# Koszul homomorphisms and universal resolutions in local algebra

Benjamin Briggs  Benjamin Briggs, Department of Mathematical Sciences,
University of Copenhagen, Universitetsparken 5 DK-2100 Copenhagen Ø
<EMAIL_ADDRESS>,  James C. Cameron  James C. Cameron, Department of
Mathematics, University of Utah, Salt Lake City, UT 84112, U.S.A.
<EMAIL_ADDRESS>,  Janina C. Letz  Janina C. Letz, Faculty of
Mathematics, Bielefeld University, PO Box 100 131, 33501 Bielefeld, Germany
Math Sciences Building, 520 Portola Plaza, Box 951555, Los Angeles, CA 90095,
United States<EMAIL_ADDRESS>and  Josh Pollitz  Josh Pollitz,
Mathematics Department, Syracuse University, Syracuse, NY 13244 U.S.A.
<EMAIL_ADDRESS>
###### Abstract.

We define a local homomorphism $(Q,k)\to(R,\ell)$ to be Koszul if its derived
fiber $R\otimes^{\mathsf{L}}_{Q}k$ is formal, and if
$\operatorname{Tor}^{Q}(R,k)$ is Koszul in the classical sense. This recovers
the classical definition when $Q$ is a field, and more generally includes all
flat deformations of Koszul algebras. The non-flat case is significantly more
interesting, and there is no need for examples to be quadratic: all complete
intersection and all Golod quotients are Koszul homomorphisms. We show that
the class of Koszul homomorphisms enjoys excellent homological properties, and
we give many more examples, especially various monomial and Gorenstein
examples. We then study Koszul homomorphisms from the perspective of
$\mathrm{A}_{\infty}$-structures on resolutions. We use this machinery to
construct universal free resolutions of $R$-modules by generalizing a
classical construction of Priddy. The resulting (infinite) free resolution of
an $R$-module $M$ is often minimal, and can be described by a finite amount of
data whenever $M$ and $R$ have finite projective dimension over $Q$. Our
construction simultaneously recovers the resolutions of Shamash and Eisenbud
over a complete intersection ring, and the bar resolutions of Iyengar and
Burke over a Golod ring, and produces analogous resolutions for various other
classes of local rings.

###### Key words and phrases:

Free resolutions, Koszul algebras, Koszul duality, bar resolution, twisted
tensor products, A-infinity algebras

###### 2020 Mathematics Subject Classification:

13D02 (primary), 16S37, 16E45, 13H10, 13F55

###### Contents

  1. 1 Introduction
  2. 2 Koszul homomorphisms
    1. 2.8 Cohen Koszul local rings
    2. 2.15 Properties of Koszul homomorphisms
  3. 3 Examples of Koszul homomorphisms
    1. 3.8 Rings of small codepth
    2. 3.13 Almost Golod Gorenstein rings
    3. 3.20 Monomial rings
  4. 4 Background on $\mathrm{A}_{\infty}$-algebras and coalgebras
  5. 5 Transfer of $\mathrm{A}_{\infty}$-algebra structures
    1. 5.6 Cyclic $\mathrm{A}_{\infty}$-algebras
  6. 6 Twisted tensor products
  7. 7 $\mathrm{A}_{\infty}$-algebra presentations for Koszul homomorphisms
    1. 7.2 Strictly Koszul presentations
    2. 7.6 The Priddy resolution
  8. 8 Examples of strictly Koszul presentations
    1. 8.7 Complete intersection homomorphisms
    2. 8.15 Almost Golod Gorenstein rings

##  1\. Introduction

The phenomenon of Koszul duality has been observed in many forms across
algebra, geometry and topology. It provides explicit computational tools for
answering homological questions and opens up deep connections between a number
of seemingly unrelated areas of mathematics. The goal of the present work is
to develop a relative theory of Koszul duality in local commutative algebra,
and to give concrete applications for understanding infinite free resolutions.

For a finite homomorphism $\varphi\colon Q\to R$ of commutative noetherian
local rings, the derived fiber $F=R\otimes^{\mathsf{L}}_{Q}k$, where $k$ is
the residue field of $Q$, is a differential graded $k$-algebra that encodes
important ring theoretic properties of $\varphi$. We define $\varphi$ to be
_Koszul_ if $F$ is formal (see 2.3), and if its homology
$\mathrm{H}(F)=\operatorname{Tor}^{Q}(R,k)$ is a Koszul $k$-algebra (see 2.1).
This recovers the classical definition when the source is a field. Through the
looking glass that connects local algebra with rational homotopy theory, the
definition is directly analogous to Berglund’s notion of a Koszul space.

Flat local maps that have a Koszul fiber are natural examples of Koszul
homomorphisms, but the non-flat case is significantly more interesting: all
complete intersection and all Golod quotient homomorphisms are Koszul, and we
give many other monomial and Gorenstein examples. In particular, there is no
need for the homomorphism to be quadratic in any sense.

The definition also has structural consequences connecting the homological
algebra over $R$ and $Q$. Our main theorem provides an algorithmic way to
transfer free resolutions over $Q$ into free resolutions over $R$. To achieve
this we introduce a slightly stronger “strictly Koszul” property (see 7.2)
that is satisfied in our main examples. These ideas borrow from a long
history, and we will discuss the context and technology behind the
construction following this summary of our main results.

######  Theorem A.

For any strictly Koszul local homomorphism $\varphi\colon Q\to R$ there is a
non-negatively graded, degreewise finite rank free $Q$-module $C$ such that:

  1. (1)

For each finitely generated $R$-module $M$ with a minimal $Q$-free resolution
$G\to M$, there is a differential $\partial^{\tau}$ on the graded $R$-module
$R\otimes C\otimes G$ such that the resulting “twisted tensor product” complex

$(R\otimes C\otimes G,\partial^{\tau})\xrightarrow{\ \simeq\ }M$

is an $R$-free resolution of $M$. If $R$ and $M$ have finite projective
dimension over $Q$, then both $C$ and the twisted tensor product differential
can be explicitly described in their entirety with a finite amount of data.

  2. (2)

Assume that $\varphi$ is small (a central case of interest is
$(Q,\mathfrak{m}_{Q})$ regular and
$\ker(\varphi)\subseteq\mathfrak{m}^{2}_{Q}$). The twisted tensor product
complex is minimal for the residue field $k$ of $R$. More generally, the
resolution is minimal whenever $M$ is inert with respect to $\varphi$, in the
sense of Lescot. Moreover,

$\textstyle{\sum_{i}}\operatorname{rank}_{Q}(C_{i})t^{i}=\frac{\operatorname{P}^{R}_{k}(t)}{P^{Q}_{k}(t)}\,.$

The following homomorphisms are strictly Koszul:

  1. (a)

Surjective complete intersection homomorphisms.

  2. (b)

Surjective Golod homomorphisms.

  3. (c)

Surjective Gorenstein homomorphisms of projective dimension three or less.

  4. (d)

Cohen presentations of compressed artinian Gorenstein rings having
characteristic zero and odd embedding dimension.

Part Item 1 of the Section, with an explicit description of the twisted tensor
product differential, is Theorem 7.7, while part Item 2 is contained in
Theorem 7.10. The examples Items a, b, c and d, and several more, are
introduced in Section 3 and treated again in Section 8, with a complete
description of the corresponding $C$ in each case.

Universal resolutions, that is, free resolutions over a ring that are defined
in a uniform way for all finitely generated modules, have been of central
interest in homological commutative algebra since at least the 60s, often
importing tools such as Massey operations and bar resolutions from algebraic
topology.

Let $\varphi\colon Q\to R$ be a local homomorphism. Shamash constructed
universal resolutions for $R$-modules when $\varphi$ is a hypersurface
quotient [Sha69], and these were clarified and extended to complete
intersection quotients by Eisenbud using the theory of higher homotopies
[Eis80]. Burke recognized in [Bur15] that the higher homotopies are a
manifestation of certain $\mathrm{A}_{\infty}$-structures (we will return to
these later in the introduction). In the presence of a $Q$-free differential
graded algebra resolution $A\to R$, and a $Q$-free differential graded
$A$-module resolution $G$ of an $R$-module $M$, Iyengar constructed a bar
resolution for $M$ over $R$ [Iye97]. By endowing $A$ with an
$\mathrm{A}_{\infty}$-algebra structure, and $G$ with an
$\mathrm{A}_{\infty}$-module structure over $A$, Burke constructed a bar
resolution even when associative multiplicative resolutions do not exist
[Bur15]. The resulting resolution is minimal when $M$ is Golod with respect to
$\varphi$, and is otherwise typically far from minimal. A recovers both the
resolutions of Shamash and Eisenbud, when $\varphi$ is a complete intersection
quotient, and the bar resolutions of Iyengar and Burke, when $\varphi$ is a
Golod quotient.

