller than the multiplicity of $R$.” In the same paper, they show that
for a graded module $M$ with $\operatorname{projdim}_{R}(M)<\infty$, we have
an equality $e(M)=ce(R)$ for $c$ some positive constant depending on $M$.
Hence the conjecture holds in the graded setting.

######  3.2.

The New Intersection Theorem [31, Theorem 13.4.1] implies that a local ring
$R$ admits a nonzero module of finite length and finite projective dimension
if and only if $R$ is Cohen-Macaulay. Over an arbitrary local ring $R$ we
consider “short complexes” defined as follows: A short complex $F$ over $R$ is
a complex

$F:0\to F_{d}\to\cdots\to F_{0}\to 0$

such that $d=\dim R$, each $F_{i}$ is a finite free $R$-module, and
$\ell_{R}(\operatorname{H}(F))<\infty$. When $R$ is Cohen-Macaulay, any short
complex is a finite free resolution of a finite length $R$-module by the
acyclicity lemma of Peskine-Szpiro [27, Lemma 1.8]. In this sense, these
objects naturally generalize the notion of a finite, free resolution of a
finite length module to all local rings. Roberts has shown the existence of a
short complex $F$ over a local ring with $\chi(F)<0$ [30]. To study the
analogue of Conjecture 1 for all local rings, the Dutta multiplicity is a more
suitable invariant.

######  3.3.

We will restrict ourselves to the case of prime characteristic $p>0$ but, due
to Kurano [18], one may define $\chi_{\infty}$ in a characteristic free way
that generalizes the definition given below in prime characteristic $p>0$; see
[18, 19, 20] for more details. We plan to address the mixed characteristic
case in future work. Let $R$ be an $F$-finite local ring of prime
characteristic $p>0$ and $\dim R=d$. If $F$ is a short complex over $R$, we
define

$\chi_{\infty}(F)=\lim_{e\to\infty}\frac{\chi(F^{e}(F))}{p^{de}}.$

This limit exists [31, Theorem 7.3.3]. Furthermore, a fundamental result of
Roberts and Kurano-Roberts [20] states that if $R$ is an equicharacteristic
complete local domain and $F$ is a short complex over $R$ with
$\operatorname{H}_{0}(F)\not=0$, then $\chi_{\infty}(F)>0$. This positivity
property plays a key role in the proof of the New Intersection Theorem in
mixed characteristic [31, Theorem 13.4.1] and Walker’s proof of Total Rank
Conjecture in characteristic $p>0$ [38, Theorem 2]. However, it is an open
conjecture of Kurano [18, Conjecture 3.1] that this also holds in mixed
characteristic.

On the other hand, Iyengar, Ma, and Walker have recently conjectured an even
tighter lower bound. An affirmative answer to Conjecture 2 implies Lech’s
conjecture holds in full generality [14, Proposition 8.3].

######  Conjecture 2.

[14, Conjecture 2] If $F$ is a short complex over a local ring $R$ with
nonzero homology, then $\chi_{\infty}(F)\geq e(R)$.

######  Example 3.4.

Suppose $R$ is a local ring of characteristic $p>0$ and
$\bm{x}=x_{1},\ldots,x_{d}$ system of parameters on $R$. Note that
$K=\text{Kos}(\bm{x})$ is a short complex over $R$. Furthermore,

$\chi_{\infty}(K)=\lim_{e\to\infty}\frac{\chi(\text{Kos}(x_{1}^{p^{e}},\ldots,x_{d}^{p^{e}})}{p^{de}}=\chi(K)$

The first equality follows by definition and the second by [31, Proposition
5.2.3]. By a theorem of Serre [33, Ch. IV, Thm 1], $\chi(K)\geq e(R)$.

######  3.5.

We now briefly discuss how Conjectures 1 and 2 are related to one another.
This leads us to the class of numerically Roberts rings. We have noted that
Conjecture 1 only concerns Cohen-Macaulay rings; see 3.2. Even if $R$ is
Cohen-Macaulay, in general, $\ell_{R}(M)\not=\chi_{\infty}(M)$ and there is no
inequality that relates these two invariants. So, as far as we know, one
conjecture does not imply the other in the Cohen-Macaualy setting where both
are sensible. However, there is a class of rings, called _numerically Roberts
rings_ [19] over which $\chi(F)=\chi_{\infty}(F)$ for any short complex $F$.
In his thesis, Ma proved the following [23, Proposition V.45]: Suppose $R$ is
a numerically Roberts ring of prime characteristic $p>0$ and of dimension at
most three. If $\operatorname{projdim}_{R}(R/I)<\infty$, then
$\ell_{R}(R/I)\geq e(R)$. Since, for example, complete intersections, Cohen-
Macaulay rings of dimension at most two, and Gorenstein rings of dimension at
most three are known to be numerically Roberts, Ma’s result establishes the
cyclic case of Conjecture 1 and 2 for these class of rings of dimension at
most three in prime characteristic $p>0$. Recall that we define for a local
ring of prime characteristic $p>0$ of dimension $d$ with maximal ideal
$\mathfrak{m}$, the Hilbert-Kunz multiplicity is defined to be the following
limit:

$e_{HK}(R)\coloneqq\lim_{e\to\infty}\frac{\ell_{R}(R/\mathfrak{m}^{[p^{ed}]})}{p^{ed}}.$

######  Proposition 3.6.

Suppose $R$ is a local ring of prime characteristic $p>0$ and $\dim R=d$. If
$F$ is a short complex with nonzero homology over $R$, then

$\chi_{\infty}(F)\geq\beta_{0}(F)e_{HK}(R)\geq\frac{\beta_{0}(F)}{d!}e(R).$

###### Proof.

Let $\mathfrak{m}$ be the maximal ideal of $R$. We have

$\chi_{\infty}(F)=\lim_{e\to\infty}\frac{\ell_{R}(F^{e}\operatorname{H}_{0}(F))}{p^{ed}}\geq\beta_{0}(F)\lim_{e\to\infty}\frac{\ell_{R}(F^{e}(R/\mathfrak{m}))}{p^{ed}}$

where the first equality is due to Roberts [31, Theorem 7.3.5] and the second
inequality holds by the fact $F^{e}\operatorname{H}_{0}(F)\twoheadrightarrow
F^{e}(R/\mathfrak{m})$ for all $e\geq 0$. We recall that the limit on the
right hand side above is the Hilbert Kunz multiplicity $e_{HK}(R)$ of the
maximal ideal $\mathfrak{m}$. Therefore,
$\chi_{\infty}(F)\geq\beta_{0}(F)e_{HK}(R)$. The final inequality follows
because $e_{HK}(R)\geq e(R)/d!$ by definition. ∎

######  Corollary 3.7.

(Ma) Conjectures 1 and 2 hold for two dimensional Cohen-Macaulay rings of
prime characteristic $p>0$.

###### Proof.

Since $R$ is Cohen-Macaulay of dimension two, it is Numerically Roberts. So,
for a nonzero $R$-module of finite length and finite projective dimension,
$\chi_{\infty}(M)=\ell_{R}(M)$. Now note that the case $\beta_{0}(M)=1$ is the
result of Ma, so we may assume $\beta_{0}(M)\geq 2$. Suppose $F\to M$ is the
minimal $R$-free resolution. By the proposition above,

$\ell_{R}(M)=\chi_{\infty}(F)\geq\beta_{0}(M)e(R)/2!\geq e(R).\qed$

######  3.8.

More recently, Iyengar, Ma, and Walker prove these length conjectures for the
following class of rings. In fact, their results say more.

######  Theorem 3.9.

[14, Theorem 3.8] Suppose $R$ is a strict complete intersection of $\dim R=d$.
If $M$ is an $R$-module of finite length and finite projective dimension, then
for each $i$,

$\begin{pmatrix}d\\\ i\ \end{pmatrix}\ell_{R}(M)\geq\beta_{i}(M)e(R).$

######  Theorem 3.10.

[14, Corollary 7.2] Suppose $A$ is a standard graded $k$-algebra of $\dim A=d$
where $k$ is a perfect field of characteristic $p>0$ and
$\mathfrak{m}=\bigoplus_{i>0}A_{i}$ is the irrelevant maximal ideal. If
$R=A_{\mathfrak{m}}$ and $F$ is a short complex over $R$ with nonzero
homology, then for each $i$,

$\begin{pmatrix}d\\\ i\ \end{pmatrix}\chi_{\infty}(F)\geq\beta_{i}(F)e(R).$

These results follow from the existence of the Ulrich modules or more
generally lim Ulrich sequence of modules with special properties; see [14] for
details. Recently, Iyengar, Ma, Walker, and Zhuang have shown that even local
complete intersections of codimension two do not admit any Ulrich module [15].

###  3.2. Lower bounds on lengths.

We will now show that modules, not necessarily of finite projective dimension,
that are dimly liftable to an unramified regular local ring must have length
at least the Hilbert-Samuel multiplicity of the ring. Below, whenever we say
an $R$-module $M$ dimly lifts to a regular ring $Q$, we assume that there is
some surjection $Q\twoheadrightarrow R$ as in Section 2.

######  Theorem 3.11.

Suppose a nonzero $R$-module $M$ of finite length dimly lifts to a $Q$-module
$L$ where $Q$ is an unramified regular local ring. Then $\ell_{R}(M)\geq
e(L)e(R)$.

###### Proof.

We have $\ell(L\otimes_{Q}R)=\ell(M)<\infty$ and $\dim L+\dim R=\dim Q$ by the
dim lift assumption. Hence, by Serre’s positivity theorem,
$\chi(L\otimes_{Q}^{\mathsf{L}}R)>0$. Furthermore,

$\chi(L\otimes_{Q}^{\mathsf{L}}R)=\ell_{R}(M)-\chi_{1}(L\otimes_{Q}^{\mathsf{L}}R).$

We have $\chi_{1}(L\otimes_{Q}^{\mathsf{L}}R)\geq 0$ due to Lichtenbaum [22,
Theorem 2]. Therefore,

$\ell_{R}(M)\geq\chi(L\otimes_{Q}^{\mathsf{L}}R)\geq e(L)e(R)$

where the final inequality follows by Theorem 1.3. ∎

######  Corollary 3.12.

Suppose $M$ is a nonzero Cohen-Macaulay module of finite projective dimension
over a local ring $R$. If $M$ dimly lifts to a $Q$-module $L$ where $Q$ is an
unramified regular local ring, then $e(M)\geq e(L)e(R)$.

###### Proof.

We claim that $R$ must be Cohen-Macaulay if it admits a nonzero Cohen-Macaulay
module of finite projective dimension. Recall that by Auslander’s zero-divisor
theorem [31, Theorem 6.2.3]: a regular element on a module of finite
projective dimension is a regular element on $R$. Choose
$\bm{x}=x_{1},\ldots,x_{r}$ a maximal regular sequence on $M$. Then
$M/(\bm{x})M$ is a module of finite length and finite projective dimension
over $R/(\bm{x})$. Therefore, by the New Intersection theorem, $R/(\bm{x})$ is
Cohen-Macaulay. Since $\bm{x}$ is a regular sequence on $R$, this implies $R$
is also Cohen-Macaulay. Furthermore, we may assume $R$ has an infinite residue
field.

We have a surjection $Q\twoheadrightarrow R$. Let $\mathfrak{m}$ denote the
maximal ideal of $Q$ and, by slight abuse of notation, also of $R$. We have
the following commutative square.

${Q}$${{Q[t]_{\mathfrak{m}Q[t]}}}$${R}$${{R[t]_{\mathfrak{m}R[t]}}}$

Note that $Q\to Q^{\prime}=Q[t]_{\mathfrak{m}Q[t]}$ and $R\to
R^{\prime}=R[t]_{\mathfrak{m}R[t]}$ are faithfully flat maps with fields as
their closed fibers. Hence, $Q^{\prime}$ is an unramified regular local ring,
$L\otimes_{Q}Q^{\prime}$ is a dim lift of the module $M\otimes_{R}R^{\prime}$,
and all the invariants of interest remain unchanged. Thus, we may now choose a
sufficiently general sequence of elements
$\bm{x}=x_{1},\ldots,x_{r}\in\mathfrak{m}\setminus\mathfrak{m}^{2}$ such that
$e(M)=\ell_{R}(M/(\bm{x})M)$ and $e(R)=e(R/(\bm{x})R)$ [2, Corollary 4.6.10].
We have ${L\otimes_{Q}R/(\bm{x})\cong M/(\bm{x})M}$ and $\dim L+\dim
R/(\bm{x})=\dim Q$. It now remains to apply Theorem 3.11. ∎

######  Corollary 3.13.

Suppose $R$ is a Cohen-Macaulay ring and $I$ is a perfect ideal in the linkage
class of a complete intersection. Then $e(R/I)\geq e(R)$.

###### Proof.

Indeed, by Example 2.1, $R/I$ is liftable so we may apply Corollary 3.12. ∎

######  Example 3.14.

Theorem 3.11 provides an obstruction to dim liftability of finite length
$R$-modules (and for perfect $R$-modules by Corollary 3.12) to an unramified
regular ring. For example, if $R$ is a local ring such that $e(R)>1$, then the
residue field $k$ of $R$ does not dimly lift to an unramified regular ring.

######  3.15.

The notion of lifting modules of finite length and finite projective dimension
is restricted to Cohen Macaulay rings; see 3.2. For all local rings, we will
now consider the lifting of short complexes. Let us fix a surjection
$\pi:Q\twoheadrightarrow R$, where $Q$ is a regular local ring. A finite free
$R$-complex $F$ lifts to $Q$ if there exists a finite free $Q$-complex $G$
such that $G\otimes_{Q}R\cong F$. We say $F$ is liftable to $Q$ and that $G$
is a lift of $F$. For example, a Koszul complex on a sequence of elements of
$R$ is liftable. We utilize the following result of Kurano [19, Theorem 1.2]:
Suppose $F$ is a short complex over a local ring $R$ that lifts to a finite
free $Q$-complex $G$ over a regular local ring $Q$. Then
$\operatorname{H}_{i}(G)=0$ for $i>0$ and $\dim\operatorname{H}_{0}(G)+\dim
R=\dim Q$.