In parallel, the universal resolutions introduced by Priddy over Koszul
algebras have had far-reaching impact [Pri70], not least as a computational
tool. Our theory directly builds on and recovers his construction, while
providing a common framework for the universal resolutions above.

The technical foundation for our universal resolutions is in Section 6. Here
we develop a general theory of twisted tensor products over a commutative ring
$Q$. The data that goes into this construction is a curved differential graded
coalgebra $C$ over $Q$, a quasi-isomorphism $\mathsf{\Omega}(C)\to R$ from the
cobar construction of $C$ to $R$, and a differential graded module structure
over $\mathsf{\Omega}(C)$ on $G$. These terms are defined in Section 4. From
this, in Theorem 6.5, we construct a canonical resolution

$R\otimes^{\tau}C\otimes^{\tau}G=(R\otimes C\otimes
G,\partial^{\tau})\xrightarrow{\ \simeq\ }M\,.$

The key to proving A is to show that $C$ can be defined in an explicit,
canonical, and minimal way when $\varphi$ is strictly Koszul. We will return
to this at the end of the introduction, with more context in hand.

÷

We turn our attention back to the Koszul homomorphisms. The first half of this
work develops the theory of these maps; this part of the paper does not
involve $\mathrm{A}_{\infty}$-structures, using only ordinary differential
graded algebras.

Similar Koszul-type conditions have been considered by other authors [CDG+20,
HI05, HRW98, Mye21], and we compare our definition with theirs in Remark 2.18.
We motivate our condition as well by drawing connections with other areas,
such as rational homotopy theory (Remarks 2.5 and 3.18) and toric topology
(Remark 3.27).

We pay particular attention to the case that $Q$ is regular. In this
situation, the resolutions constructed in A essentially depend only on the
ring $R$, and they are always finitely determined. When $R$ is a local ring
such that every Cohen presentation $\varphi\colon Q\to\widehat{R}$ is a Koszul
homomorphism, we say that $R$ is _Cohen Koszul_ ; see Section 2.8. These rings
enjoy excellent homological properties while being surprisingly abundant; they
behave in many ways like classical Koszul algebras despite not necessarily
being quadratic.

We show that Cohen Koszul local rings have rational Poincaré series that can
be computed explicitly from their Koszul homology:

$\operatorname{P}^{R}_{k}(t)=\frac{(1+t)^{e}}{\sum_{i,w}(-1)^{w}\operatorname{rank}_{k}(\mathrm{H}_{i}(K^{R})_{(w)})t^{i+w}}\,;$

see Proposition 2.11, where the notation is explained.

Section 3 is devoted entirely to examples, and we prove that surjective
complete intersection homomorphisms are Koszul (Example 3.2), along with
surjective Golod homomorphisms (Example 3.5), and Gorenstein homomorphisms of
projective dimension three (Example 3.12). We exactly determine the local
rings of codepth three or less that are Cohen Koszul in terms of the
classification into the types described in [AKM88]; we find that in every type
except one the local ring is Cohen Koszul (Theorem 3.10). Graded local rings
having an almost linear resolution in the sense of [DE22] are also Cohen
Koszul (Remark 3.17). We treat monomial rings in Section 3.20, making
connections with combinatorial commutative algebra and with the topology of
moment angle complexes. Monomial rings are classically Koszul exactly when
they are quadratic [Frö75], while Cohen Koszul monomial rings need not be, and
we produce many nontrivial examples in Proposition 3.22. We further give
examples that illuminate how the Koszul condition relates with classical
Koszulity, formality, being quadratic, and various other technical conditions.

One of our main examples is a class of rings that we call _almost Golod
Gorenstein_ , treated in Section 3.13. Within the class of Gorenstein local
rings, these display extremal homological behavior analogous to Golod rings
within the class of all local rings; cf. Proposition 3.19. In Theorem 3.16 we
establish a characterization in terms of the derived fiber that is similar to
Avramov’s characterization of Golod rings [Avr86], and, under some additional
technical assumptions, we deduce that almost Golod Gorenstein rings are Cohen
Koszul.

$\boldsymbol{\vee\mid}$

In the second half of the paper we study Koszul homomorphims from the
perspective of $\mathrm{A}_{\infty}$-structures on resolutions. An
$\mathrm{A}_{\infty}$-algebra is a complex $A$ equipped with multilinear
operations $m_{n}\colon A^{\otimes n}\to A$ for $n\geqslant 2$ that together
satisfy certain associativity conditions generalizing the definition of a
differential graded algebra (which one recovers by assuming $m_{n}=0$ for
$n\geqslant 3$). These objects were introduced by Stasheff to characterize
loop spaces in algebraic topology [Sta63a].

Koszulity is well-known to be connected with formality (Remark 2.5), and in
turn it has been understood since [Kad82] that formality can be made visible
through $\mathrm{A}_{\infty}$-structures. In the present context, these
structures are important because they carry the information necessary to
construct the universal resolutions in A while being flexible enough that all
resolutions can always be given $\mathrm{A}_{\infty}$-structures.

An introduction to $\mathrm{A}_{\infty}$-algebras and
$\mathrm{A}_{\infty}$-modules over commutative rings is given in Section 4. In
Section 5 we prove some quite general transfer results, in particular,
constructing $\mathrm{A}_{\infty}$-structures on minimal resolutions of local
rings and modules. Burke was one of the first to develop and apply the theory
of $\mathrm{A}_{\infty}$-algebras over a commutative ring (rather than over a
field) [Bur15, Bur18], and our treatment owes a substantial intellectual debt
to his work.

We develop the theory of cyclic $\mathrm{A}_{\infty}$-algebras over
commutative rings in Section 5.6. These were introduced by Kontsevich as part
of his homological mirror symmetry program [Kon94]. These
$\mathrm{A}_{\infty}$-algebras possess extra structure that takes advantage of
the Poincaré duality on the minimal resolution of a Gorenstein ring, and we
apply this theory to almost Golod Gorenstein rings.

With this perspective in hand we return to Koszul homomorphisms in Section 7.
The next result shows that, at the derived level, Koszul homomorphisms may be
thought of as deformations of classical Koszul algebras. A more precise and
more general statement is given in Theorem 7.1. We write ${\rm T}(V)$ for the
tensor algebra $\bigoplus_{n\geqslant 0}V^{\otimes_{Q}n}$ on a graded
$Q$-module $V$.

######  Theorem B.

A surjective local homomorphism $\varphi\colon Q\to R$ is Koszul if and only
if there is a positively graded, degreewise finite rank free $Q$-module $V$, a
direct summand $W\subseteq V\otimes_{Q}V$, and an
$\mathrm{A}_{\infty}$-structure $\\{m_{n}\\}$ on $A={\rm T}(V)/(W)$ such that

  1. (1)

the induced quotient $A\to R$ is an $\mathrm{A}_{\infty}$-algebra quasi-
isomorphism,

  2. (2)

modulo the maximal ideal of $Q$, the $\mathrm{A}_{\infty}$-stucture
$\\{m_{n}\\}$ on $A$ agrees with the usual algebra structure on ${\rm
T}(V)/(W)$, that is,

$m_{2}\otimes_{Q}k=\mu\otimes_{Q}k\quad\text{and}\quad m_{n}\otimes_{Q}k=0\
\text{ for }\ n\neq 2\,,$

where $\mu$ is the usual product on the quotient of a tensor algebra,

  3. (3)

the $k$-algebra ${\rm T}(V\otimes_{Q}k)/(W\otimes_{Q}k)$ is Koszul with this
algebra structure.