######  Remark 3.16.

Suppose $R$ is Cohen Macaulay and $M$ is an $R$-module of finite length and
finite projective dimension. If $F\xrightarrow{\simeq}M$ is the minimal free
resolution of $M$, then the complex $F$ lifts to $Q$ if and only if the module
$M$ lifts to $Q$. Indeed, the forward direction follows by Kurano’s result and
the converse is clear. This also highlights the fact that the main obstruction
to lifting an $R$-module of finite length and finite projective dimension is
lifting the complex structure of its free resolution. The acylicity comes for
free.

######  Proposition 3.17.

Suppose $F$ is a short complex over $R$ that lifts to $Q$. Then the $R$-module
$\operatorname{H}_{0}(F)$ dimly lifts to $Q$.

###### Proof.

Let $G$ be the lift of $F$. By Kurano’s result, $\operatorname{H}_{i}(G)=0$
for $i>0$ and $\dim\operatorname{H}_{0}(G)+\dim R=\dim Q$. Now since
$\operatorname{H}_{0}(G)\otimes_{Q}R=\operatorname{H}_{0}(F)$ by the right
exactness of tensor product, $\operatorname{H}_{0}(G)$ is a dim lift of
$\operatorname{H}_{0}(F)$. ∎

######  Definition 3.18.

A short complex $F$ over a local ring $R$ dimly lifts to $Q$ if the $R$-module
$\operatorname{H}_{0}(F)$ dimly lifts to $Q$.

######  Theorem 3.19.

Suppose $R$ is a local ring of prime characteristic $p>0$ and $F$ is a short
complex over $R$ with nonzero homology. If $F$ is dimly liftable to an
unramified regular local ring $Q$, then

$\chi_{\infty}(F)\geq e(R).$

###### Proof.

Let the $Q$-module $L$ denote the dim lift of $H_{0}(F)$ and let
$G\xrightarrow{\simeq}L$ denote its minimal $Q$-free resolution. For $e\geq
0$,
$\chi(F^{e}G\otimes_{Q}R)=\ell(\operatorname{H}_{0}(F^{e}F))-\chi_{1}(F^{e}G\otimes_{Q}R)$.
We have $\operatorname{codim}L=\dim R=d$ and since $Q$ is regular
$\chi_{\infty}(L,R)=\chi(L,R)$ [5, 1.3 Remark]. Taking limits,

$\chi(L,R)=\chi_{\infty}(L,R)=\lim_{e\to\infty}\frac{\ell(\operatorname{H}_{0}(F^{e}F))}{p^{ed}}-\lim_{e\to\infty}\frac{\chi_{1}(F^{e}G\otimes_{Q}R)}{p^{ed}}.$

Due to Roberts (proof of [31, Theorem 7.3.5]), as $F$ is a short complex, the
first term on the left hand side is equal to $\chi_{\infty}(F)$ and due to
Lichtenbaum [22, Theorem 2] we know $\chi_{1}(F^{e}G\otimes_{Q}R)\geq 0$ and
therefore the second limit is at least zero. This shows that
$\chi_{\infty}(F)\geq\chi(L,R)>0$. The proof is then complete by Theorem 1.3.
∎

######  Remark 3.20.

Suppose $R$ is a local ring of characteristic $p>0$ and dimension $d$. If $M$
is a finite length $R$-module, the following limit exists.

$\ell_{\infty}(M)=\lim_{e\to\infty}\frac{\ell_{R}(F^{e}(M))}{p^{ed}}.$

The proof of Theorem 3.19 shows that if $M$ is dimly liftable to an unramified
regular local ring, then $\ell_{\infty}(M)\geq e(R)$.

###  3.3. Extremal case

What modules of finite length and finite projective dimension (resp. short
complexes) have length (resp. $\chi_{\infty})$ that equals $e(R)$? We will
provide a new proof to the following result of Ma [23, Theorem V.28].

######  Proposition 3.21.

Suppose $R$ is a strict complete intersection of dimension $d$. If $M$ is an
$R$-module of finite projective dimension with ${\ell_{R}(M)=e(R)}$, then for
some system of parameters $\bm{x}=x_{1},\ldots,x_{d}$ on $R$, we have $M\cong
R/(\bm{x}).$

###### Proof.

Given Theorem 3.9, if we sum up both sides in the inequalities, we get
$2^{d}\ell_{R}(M)\geq\sum_{i=1}^{d}\beta_{i}(M)e(R)$. By cancelling $e(R)$, we
get $2^{d}\geq\sum_{i=1}^{d}\beta_{i}(M)$. If the characteristic of the
residue field is not two, Walker’s total rank theorem [38, Theorem 2] implies
$2^{d}=\sum_{i=1}^{d}\beta_{i}(M)$. Again by Walker’s theorem, this condition
forces $M\simeq\text{Kos}(x_{1},\ldots,x_{d})$ for some system of parameters
$x_{1},\ldots,x_{d}$. If the characteristic is two, the same conclusion
follows due to [37, Theorem 1.12]. ∎

An analogous argument can be carried out for short complexes over standard
graded algebras as in Theorem 3.10 but we do not know whether for a short
complex $F$, the equality $\sum_{i=1}^{d}\beta_{i}(F)=2^{d}$ forces
$F\cong\text{Kos}(x_{1},\ldots,x_{d})$ for some system of parameters. This is
formulated as a conjecture by VandeBogert and Walker [37, Conjecutre 1.11].

######  Proposition 3.22.

Suppose $R$ is a local ring and $M$ is a cyclic $R$-module of finite length
that is liftable to an unramified regular local ring $Q$. Then, if
$\ell_{R}(M)=e(R)$, then $M\cong R/(\bm{x})$ for some system of parameters
$\bm{x}=x_{1},\ldots,x_{d}$ on $R$.

###### Proof.

Due to Theorem 3.11, we have $e(R)=\ell_{R}(M)\geq e(L)e(R)$ where $L$ is the
lift of $M$. Therefore, $e(L)=1$. Since $M$ is cyclic, we may assume $L$ is
also cyclic. Assume $L\cong Q/J$ for some ideal $J$. Since $Q/J$ is a Cohen
Macaulay algebra (see 2.6) such that $e(Q/J)=1$, it follows that $Q/J$ must be
regular. So $J$ is genereated by a part of a regular system of parameters on
$Q$. Therefore, $M\cong R/(\bm{x})$ where $\bm{x}=x_{1},\ldots,x_{d}$ is the
image of $J$ on $R$ and $d=\dim R$. ∎

We do not know of any $R$-module of finite projective dimension with
$\ell_{R}(M)=e(R)$ which is not isomorphic to $R/(\bm{x})$ for some system of
parameters $\bm{x}$ on R.

##  4\. Unliftable modules

In general, there are many obstructions for a module of finite projective
dimension over a singular ring to lift to a regular ring. We report several
such examples. Notably, we will also show there exists a module of finite
projective dimension over a singular ring that cannot dimly lift to a regular
ring.

######  Example 4.1.

Suppose $R$ is a local ring of dimension $d$ and $M$ is an $R$-module of
finite length and finite projective dimension. If for some $R$-module $N$ such
that $\dim N<\dim R$, $\chi(M,N)\not=0$, then $M$ cannot lift to any regular
ring $Q$. Indeed, if $L$ is a lift of $M$, then
$\chi^{Q}(L,N)=\chi^{R}(M,N)\not=0$. But this contradicts the vanishing
theorem over regular local rings (see 1.2) as $\dim L+\dim N<\dim Q.$

Dutta, Hochster, and McLaughlin produced the following example [6]. Let
${R=k[[x,y,z,w]]/(xy-zw)}$ and $\mathfrak{p}=(x,w)R$. There exists a module
$M$ such that $\ell_{R}(M)=15$, $\operatorname{projdim}_{R}(M)=3$ such that
$\chi^{R}(M,N)=-1$. Since $\dim N=2$, $M$ cannot lift to a regular ring by the
discussion above. Building on this example, Smoke shows there exists a cyclic
$R$-module $R/I$ of finite length and finite projective dimension such that
$\chi^{R}(R/I,R/\mathfrak{p})=\chi^{R}(M,R/\mathfrak{p})=-1$ [35, Section 6].
Therefore, this cyclic $R$-module of projective dimension three does not lift
either.

######  Example 4.2.

In this example, we show there exists a module of finite length and finite
projective dimension that is not dimly liftable to any regular local ring. Let
$R$ be as in the example above. Suppose an $R$-module $M$ dimly lifts to some
regular ring $Q$. Since $R$ is complete and of equal characteristic, we may
assume so is $Q$. Since $R$ is a complete, complete intersection, we have
$R=Q/(g_{1},\ldots,g_{c})$. As $R$ is a hypersurface, we can assume
$g_{1},\ldots,g_{c-1}\in\mathfrak{m}\setminus\mathfrak{m}^{2}$ where
$\mathfrak{m}$ is the maximal ideal of $Q$. Let $L$ be the $Q$-module that is
a dim lift of $M$. Choosing a sufficiently general sequence of elements in
$g_{1}^{\prime},\ldots,g_{c-1}^{\prime}\in(g_{1},\ldots,g_{c-1})\setminus\mathfrak{m}^{2}$,
we can consider the regular ring
$Q^{\prime}=Q/(g_{1}^{\prime},\ldots,g_{c-1}^{\prime})$. Then the
$Q^{\prime}$-module $L^{\prime}=L\otimes_{Q}Q^{\prime}$ has dimension $\dim
L-(c-1)$ and $L^{\prime}\otimes_{Q^{\prime}}R=M$. Therefore, if $M$ dimly
lifts to some regular ring $Q$, we may assume $R=Q/(f)$.