Coming full circle, we are now able to describe the coalgebra $C$ that appears
in A. Fixing a Koszul homomorphism $\varphi$ with $V$ and $W$ as in B, we
define

$C\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\bigoplus_{n}\Big{(}{\textstyle\bigcap\limits_{i+2+j=n}V^{\otimes_{Q}i}\otimes_{Q}W\otimes_{Q}V^{\otimes_{Q}j}}\Big{)}\,.$

This is modeled on the work of Priddy [Pri70]. By construction, the $Q$-dual
$C^{\vee}$ is the quadratic dual ${\rm T}(V^{\vee})/(W^{\perp})$ of the
algebra ${\rm T}(V)/(W)$. The strict Koszul condition introduced in Section
7.2 guarantees that the $\mathrm{A}_{\infty}$-structure on $A$ induces the
structure of a curved differential graded coalgebra on $C$; see Definition
7.3. We think of $C$ as Koszul dual to $R$ _relative to $Q$_, as justified by
Theorem 7.5.

We conclude the paper with a reexamination of examples, in Section 8. We start
by showing that certain deformations of classical Koszul algebras yield
strictly Koszul homomorphisms, and we obtain resolutions that directly deform
the original resolutions of Priddy. We study Golod homomorphisms in Example
8.2; in this case $C=\mathsf{B}(A)$ is the bar construction of the
$\mathrm{A}_{\infty}$-algebra $A$, and we recover the bar resolution of
Iyengar and Burke. In Example 8.4 we show that Gorenstein homomorphisms of
projective dimension three are strictly Koszul, and describe $C$ as the dual
of a noncommutative hypersurface. We show that complete intersection
homomorphisms are strictly Koszul in Section 8.7, and we show that $C$ is the
free divided power algebra on $V$; our twisted tensor products encode the
theory of higher homotopies, and the resulting resolutions recover those of
Shamash and Eisenbud. We end by treating almost Golod Gorenstein rings in
Section 8.15; this requires a substantial amount of machinery, and the result
is a large class of interesting Gorenstein rings over which we have explicit,
small, universal resolutions.

######  Acknowledgements.

We owe thanks to Steven Amelotte for many useful conversations, in particular
improving our understanding of moment angle manifolds, the combinatorics of
simplicial complexes, and almost linear resolutions. We thank Alexander
Berglund for sharing important insights on Koszul-type phenomena in algebra
and topology. For many fruitful discussions on systems of higher homotopies,
and for allowing us to share some of these ideas here, we are grateful to
Eloísa Grifo. We thank Srikanth Iyengar for his encouragement and many useful
discussions on $\mathrm{A}_{\infty}$-structures. We also thank Trung Chau,
Michael DeBellevue, and Keller VandeBogert for helpful conversations regarding
concrete examples.

Part of this work was done at the Hausdorff Research Institute for
Mathematics, Bonn, when Briggs, Letz, and Pollitz were at the “Spectral
Methods in Algebra, Geometry, and Topology” trimester, funded by the Deutsche
Forschungsgemeinschaft under Germany’s Excellence
Strategy–EXC-2047/1–390685813. Briggs is supported by the European Union under
the Grant Agreement no. 101064551, and part of this work was completed when he
was funded by NSF grant DMS-1928930. Letz was partly supported by the Deutsche
Forschungsgemeinschaft (SFB-TRR 358/1 2023 - 491392403) and by the Alexander
von Humboldt Foundation in the framework of a Feodor Lynen research fellowship
endowed by the German Federal Ministry of Education and Research. Pollitz was
supported by NSF grants DMS-1840190, DMS-2002173, and DMS-2302567.

##  2\. Koszul homomorphisms

In this section we discuss the Koszul property in various settings, starting
from the classical notion for an algebra over a field and leading up to a
definition of a Koszul local homomorphism. Examples have been collected in the
Section 3. While later sections exploit the machinery of
$\mathrm{A}_{\infty}$-algebras, this section requires only knowledge of
differential graded (dg) algebras; a suitable reference for the latter is
[Avr98].

We fix a field $k$, and work with local rings having residue field $k$. We
also consider modules with two $\mathbb{Z}$-gradings, the _weight grading_ and
the _homological grading_. The weight grading is denoted $M=M_{(\star)}$ and
the corresponding shift $M(d)$ is given by $M(d)_{(w)}=M_{(w+d)}$. For the
homological grading we write $M=M_{\bullet}$ and the suspension $\Sigma^{d}M$
is given by $(\Sigma^{d}M)_{i}=M_{i-d}$. The homological degree of an element
$m\in M$ is denoted $|m|$. We assume that the two gradings are _compatible_ in
the sense that $M$ is bigraded by its submodules
$M_{i,(w)}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}M_{i}\cap
M_{(w)}$. If $M$ is a complex, the differential $\partial^{M}$ should preserve
the weight grading and decrease the homological grading by one, and we equip
$\Sigma M$ with the differential $\partial^{\Sigma
M}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}-\partial^{M}$.

######  Definition 2.1.

An augmented $k$-algebra $K$ is _Koszul (over $k$)_ if it admits an algebra
grading $K=\bigoplus_{w\geqslant 0}K_{(w)}$, known as a _weight grading_ ,
such that $K_{(0)}=k$ and such that the minimal resolution of $k$ is linear
with respect to this grading.

######  Remark 2.2.

Definition 2.1 is essentially the classical definition of a Koszul algebra due
to Priddy [Pri70, Section 2], see also [Löf86, Frö99], except that we do not
consider the weight grading to be part of the given data. We emphasize that
the additional grading may be _different_ to the given one; cf. Remark 3.3. By
[BGS96, Secton 2.3] a Koszul algebra is quadratic with respect to this new
grading; that is, generated as an associative $k$-algebra by elements of
weight one, subject to relations of weight two. For example if $U$ is a graded
vector space then the trivial extension algebra $K=k\ltimes U$ is a graded
augmented algebra, and the weight grading $K_{(0)}=k$ and $K_{(1)}=U$ makes
$K$ Koszul; see Example 3.4.

Next we define a Koszul property for dg algebras. This definition appears
implicitly in the work of Berglund [Ber14], where it is applied to Sullivan
models for topological spaces; it is _not_ the same as the Koszul property
introduced in [HW08], which is too general for our purposes.

######  2.3.

Let $A$ and $B$ be dg $k$-algebras. Recall that $A$ is _quasi-isomorphic_ to
$B$, denoted $A\simeq B$, if there exists a zig-zag of quasi-isomorphisms of
dg $k$-algebras connecting $A$ and $B$. A dg $k$-algebra $K$ is called
_formal_ if it is quasi-isomorphic to $\mathrm{H}(K)$.

######  Definition 2.4.

An augmented dg $k$-algebra $K$, such that $\mathrm{H}(K)$ is _Koszul_ if it
is formal and $\mathrm{H}(K)$ is Koszul in the sense of Definition 2.1.

######  Remark 2.5.

It is well-known that formality is closely related with the Koszul property.
In fact, an augmented $k$-algebra $K$ is Koszul in the sense of Definition 2.1
if and only if the dg $k$-algebra $\operatorname{\mathsf{R}Hom}_{K}(k,k)$ is
formal; see [GM74] and also [Kel02, 2.2] and [Ber14, Theorem 2.9]. This
condition is sometimes called _coformality_ of $K$. From this perspective, a
dg algebra $K$ is Koszul in the sense of Definition 2.4 if and only if it is
both formal and coformal.

Before introducing the Koszul property for local homomorphisms, we need to
recall the notion of the derived fiber. Let $\varphi\colon Q\to R$ be a local
homomorphism of commutative noetherian local rings having maximal ideals
$\mathfrak{m}_{Q}$ and $\mathfrak{m}_{R}$, respectively, and common residue
field $k$.

Let $A\to R$ be a dg algebra resolution of $R$ over $Q$; that is, $A$ is a dg
algebra concentrated in non-negative degrees, such that $A$ is degreewise a
free $Q$-module, and $A\to R$ is a morphism of dg algebras inducing an
isomorphism in homology. The _derived fiber_ of $\varphi$ is the dg
$k$-algebra

$R\otimes^{\mathsf{L}}_{Q}k\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}A\otimes_{Q}k\,.$

Up to a zig-zag of quasi-isomorphisms of dg $k$-algebras,
$R\otimes^{\mathsf{L}}_{Q}k$ is independent of the choice of $A$. For more
information see [Avr86]. We remark that one can equally well use a different
species of model for the resolution $A$, such as simplicial algebras or
$\mathrm{A}_{\infty}$-algebras. We use the latter in Section 7.

######  Definition 2.6.

Let $\varphi\colon Q\to R$ be a finite local homomorphism. We say that
$\varphi$ is _Koszul_ if $R\otimes^{\mathsf{L}}_{Q}k$ is a Koszul dg
$k$-algebra; that is, $R\otimes^{\mathsf{L}}_{Q}k$ is formal and its homology
$\operatorname{Tor}^{Q}(R,k)$ is Koszul in the sense of Definition 2.1.

According to Definition 2.1, when $\varphi$ is Koszul, the Tor algebra
$\operatorname{Tor}^{Q}(R,k)$ admits a quadratic presentation, albeit not
necessarily generated by elements in homological degree one.

Taking $Q=k$ to be a field, we recover the classical definition: The
homomorphism $k\to R$ is Koszul if and only if $R$ is a Koszul $k$-algebra;
cf. Definition 2.1.

The examples given in the next section show that Koszul homomorphisms are
extremely common. In particular, this class includes all flat local
homomorphisms whose fibers are (classically) Koszul; all complete intersection
and all Golod homomorphisms; Cohen presentations of most local rings having
codepth at most $3$, and of generic Gorenstein rings.

######  Remark 2.7.

When the minimal $Q$-free resolution $A$ of $R$ admits a dg algebra structure,
the derived fiber $R\otimes^{\mathsf{L}}_{Q}k=A\otimes_{R}k$ has zero
differential, and is automatically formal. This is the case for complete
intersection homomorphisms (Example 3.2) and when
$\operatorname{proj\,dim}_{Q}(R)\leqslant 3$ (Theorem 3.10). When $\varphi$ is
Golod the minimal resolution typically does not support a dg algebra
structure, cf. Remark 8.3, but nonetheless $R\otimes^{\mathsf{L}}_{Q}k$ is
formal (Example 3.5). The monomial rings presented in Examples 3.24 and 3.6
yield non-Koszul homomorphisms; the corresponding minimal resolution does not
admit a dg algebra structure in the former example (see [Avr81, 2.2]), but it
does in the latter example.

###  2.8. Cohen Koszul local rings

Recall that every local ring $R$ admits a Cohen presentation, that is, a
surjection $\varphi\colon Q\to\widehat{R}$ from a regular local ring $Q$, and
one may assume that $\varphi$ is minimal in the sense that
$\ker(\varphi)\subseteq\mathfrak{m}_{Q}^{2}$.

######  Definition 2.9.

A local ring $R$ is _Cohen Koszul_ if every homomorphism $\varphi\colon
Q\to\widehat{R}$ is Koszul, where $Q$ is a regular local ring and $\varphi$ is
surjective with $\ker(\varphi)\subseteq\mathfrak{m}_{Q}^{2}$. In other words,
every minimal Cohen presentation of $R$ is Koszul.

######  Remark 2.10.

For equicharacteristic rings the minimal Cohen presentation $Q\to\widehat{R}$
is essentially unique. In this situation, if $R$ is already a quotient of a
regular local ring, then by Proposition 2.16 there is no need to complete $R$
to determined whether $R$ is Cohen Koszul.

Also in this equicharacteristic case, $R$ can be given the structure of an
algebra over its residue field $k$, and then the Koszul complex $K^{R}$ on the
maximal ideal of $R$ is a dg $k$-algebra as well. It is well-known that

$K^{R}\simeq\widehat{R}\otimes^{\mathsf{L}}_{Q}k$

as dg $k$-algebras, see for example [ABIM10, Theorem 8.1]. Therefore in this
situation we can say that $R$ is Cohen Koszul exactly when $K^{R}$ is a Koszul
dg $k$-algebra.

In the mixed characteristic case, the fact that $K^{R}$ is not a dg
$k$-algebra introduces subtleties. The distinction between formality of dg
$k$-algebras and formality of dg rings means that it is not clear whether
Definition 2.9 is independent of the choice of Cohen presentation. In all of
our examples however, the choice will be irrelevant.

Complete intersection rings are Cohen Koszul by Example 3.2, Golod rings are
Cohen Koszul by Example 3.5, and most rings of codepth $3$ are Cohen Koszul
according to Theorem 3.10.

Cohen Koszul local rings have rational Poincaré series. Recall that the
Poincaré series of a finitely generated $R$-module $M$ is

$\operatorname{P}^{R}_{M}(t)=\sum_{n\in\mathbb{Z}}\operatorname{rank}_{k}(\operatorname{Tor}_{n}^{R}(M,k))t^{n}\,.$

######  Proposition 2.11.

Let $R$ be a Cohen Koszul local ring with embedding dimension $e$ and residue
field $k$. Fix a weight grading making the Koszul homology $\mathrm{H}(K^{R})$
a Koszul $k$-algebra. Then

$\operatorname{P}^{R}_{k}(t)=\frac{(1+t)^{e}}{\sum_{i,w}(-1)^{w}\operatorname{rank}_{k}(\mathrm{H}_{i}(K^{R})_{(w)})t^{i+w}}\,.$

######  Remark 2.12.

We note that since $\mathrm{H}_{*}(K^{R})$ is generated in weight $1$, the
rank of $\mathrm{H}_{i}(K^{R})_{(w)}$ is equal to the rank of
$[\mathrm{H}_{>0}(K^{R})^{w}/\mathrm{H}_{>0}(K^{R})^{w+1}]_{i}$. Therefore is
is not necessary to _choose_ a weight grading to calculate the Poincaré series
above.

###### Proof.

Let $T=\mathrm{H}(K^{R})$, bigraded by homological degree and by weight, and
write

$\mathrm{H}_{T}(s,t)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\sum_{i,w}\operatorname{rank}_{k}(T_{i,(w)})t^{i}s^{w}\quad\text{and}\quad\operatorname{P}^{T}_{k}(s,t)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\sum_{i,w}\operatorname{rank}_{k}(\operatorname{Tor}_{w}^{T}(k,k)_{i})t^{i}s^{w}\,.$

In $\operatorname{Tor}_{w}^{T}(k,k)_{i}$, the index $w$ is the usual
homological grading of $\operatorname{Tor}$, and $i$ is the extra grading that
comes from the homological grading on $T$. Since $T$ is Koszul with respect to
its weight grading, the usual computation of the Poincaré series of a Koszul
algebra shows that $\mathrm{H}_{T}(s,t)\operatorname{P}^{T}_{k}(-s,t)=1$; see
[Löf86].

Let $Q\to\widehat{R}$ be a minimal Cohen presentation. Formality of
$\widehat{R}\otimes^{\mathsf{L}}_{Q}k$ implies that the spectral sequence
[Avr81, 6.2.1] is degenerate, and so from [Avr81, 6.2 (b’)] we obtain the
first equality below, which yields the desired series

$\operatorname{P}^{R}_{k}(t)=(1+t)^{e}\operatorname{P}^{T}_{k}(t,t)=\frac{(1+t)^{e}}{\mathrm{H}_{T}(-t,t)}\,.\qed$

Proposition 2.11 recovers the known Poincaré series for complete intersection
rings, Golod rings, and almost Golod Gorenstein rings; see Section 3 for the
latter.

There are a number of results that apply to certain subsets of Cohen Koszul
rings, motivating the study of whether such a property holds for these rings
in general. We highlight a couple instances below.

######  Remark 2.13.

Recently, Brown–Dao–Sridhar have shown that over complete intersection and
Golod rings, the ideals of minors of differentials in minimal free resolutions
are eventually two-periodic [BDS23]. It would be worthwhile, and seems
plausible (in light of the structural result in Theorem 7.7), to determine
whether (strictly) Cohen Koszul rings satisfy this property more generally.

######  Remark 2.14.

Lower bounds on the Loewy length of the homology module of perfect complexes
are of interest in both algebra and topology; see, for example, [ABIM10, AP93,
Car83, IW18, Wal17]. For Cohen Koszul rings one can establish such bounds.