Now consider $M$ and $\mathfrak{p}$ as in Example 4.1 above. Assume $L$ is a
dim lift of $M$. Consider the exact sequence

$0\to{\operatorname{ann}}_{L}(f)\to L\xrightarrow{f}L\to M\to 0.$

Note $M_{1}\coloneqq{\operatorname{ann}}_{L}(f)$ is a nonzero $R$-module as
$M$ is not liftable. Using the standard change of ring long exact sequence of
Tors, we have $\operatorname{projdim}_{R}(M_{1})<\infty$. On the other hand,
since $\dim L=1$, $\chi(f;L)>0$ whence $\ell_{R}(M)>\ell_{R}(M_{1})$.
Furthermore,

$0=\chi(L,R/\mathfrak{p})=\chi(M,R/\mathfrak{p})-\chi(M_{1},R/\mathfrak{p})$

where the first equality follows by the vanishing theorem over regular rings
and the second by Lemma 2.2. Therefore, $\chi(M_{1},R/\mathfrak{p})=-1$ and
$\ell_{R}(M_{1})<\ell_{R}(M)$. In particular, $M_{1}$ is not liftable by the
argument in Example 4.1.

If $M_{1}$ is not dimly liftable, we have found our desired example.
Otherwise, we run the same argument to deduce an existence of an unliftable
nonzero module $M_{2}$ of finite projective dimension such that
$\ell_{R}(M_{2})<\ell_{R}(M_{1})$ and $\chi_{R}(M_{2},R/\mathfrak{p})=-1$.
Clearly, this process must terminate at some point giving us the desired
example.

######  Example 4.3.

Due to Kurano [19, Theorem 1.2], if $R$ is a local ring and $F$ is a short
complex over $R$ that is liftable to a regular ring, then
$\chi_{\infty}(F)=\chi(F)$. Miller-Singh construct a five dimensional
Gorenstein ring and module of finite length and finite projective dimension
such that $\chi(F)\not=\chi_{\infty}(F)$ where $F$ is the module’s minimal
free resolution [26]. Hence, this module is also unliftable.

######  Example 4.4.

The following example is due to Hietmann and communicated to us by Walker. We
will construct a class of Cohen Macaulay rings of dimension two and a module
of finite length and finite projective dimension with betti numbers $(2,4,2)$
such that they do not lift to a regular ring. Consider the “generic” complex
with betti numbers $(2,4,2)$.

(1) $0\to S^{2}\xrightarrow{\begin{pmatrix}x_{1}&x_{2}\\\ x_{3}&x_{4}\\\
x_{5}&x_{6}\\\
x_{7}&x_{8}\end{pmatrix}}S^{4}\xrightarrow{\begin{pmatrix}y_{1}&y_{2}&y_{3}&y_{4}\\\
y_{5}&y_{6}&y_{7}&y_{8}\end{pmatrix}}S^{2}\to 0$

where $S=\mathbb{Z}[X;Y]$ with $x_{i}$, $y_{j}$ considered indeterminates. We
want to introduce relations on $S$ to make $(1)$ a complex. First, let
$I=I_{1}(YX)$, i.e. the ideal generated by entries of the product $YX$.
Modding out $S$ by $I$ makes $(1)$ a complex. Now let
$P\subseteq\\{1,2,3,4\\}$ be an ordered subset of two elements and let $P^{c}$
denote its complement. Let $\Delta^{Y}_{P}$ denote the two by two minor of $Y$
with columns indexed by $P$ and let $\Delta^{X}_{P^{c}}$ denote the two by two
minor of $X$ with with rows indexed by $P^{c}$. Let $(P\mid P^{c})$ denote the
permutation defined by the sets $P$ and $P^{c}$, and let
$\sigma_{P}=\sigma(P\mid P^{c})$ denote the sign of this permutation. We let
$J$ denote the ideal generated by elements
$\Delta^{Y}_{P}-\sigma_{P}\Delta^{X}_{P^{c}}$ as $P$ ranges over all ordered
subsets of $\\{1,2,3,4\\}$ consisting of two elements. From the theory of
generic resolutions of length two, over $R=S/(I+J)$, we have that $(1)$ is an
acyclic complex [36, Section 2]. Furthermore, $R$ is Cohen Macaulay. Let $M$
denote the zeroth homology module of the acyclic complex above. By the
symmetry of the construction, the $R$-linear dual of $(1)$ is also acyclic.
Hence, $M$ is a Cohen Macaulay module since it is perfect module over a Cohen
Macaulay ring.