Let $R$ be a local ring with residue field $k$, and let
$k[\chi_{1},\ldots,\chi_{n}]$ denote a maximal polynomial subalgebra of the
graded $k$-algebra $\operatorname{Ext}_{R}(k,k)$, generated by elements in
even degree. For example, if $R$ is complete intersection, then $n$ is the
codimension of $R$. If $R$ is Cohen Koszul, then for any finite free
$R$-complex $F$ with $\mathrm{H}(F)\neq 0$ one has the inequality

$\sum_{n\in\mathbb{Z}}\ell\ell_{R}\mathrm{H}_{n}(F)\geqslant n+1\,.$

One can use similar ideas to those in [ABIM10], as well as [BGP23], to
establish this bound; here, however, formality of the derived fiber of a Cohen
presentation of $R$ is a main ingredient. Moreover, when $R$ is complete
intersection it agrees with the common bounds from [ABIM10, BGP23].

###  2.15. Properties of Koszul homomorphisms

Before moving on to examples we establish some basic change of rings
properties for Koszul homomorphisms. First, we note that being Koszul is
invariant under certain flat base changes, and in particular under completion.

######  Proposition 2.16.

Given a finite local homomorphism $\varphi\colon Q\to R$ and a flat local
homomorphism $\psi\colon Q\to Q^{\prime}$ inducing an isomorphism on residue
fields, $\varphi$ is Koszul if and only if $\varphi\otimes Q^{\prime}\colon
Q^{\prime}\to R\otimes_{Q}Q^{\prime}$ is Koszul.

###### Proof.

The natural map
$R\otimes^{\mathsf{L}}_{Q}k\to(R\otimes_{Q}Q^{\prime})\otimes^{\mathsf{L}}_{Q^{\prime}}k$
is a quasi-isomorphism of dg $k$-algebras. Indeed, if $A$ is a dg algebra
resolution of $R$ over $Q$, then $A\otimes_{Q}Q^{\prime}$ is a dg algebra
resolution of $R\otimes_{Q}Q^{\prime}$ over $Q^{\prime}$, and
$(A\otimes_{Q}Q^{\prime})\otimes_{Q^{\prime}}k\cong A\otimes_{Q}k$. ∎

The next proposition will often be useful in reducing the dimension of $Q$ or
$R$.

######  Proposition 2.17.

Let $\varphi\colon Q\to R$ be a finite local homomorphism, and let
$x\in\mathfrak{m}_{Q}$ and $y\in\mathfrak{m}_{R}$.

  1. (1)

If $x$ is regular on $Q$ and $y$ is regular on $R$, with $\varphi(x)=y$, then
$\varphi$ is Koszul if and only if the map of quotients $Q/(x)\to R/(y)$ is
Koszul.

  2. (2)

If $y$ is regular on $R$ then $\varphi$ is Koszul if and only if the
composition $Q\to R/(y)$ is Koszul.

  3. (3)

If $\varphi(x)=0$ and $x$ generates a free $R$-module summand of
$\ker(\varphi)/\ker(\varphi)^{2}$, then $\varphi$ is Koszul if and only if the
induced map $Q/(x)\to R$ is Koszul.

###### Proof.

For part Item 1, let $A\xrightarrow{\simeq}R$ be a dg algebra resolution of
$R$ over $Q$. The assumptions on $x$ and $y$ imply that $A\otimes_{Q}Q/(x)$ is
a dg algebra resolution of $R/(y)$ over $Q/(x)$. In particular there are
quasi-isomorphisms

$R\otimes^{\mathsf{L}}_{Q}k\simeq
A\otimes_{Q}k\cong(A\otimes_{Q}Q/(x))\otimes_{Q/(x)}k\simeq
R/(y)\otimes^{\mathsf{L}}_{Q/(x)}k$

of dg $k$-algebras, and the claim follows.

For part Item 2, if $A$ is is a dg algebra resolution of $R$ over $Q$, as
above, then there is an element $\tilde{y}\in\mathfrak{m}_{Q}A_{0}$ mapping to
$y\in R$. Taking an exterior variable $e$ of degree $1$, and setting
$\partial(e)=\tilde{y}$, the extension $A\langle e\rangle$ is then a dg
algebra resolution of $R/(y)$ over $Q$; see [Avr98, 6.1]. We see that

$R/(y)\otimes^{\mathsf{L}}_{Q}k\simeq A\langle
e\rangle\otimes_{Q}k\cong(A\otimes_{Q}k)\otimes_{k}\Lambda_{k}(e)\,,$

where $\Lambda_{k}(e)$ is the exterior algebra over $k$ on the degree $1$
variable $e$. Hence it reamins to note that the tensor product of dg
$k$-algebras is formal if and only if both of its factors are formal, and
Koszul if and only if both of its factors are Koszul; see [Frö99, Theorem 2]
for the latter.

For part Item 3 we invoke [Iye01, Proposition 2.1] to obtain a dg algebra
resolution $A$ of $R$ over $Q$ and an isomorphism of dg $k$-algebras
$A\otimes_{Q}k\cong W\otimes_{k}\Lambda_{k}(e)$, where $W$ is a dg subalgebra
of $A\otimes_{Q}k$ and $\Lambda_{k}(e)$ is the exterior algebra on a generator
of degree $1$ (this result is based upon André’s theory of special cycles
[And82]). Moreover, the proof in [Iye01] identifies the inclusion
$\Lambda_{k}(e)\to A\otimes_{Q}k$ with the natural map
$Q/(x)\otimes^{\mathsf{L}}_{Q}k\to R\otimes^{\mathsf{L}}_{Q}k$. It follows
that

$R\otimes^{\mathsf{L}}_{Q/(x)}k\simeq(R\otimes^{\mathsf{L}}_{Q}k)\otimes^{\mathsf{L}}_{Q/(x)\otimes^{\mathsf{L}}_{Q}k}k\simeq(W\otimes_{k}\Lambda_{k}(e))\otimes_{\Lambda_{k}(e)}k\cong
W\,.$

As in part Item 2 we may deduce that $R\otimes^{\mathsf{L}}_{Q/(x)}k$ is
Koszul if and only if
$R\otimes^{\mathsf{L}}_{Q}k\simeq(R\otimes^{\mathsf{L}}_{Q/(x)}k)\otimes_{k}\Lambda_{k}(e)$
is Koszul. ∎

######  Remark 2.18.

Many other Koszul-like properties have appeared in the literature. Within
local commutative algebra, Herzog, Reiner, and Welker introduced a notion of
Koszul local ring in [HRW98], and the same condition is investigated in
[HI05]. The local ring $k\llbracket x,y\rrbracket/(x^{2}-y^{3})$ is Koszul in
the sense of these references, but it is not a Koszul $k$-algebra according to
Definition 2.1, since it does not admit a quadratic presentation. However, the
same ring $k\llbracket x,y\rrbracket/(x^{2}-y^{3})$ is Koszul as a
$k\llbracket y\rrbracket$-algebra by Example 3.1, and it is Koszul as a
$k\llbracket x,y\rrbracket$-algebra by Example 3.2—in other words, it is Cohen
Koszul.

Myers studied a Koszulity condition in [Mye21] that is closely related to
ours. That work begins with a standard graded $k$-algebra $R$, and its Koszul
homology $\mathrm{H}(K^{R})$ is said to be _strand Koszul_ if it is Koszul
with respect to the induced weight grading by strands:
$\mathrm{H}(K^{R})_{(w)}=\bigoplus_{i+j=w}\mathrm{H}_{i}(K^{R})_{j}$ (the
total of the homological and internal gradings). According to [Mye21, Theorem
B] the Koszul complex $K^{R}$ is automatically quasi-formal in this situation.
In contrast, $R$ is Cohen Koszul if $K^{R}$ is formal and $\mathrm{H}(K^{R})$
is Koszul with respect to _any_ weight grading. In the next section we see
that there are many natural examples for which $\mathrm{H}(K^{R})$ is Koszul
with respect to a different grading than the strand grading.

The authors of [CDG+20] have also investigated how the Koszul condition on a
local ring affects the algebra structure of the Koszul homology
$\mathrm{H}(K^{R})$. While this is connected to the present work, we note that
there are many examples of local $k$-algebras that are Cohen Koszul but not
Koszul as $k$-algebras.