We localize $R$ appropriately, say at $(p,X,Y)$ for a mixed characteristic
example. Then we may choose a maximal regular sequence $r_{1},\ldots,r_{\dim
M}$ on $M$. Taking powers, we may assume $r_{1},\ldots,r_{\dim
M}\subseteq\mathfrak{m}_{R}^{2}$. Now let
$\overline{R}=R/(r_{1},\ldots,r_{\dim M})$ and
$\overline{M}=M/(r_{1},\ldots,r_{\dim M})M$. We have $\overline{R}$ is Cohen
Macaulay and $\dim\overline{R}=\dim R-\dim M=\operatorname{projdim}_{R}(M)=2$
and $\overline{M}$ is a finite length $\overline{R}$-module of finite
projective dimension. We claim that this module cannot lift to a regular ring.

###### Proof.

For contradiction, assume it does lift to some regular ring $Q$. Then we may
assume $Q\twoheadrightarrow\overline{R}$ minimally, i.e.
$\mathfrak{m}_{Q}/\mathfrak{m}_{Q}^{2}=\mathfrak{m}_{\bar{R}}/\mathfrak{m}_{\bar{R}}^{2}$.
We have $\bar{x_{1}},\ldots,\bar{x_{8}},\bar{y_{1}},\ldots,\bar{y_{8}}$
minimally generates the maximal ideal $\mathfrak{m}_{\bar{R}}$. Their lifts to
$Q$ minimally generate the maximal ideal of $Q$, but as $Q$ is a regular local
ring, they must form a regular system of parameters on $Q$. If we can lift the
resolution of $\bar{M}$, we have that

$0\to Q^{2}\xrightarrow{\widetilde{X}}Q^{4}\xrightarrow{\widetilde{Y}}Q^{2}\to
0$

is in particular a $Q$-complex. But $\widetilde{Y}\widetilde{X}=0$ implies
there are non-Koszul relations amongst lifts of
$\bar{x_{1}},\ldots,\bar{x_{8}},\bar{y_{1}},\ldots,\bar{y_{8}}$ to $Q$. This
is a contradiction as these lifts form a regular sequence on $Q$. ∎

One may generalize this example to produce many non-cyclic modules of finite
length and finite projective dimension over a two dimensional Cohen Macaulay
ring that do not lift to a regular ring. The length conjecture for such
modules remains open in mixed characteristic.