######  Remark 2.19.

We end this section with remarks on the generality of Definition 2.6.

We have chosen to focus on the setting of finite $Q$-algebras because this is
necessary to meaningfully talk about transferring homological information from
$Q$ to $R$. However, the notion of Koszul homomorphism can be extended
fruitfully to all local homomorphisms, with some additional technicalities. In
particular, to accommodate non-finite algebras, Definition 2.1 should be
adapted to require an isomorphism on the completion at the augmentation ideal
$\widehat{K}\cong\prod_{w\geqslant 0}K_{(w)}$, and that $k$ admit a linear
resolution over the corresponding graded ring $\bigoplus_{w\geqslant
0}K_{(w)}$.

For $Q$ non-local, one can say a $Q$-algebra $R$ is _Koszul at a prime_
$\mathfrak{p}\in\operatorname{Spec}(Q)$ if
$\kappa(\mathfrak{p})\otimes^{\mathsf{L}}_{Q}R$ is a Koszul dg algebra over
$\kappa(\mathfrak{p})=Q_{\mathfrak{p}}/\mathfrak{p}Q_{\mathfrak{p}}$. From
this perspective it is natural to replace $R$ with a sheaf of algebras on some
scheme; examples related to this have appeared in the literature, such as the
sheaf of Clifford algebras constructed by Buchweitz in [BEH87, Appendix].

In this work we focus on applications to commutative algebra. The natural
generalization to non-commutative algebras is interesting as well, using
exactly the same definitions.

##  3\. Examples of Koszul homomorphisms

This section contains examples (and counterexamples) demonstrating the
ubiquity of the Koszul condition. The first class of examples generalizes the
class of Koszul algebras over a field in a straightforward manner.

######  Example 3.1 (Flat homomorphisms with Koszul fiber).

A flat finite local homomorphism $\varphi\colon Q\to R$ is Koszul if and only
if its fiber $R\otimes_{Q}k$ is a Koszul $k$-algebra. Such examples are
readily constructed by deforming presentations of known Koszul algebras. For
example, the $k$-algebra $k[x]/(x^{2})$ is Koszul, and so the homomorphism

$Q\to Q[x]/(x^{2}-ax-b)$

is Koszul for any $a,b\in\mathfrak{m}_{Q}$.

We will see that the non-flat case is significantly more interesting, and
crucially _there is no need for the map $\varphi\colon Q\to R$ to be quadratic
in any sense_, as the following examples demonstrate. Nonetheless, later we
return to the idea that Koszul homomorphisms look like deformations of
classical Koszul presentations; cf. Theorem 7.1.

######  Example 3.2 (Complete intersection homomorphisms).

Let $\varphi\colon Q\to R$ be a surjective, local, complete intersection
homomorphism of codimension $c$. That is, $\ker(\varphi)$ is generated by a
$Q$-regular sequence $\bm{f}=f_{1},\ldots,f_{c}$. In this case, the Koszul
complex $A=\operatorname{Kos}^{Q}(\bm{f})$ is a dg algebra resolution of $R$
over $Q$. Then

$R\otimes^{\mathsf{L}}_{Q}k=A\otimes_{Q}k=\Lambda_{k}(\Sigma
A_{1}\otimes_{Q}k)$

is the exterior algebra on a $k$-space of rank $c$ in homological degree one,
with zero differential. Thus the derived fiber of $\varphi$ is clearly formal,
and it is well-known to be a Koszul $k$-algebra with its weight homological
gradings coinciding; cf. [Pri70, Examples 2.2(2)]. In particular, a local
complete intersection ring is Cohen Koszul.

######  Remark 3.3.

For a Cohen Koszul ring $R$, the homological and weight grading on
$\mathrm{H}(K^{R})$ coincide if and only if $R$ is complete intersection.
Indeed, the reverse implication was indicated in Example 3.2, and the forward
implication follows from a Theorem of Wiebe [Wie69]; see also [BH98, Theorem
2.3.15].

######  Example 3.4 (Trivial extension algebras).

Given a graded ring $B$ and a graded $B$-module $U$, let $B\ltimes U$ denote
the trivial extension of $B$ by $U$. This is the graded module $B\oplus U$
with multiplication

$(b,u)\cdot(b^{\prime},u^{\prime})=(bb^{\prime},bu^{\prime}+b^{\prime}u)\,.$

Tha main case of interest is that $B$ is augmented to $k$, and $U$ is a graded
$k$-space thought of as a trivial $B$-module. If $B$ is also a $k$-algebra,
then $B$ is a Koszul if and only if $B\ltimes U$ is by [Che17].

In particular, for any $k$-space $U$ the local $k$-algebra $k\ltimes U$ is
Koszul. It is also Cohen Koszul, according to Example 3.5.

For a surjective local map $\varphi\colon Q\to R$ and $R$-module $M$ there is
the following coefficientwise inequality of Poincaré series:

(3.4.1)
$\operatorname{P}^{R}_{M}(t)\preccurlyeq\dfrac{\operatorname{P}^{Q}_{M}(t)}{1-t(\operatorname{P}^{Q}_{R}(t)-1)}\,.$

This fact is due to Serre; see, for example, [Avr98, Proposition 3.3.2].

######  Example 3.5 (Golod homomorphisms).

Let $\varphi\colon Q\to R$ be a surjective, local, Golod homomorphism. That
is, the Serre bound Eq. 3.4.1 is an equality for the residue field $M=k$. For
other, equivalent definitions, see [Avr86, Theorem 2.3]. In particular, a
surjective, local homomorphism is Golod if and only if there is a quasi-
isomorphism of dg algebras

$R\otimes^{\mathsf{L}}_{Q}k\simeq k\ltimes U\,,$

where $U$ is a positively graded vector space over $k$. The trivial algebra
$k\ltimes U$ is Koszul by [LV12, Proposition 3.4.9], see also Example 3.4, and
hence $\varphi$ is Koszul.

The examples above provide many instances of Cohen Koszul $k$-algebras
appearing in local commutative algebra, and _some_ of these examples are
Koszul in the classical sense: for example a local complete intersection
$k$-algebra is Koszul if and only if it is quadratic; see [Frö99, 3.1]. We
give a small example of a local $k$-algebra that is neither Koszul nor Cohen
Koszul.

######  Example 3.6.

Suppose $R=k\llbracket a,b,c\rrbracket/(a^{2},bc,ac+b^{2})$. A computation
shows $\Lambda_{k}(e_{1},e_{2},e_{3})/(e_{1}e_{2}e_{3})$ is an algebra retract
of $\mathrm{H}(K^{R})$; one could, for example, use Macaulay2 [GS] for this
calculation. In particular, $\mathrm{H}(K^{R})$ has a relation of weight $3$
in any weight grading, and so $\mathrm{H}(K^{R})$ cannot be a Koszul
$k$-algebra. Thus, $R$ is not Cohen Koszul. Moreover, $R$ is the completion of
a quadratic algebra that is not Koszul in the classical sense. One can see
this by computing the third differential in the minimal free resolution of $k$
over $R$; alternatively, see [BF85].

In Theorem 3.10, we see that this is part of an exceptional class of non-Cohen
Koszul local rings among rings having embedding dimension at most three.

######  Example 3.7 (Short Gorenstein local rings).

A local ring $R$ with maximal ideal $\mathfrak{m}_{R}$ is called _short
Gorenstein_ if it is Gorenstein and $\mathfrak{m}_{R}^{3}=0$. Equivalently,
these are the local rings having Hilbert series $\mathrm{H}_{R}(t)=1+nt+t^{2}$
for some $n$. This is an important class of local rings that occurs frequently
in what follows. If $R$ is also a $k$-algebra, then $R$ is Koszul by [Frö82]
or [Sch80] (this follows as well from the slightly earlier computations in
[LA78]). Short Gorenstein rings are also Cohen Koszul by Example 3.15.

###  3.8. Rings of small codepth

Recall that for a local ring $R$ with maximal ideal $\mathfrak{m}_{R}$ and
residue field $k$, its codepth is

$\operatorname{codepth}(R)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\operatorname{rank}_{k}(\mathfrak{m}_{R}/\mathfrak{m}_{R}^{2})-\operatorname{depth}(R)\,.$

This value is a measure of the singularity of $R$ in the sense that
$\operatorname{codepth}(R)=0$ if and only if $R$ is regular. The next result
illustrates that a local ring of small codepth is almost always Cohen Koszul.
First we remind the reader of the structure theorem on Koszul homology for
rings having codepth three.

######  3.9.

Assume $R$ has codepth three and fix a minimal Cohen presentation
$\varphi\colon Q\to\widehat{R}$. The minimal $Q$-free resolution $A$ of
$\widehat{R}$ supports a dg algebra structure; see, for example, [BE77].
Hence, $\widehat{R}\otimes^{\mathsf{L}}_{Q}k$ is formal and the algebra
structure of its homology
$T=\operatorname{Tor}^{Q}(\widehat{R},k)=\mathrm{H}(K^{R})$ has been
classified as follows.

Fix bases $\\{e_{1},\ldots,e_{\ell}\\}$, $\\{f_{1},\ldots,f_{m}\\}$ and
$\\{g_{1},\ldots,g_{n}\\}$ for $T_{1}$, $T_{2}$ and $T_{3}$, respectively. By
[AKM88], there are non-negative integer parameters $p,q,r$, satisfying

$p\leqslant\ell-1,\quad q\leqslant m-p,\quad r\leqslant\min\\{\ell,m\\}\,,$

such that $T$ is one of the graded-commutative algebras determined below,
where products between the basis elements not listed are zero:

CI:

$e_{1}e_{2}=f_{3},$ $e_{1}e_{3}=f_{2}$, $e_{2}e_{3}=f_{1}$, $e_{i}f_{i}=g_{1}$
for $i=1,2,3.$

TE:

$e_{1}e_{2}=f_{3},$ $e_{1}e_{3}=f_{2}$, $e_{2}e_{3}=f_{1}$.

B:

$e_{1}e_{2}=f_{3},$ $e_{1}f_{1}=g_{1}$, $e_{2}f_{2}=g_{1}$.

G$(r)$:

$e_{i}f_{i}=g_{1}$ for $i=1,\ldots,r$ and $r\geq 2$

H$(p,q)$:

$e_{p+1}e_{i}=f_{i}$ for $i=1,\ldots,p$, and $e_{p+1}f_{p+i}=g_{i}$ for
$i=1,\ldots,q$.

In each case, let $T^{\prime}$ denote the corresponding subalgebra on the
basis elements appearing in the multiplication table above. If $U$ is the
$k$-space spanned by the basis of elements of $T$ not recorded in the same
multiplication table, then note that there is an isomorphism of graded
$k$-algebras

(3.9.1) $T\cong T^{\prime}\ltimes U\,.$

Finally, as a matter of terminology, we say a local ring $R$ belongs to one of
these classes if $T=\operatorname{Tor}^{Q}(\widehat{R},k)$ has the
corresponding algebra structure.

######  Theorem 3.10.

A local ring of codepth two or less is Cohen Koszul. A local ring of codepth
three is Cohen Koszul if and only if it belongs to $\mathbf{CI}$,
$\mathbf{B}$, $\mathbf{G}(r)$ or $\mathbf{H}(p,q)$.

###### Proof.

Let $R$ be a local ring of codepth $c$ with residue field $k$. Fix a minimal
Cohen presentation $\varphi\colon Q\to\widehat{R}$ and set
$T=\operatorname{Tor}^{Q}(\widehat{R},k)$.

If $c\leqslant 2$, then $R$ must be complete intersection or Golod; that is,
$\varphi$ is a complete intersection or Golod homomorphism. This follows from
the Hilbert–Burch theorem (see [Bur68], as well [BH98, Theorem 1.4.17])
combined with a result of [Lev76, Theorem 2.3]; see also [Avr99, Proposition
5.3.4]. Therefore by Examples 3.2 and 3.5 $R$ is Cohen Koszul in either case.

Now assume $c=3$. The simplest case is when $R$ belongs to CI, since in this
case $R$ is complete intersection and so $R$ is Cohen Koszul; cf. Example 3.2.

For the remainder of the proof we adopt the notation from 3.9 and analyze the
graded algebra structure of $T$. By Examples 3.4 and 3.9.1, $T$ is Koszul if
and only if $T^{\prime}$ is, and so we replace $T$ by $T^{\prime}$ in what
follows.

If $T$ is H$(p,q)$, then it is the tensor product of a trivial extension
algebra on $e_{1},\dots,e_{p},f_{p+1},\dots,f_{p+q}$ and the exterior algebra
on $e_{p+1}$, where each of these have weight 1. Hence, $T$ is a tensor
product of Koszul $k$-algebras and so it is Koszul.

If $T$ is G$(r)$, we give $\\{e_{i}\\}$ and $\\{f_{i}\\}$ weight one and
$g_{1}$ weight two. Then $T$ is a short Gorenstein $k$-algebra and hence
Koszul; see Example 3.7.

If $T$ is B, then we give $T$ weight grading

$T_{(w)}=\begin{cases}k&w=0\\\ ke_{1}\oplus ke_{2}\oplus kf_{1}\oplus
kf_{2}&w=1\\\ kf_{3}\oplus kg_{1}&w=2\\\ 0&w\geqslant 3\,.\end{cases}$

As an algebra $T$ is the quotient of the exterior algebra

$T\cong\Lambda_{k}(e_{1},e_{2},f_{1},f_{2})/(e_{1}f_{2},e_{2}f_{1},f_{1}f_{2},e_{1}f_{1}-e_{2}f_{2})\,.$

The graded $k$-algebra $T$ is Koszul since its defining ideal has a quadratic
Gröbner basis, and is therefore Koszul by [MM22].

Finally, if $T$ is type TE, then $T$ is not quadratic with respect to any
weight grading. Indeed, the products in 3.9 force $e_{1},e_{2},e_{3}$ to all
have weight one, as well as the minimal relation $e_{1}e_{2}e_{3}=0$. Hence,
$T$ is not Koszul. ∎

######  Remark 3.11.

Theorem 3.10 deals with the ‘absolute’ case. That is to say, when a local ring
is Cohen Koszul. However given a surjective map of local rings $\varphi\colon
Q\to R$ one has that this is always Koszul for
$\operatorname{proj\,dim}_{Q}(R)\leqslant 2$. Indeed, in this case $\varphi$
is either a complete intersection homomorphism, or a Golod homomorphism. For
the case $\operatorname{proj\,dim}_{Q}(R)=3$, the structure theorem on
$T=\operatorname{Tor}^{Q}(R,k)$ discussed in 3.9 can be applied assuming that
$\ker(\varphi)$ is a perfect ideal. In this case $\varphi$ is Koszul except in
the case that $T$ belongs to TE.

A surjective local homomorphism $\varphi\colon Q\to R$ of finite projective
dimension is _Gorenstein of projective dimension $d$_ if

(3.11.1) $\mathrm{Ext}_{Q}^{i}(R,Q)=\begin{cases}R&i=d\\\ 0&i\neq
d\,.\end{cases}$

For example, Gorenstein rings of codimension $d$ are exactly those whose
minimal Cohen presentations are Gorenstein of projective dimension $d$. If
$d=3$ then $\operatorname{Tor}^{Q}(R,k)$ belongs to G$(r)$ and the dg algebra
structure on the minimal resolution of $R$ over $Q$ can be described
explicitly; one can hence verify directly, as is done below, that such maps
are Koszul.