## References

  * [1] Luchezar L. Avramov and Ragnar-Olaf Buchweitz, _Lower bounds for Betti numbers_ , Compositio Math. 86 (1993), no. 2, 147–158. MR 1214454
  * [2] Winfried Bruns and Jürgen Herzog, _Cohen-Macaulay rings_ , revised ed., Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1998. MR 1251956
  * [3] David A. Buchsbaum and David Eisenbud, _Lifting modules and a theorem on finite free resolutions_ , Ring theory (Proc. Conf., Park City, Utah, 1971), Academic Press, New York-London, 1972, pp. 63–74. MR 340343
  * [4] Hailong Dao, _On liftable and weakly liftable modules_ , J. Algebra 318 (2007), no. 2, 723–736. MR 2371969
  * [5] Sankar P. Dutta, _Frobenius and multiplicities_ , J. Algebra 85 (1983), no. 2, 424–448. MR 725094
  * [6] Sankar P. Dutta, M. Hochster, and J. E. McLaughlin, _Modules of finite projective dimension with negative intersection multiplicities_ , Invent. Math. 79 (1985), no. 2, 253–291. MR 778127
  * [7] David Eisenbud, _Homological algebra on a complete intersection, with an application to group representations_ , Trans. Amer. Math. Soc. 260 (1980), no. 1, 35–64. MR 570778
  * [8] H. Gillet and C. Soulé, _Intersection theory using Adams operations_ , Invent. Math. 90 (1987), no. 2, 243–277. MR 910201
  * [9] Lorenzo Guerrieri, Xianglong Ni, and Jerzy Weyman, _An ADE correspondence for grade three perfect ideals_ , 2024, https://math.berkeley.edu/~xlni/Preprints/ADE.pdf.
  * [10] Tor H. Gulliksen, _A change of ring theorem with applications to Poincaré series and intersection multiplicity_ , Math. Scand. 34 (1974), 167–183. MR 364232
  * [11] Melvin Hochster, _An obstruction to lifting cyclic modules_ , Pacific J. Math. 61 (1975), no. 2, 457–463. MR 412169
  * [12] by same author, _The dimension of an intersection in an ambient hypersurface_ , Algebraic Geometry (Berlin, Heidelberg) (Anatoly Libgober and Philip Wagreich, eds.), Springer Berlin Heidelberg, 1981, pp. 93–106.
  * [13] S. Iyengar, _Depth for complexes, and intersection theorems_ , Math. Z. 230 (1999), no. 3, 545–567. MR 1680036
  * [14] Srikanth B. Iyengar, Linquan Ma, and Mark E. Walker, _Multiplicities and Betti numbers in local algebra via lim Ulrich points_ , Algebra Number Theory 16 (2022), no. 5, 1213–1257. MR 4471041
  * [15] Srikanth B. Iyengar, Linquan Ma, Mark E. Walker, and Ziquan Zhuang, _Non-existence of Ulrich modules over Cohen-Macaulay local rings_ , arXiv preprint, 2024, https://arxiv.org/abs/2403.15566, pp. 1–12.
  * [16] David A. Jorgensen, _Existence of unliftable modules_ , Proc. Amer. Math. Soc. 127 (1999), no. 6, 1575–1582. MR 1476141
  * [17] by same author, _Some liftable cyclic modules_ , Comm. Algebra 31 (2003), no. 1, 493–504. MR 1969236
  * [18] Kazuhiko Kurano, _An approach to the characteristic free Dutta multiplicity_ , J. Math. Soc. Japan 45 (1993), no. 3, 369–390. MR 1219875
  * [19] by same author, _Test modules to calculate Dutta multiplicities_ , J. Algebra 236 (2001), no. 1, 216–235. MR 1808352
  * [20] Kazuhiko Kurano and Paul C. Roberts, _Adams operations, localized Chern characters, and the positivity of Dutta multiplicity in characteristic $0$_, Trans. Amer. Math. Soc. 352 (2000), no. 7, 3103–3116. MR 1707198
  * [21] Christer Lech, _Note on multiplicities of ideals_ , Ark. Mat. 4 (1960), 63–86. MR 140536
  * [22] Stephen Lichtenbaum, _On the vanishing of ${\rm Tor}$ in regular local rings_, Illinois J. Math. 10 (1966), 220–226. MR 188249
  * [23] Linquan Ma, _The Frobenius endomorphism and multiplicities_ , Ph.D. thesis, University of Michigan, 2014.
  * [24] by same author, _Lech’s conjecture in dimension three_ , Adv. Math. 322 (2017), 940–970. MR 3720812
  * [25] by same author, _Lim Ulrich sequences and Lech’s conjecture_ , Invent. Math. 231 (2023), no. 1, 407–429. MR 4526825
  * [26] Claudia M. Miller and Anurag K. Singh, _Intersection multiplicities over Gorenstein rings_ , Math. Ann. 317 (2000), no. 1, 155–171. MR 1760672
  * [27] C. Peskine and L. Szpiro, _Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck_ , Inst. Hautes Études Sci. Publ. Math. (1973), no. 42, 47–119. MR 374130
  * [28] Paul Roberts, _The vanishing of intersection multiplicities of perfect complexes_ , Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 127–130. MR 799793
  * [29] by same author, _Le théorème d’intersection_ , C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 7, 177–180. MR 880574
  * [30] by same author, _Intersection theorems_ , Commutative algebra (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 417–436. MR 1015532
  * [31] by same author, _Multiplicities and Chern classes in local algebra_ , Cambridge Tracts in Mathematics, vol. 133, Cambridge University Press, Cambridge, 1998. MR 1686450
  * [32] Paul C. Roberts, _Recent developments on Serre’s multiplicity conjectures: Gabber’s proof of the nonnegativity conjecture_ , Enseign. Math. (2) 44 (1998), no. 3-4, 305–324. MR 1659224
  * [33] Jean-Pierre Serre, _Local algebra_ , Springer Berlin, Heidelberg, 1965.
  * [34] Chris Skalit, _Intersection multiplicity of Serre in the unramified case_ , J. Algebra 428 (2015), 91–121. MR 3314287
  * [35] William Smoke, _Perfect modules over Cohen-Macaulay local rings_ , J. Algebra 106 (1987), no. 2, 367–375. MR 880963
  * [36] Alexandre B. Tchernev, _Universal complexes and the generic structure of free resolutions_ , Michigan Math. J. 49 (2001), no. 1, 65–96. MR 1827076
  * [37] Keller VandeBogert and Mark E. Walker, _The total rank conjecture in characteristic two_ , 2023, https://arxiv.org/pdf/2305.09771, pp. 1–19.
  * [38] Mark E. Walker, _Total Betti numbers of modules of finite projective dimension_ , Ann. of Math. (2) 186 (2017), no. 2, 641–646. MR 3702675