######  Example 3.12 (Gorenstein homomorphisms of projective dimension $3$).

Assume
that $\varphi$ is Gorenstein of projective dimension $3$. Buchsbaum and
Eisenbud constructed the minimal free resolution of $R$ over $Q$ in [BE77,
Theorem 2.1 & 4.1]:

$A=0\to Q\to Q^{r}\xrightarrow{\psi}Q^{r}\to Q\to 0$

where $r\geqslant 3$ is odd and the first and third differential of $A$ depend
on Pfaffians of submatrices of the alternating matrix $\psi$. Furthermore $A$
is a graded-commutative dg algebra with the following multiplication: We fix
bases $\\{e_{i}\\}_{i=1}^{r}$, $\\{f_{i}\\}_{i=1}^{r},$ and $\\{g\\}$ for
$A_{1},$ $A_{2},$ and $A_{3}$, respectively. The multiplication is determined
by

$e_{i}e_{j}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\sum_{\ell=1}^{r}(\pm
1)\operatorname{pf}(\psi_{ij\ell})f_{\ell}\quad\text{for }i<j\,,\quad
e_{i}f_{j}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\delta_{ij}g\quad\text{and}\quad
f_{i}f_{j}=0$

where $\psi_{ij\ell}$ is the submatrix of $\psi$ obtained by deleting the
$i$th, $j$th and $\ell$th row and column, and $\delta_{ij}$ is the Kronecker
delta function. The exact description is not important for the sequel, see
[Avr98, Example 2.1.3] for detail.

When $r=3$, it follows that $A$ is a Koszul complex on three elements and so
$\varphi$ is a surjective complete intersection map, hence $\varphi$ is Koszul
by Example 3.2.

When $r\geqslant 5$, we have that
$\operatorname{pf}(\psi_{ij\ell})\in\mathfrak{m}_{Q}$ for any $i$, $j$ and
$\ell$. Hence the only non-zero products in the graded algebra $A\otimes_{Q}k$
are

$e_{i}f_{i}=f_{i}e_{i}=g\quad\text{for }1\leqslant i\leqslant r\,.$

Giving $A\otimes_{Q}k$ the weight grading

$(A\otimes_{Q}k)_{(w)}=\begin{cases}A_{0}\otimes_{Q}k&w=0\\\
(A_{1}\otimes_{Q}k)\oplus(A_{2}\otimes_{Q}k)&w=1\\\ A_{3}\otimes_{Q}k&w=2\\\
0&\text{else}\,,\end{cases}$

it belongs to G($r$) in 3.9. The fact that $A\otimes_{Q}k$ is a Koszul
$k$-algebra was established in the proof of Theorem 3.10. Thus $\varphi$ is a
Koszul homomorphism.

###  3.13. Almost Golod Gorenstein rings

We discuss here a large class of local Gorenstein rings displaying interesting
homological behavior, that has been studied before in [RŞ14, Section 6].

######  Definition 3.14.

We say that an artinian local ring $R$ is _almost Golod_ if the socle quotient
$R/\operatorname{soc}(R)$ is Golod. A general local ring is _almost Golod_ if
it is Cohen–Macaulay and $R/(\bm{x})$ is an almost Golod artinian ring, where
$\bm{x}$ is a maximal regular sequence that is part of a minimal generating
set for $\mathfrak{m}_{R}$.

######  Example 3.15 (Almost Golod Gorenstein rings).

Let $R$ be an almost Golod local ring that is also Gorenstein of dimension
$d$.

Fix a minimal Cohen presentation $\varphi\colon Q\to\widehat{R}$ and set
$T=\operatorname{Tor}^{Q}(\widehat{R},k)$. Since $Q$ is regular and $R$ is
Gorenstein, $T$ is a Poincaré duality algebra by [AG71]. That is to say, for
each $0\leqslant i\leqslant d$ the multiplication maps

$T_{i}\times T_{d-i}\to T_{d}\cong k$

are perfect pairings. Furthermore, by [LA78, Theorem 1] the quotient $T/T_{d}$
is a subalgebra of the trivial extension algebra
$\operatorname{Tor}^{Q}(R/\operatorname{soc}(R),k)$, and hence is itself a
trivial extension algebra. It follows that $T$ is a short Gorenstein
$k$-algebra. Moreover, prescribing $T$ with the following weight grading

$T_{(0)}=T_{0},\quad T_{(1)}={\textstyle\bigoplus_{i=1}^{d-1}}T_{i},\ \text{
and }\ \ T_{(2)}=T_{d}$

makes $T$ a Koszul $k$-algebra, with the multiplication of $T$ being
equivalent to a perfect pairing on $T_{(1)}$; see the proof of Theorem 3.10.

We prove that these rings are Cohen Koszul under the assumption that $R$
contains a field of characteristic zero and $d$ is odd, by giving a
characterization analogous to Avramov’s characterization of Golod rings
[Avr86]. We do not know whether the assumption on the characteristic or on $d$
is necessary.

######  Theorem 3.16.

Let $R$ be a local with a minimal Cohen presentation $Q\to\widehat{R}$. If
there is a quasi-isomorphism of dg $k$-algebras
$\widehat{R}\otimes^{\mathsf{L}}_{Q}k\simeq T$, where $T$ is a short
Gorenstein graded $k$-algebra, then $R$ is almost Golod Gorenstein. Assuming
that $R$ contains a field of characteristic zero, and that
$\operatorname{codepth}(R)$ is odd, the converse holds as well. In particular,
almost Golod Gorenstein rings (of characterstic zero and odd codepth) are
Cohen Koszul.

###### Proof.

If $R$ is almost Golod Gorenstein, we have already seen that
$T=\operatorname{Tor}^{Q}(\widehat{R},k)$ is a short Gorenstein algebra, and
in particular Koszul. The proof that $\widehat{R}\otimes^{\mathsf{L}}_{Q}k$ is
formal under the stated assumptions will be given in Theorem 5.7 and Lemma
8.17.

Conversely assume that $\widehat{R}\otimes^{\mathsf{L}}_{Q}k$ is quasi-
isomorphic to a short Gorenstein algebra $T$, and write $e$ for the embedding
dimension of $R$. By Proposition 2.11,

${\operatorname{P}^{{R}}_{k}(t)}=\frac{(1+t)^{e}}{1-(\mathrm{H}_{T}(t)-1-t^{e})t+t^{e+2}}\,,$

and hence as a consequence of [RŞ14, Proposition 6.2], $R$ is almost Golod. ∎

######  Remark 3.17.

The prototypical example of an almost Golod Gorenstein ring is a short
Gorenstein local ring $R$. In this case
$R/\operatorname{soc}(R)=R/\mathfrak{m}_{R}^{2}$ is Golod by [Gol78].

Among the complete intersection local rings, the almost Golod Gorenstein rings
are exactly those having codimension two or less; see Theorem 3.16. If $R$ is
a Gorenstein local ring of codimension $3$ that is not complete intersection,
then $R$ is almost Golod Gorenstein by Example 3.12 and Theorem 3.16.

By [RŞ14, Proposition 6.3] every Gorenstein compressed local ring of socle
degree at least $4$ is almost Golod Gorenstein. Moreover for fixed emdedding
dimension and socle degree, the generic Gorenstein local $k$-algebra is
compressed by [Iar84, Theorem I]. Hence the almost Golod Gorenstein condition
is extremely common.

If $Q$ is a standard graded polynomial algebra, a homogeneous quotient $R=Q/I$
is said to have an _almost linear_ resolution over $Q$ if the ideal $I$ is
generated by forms of degree $d$, and for all
$0<i<\operatorname{proj\,dim}_{Q}(R)$ we have
$\operatorname{Tor}_{i}^{Q}(R,k)_{j}=0$ unless $j-i=d-1$ [DE22]. This yields
another large class of examples: One can show that graded Gorenstein rings
with almost linear resolutions are almost Golod Gorenstein using techniques
from later in the paper (the $\mathrm{A}_{\infty}$-structure on
$\operatorname{Tor}^{Q}(R,k)$ will satisfy Theorem 7.1 for degree reasons, and
then Theorem 3.16 can be applied; we leave these details to the interested
reader).

######  Remark 3.18.

In rational homotopy theory, Golod rings correspond to spaces that are
(rationally) homotopy equivalent to a wedge of spheres, while Gorenstein rings
are analogous to manifolds, or more generally Poincaré duality spaces; see the
looking glass [AH86] for more information.

To be more precise, if $M$ is a simply connected manifold and the punctured
space $M\smallsetminus\\{{\rm pt}\\}$ is rationally homotopy equivalent to a
wedge of spheres, then the cohomology ring $\mathrm{H}^{*}(M;\mathbb{Q})$ is
an almost Golod Gorenstein ring. In [Sta83], Stasheff proved such spaces are
formal, and therefore they are Koszul in the sense of Berglund [Ber14]. A well
studied class of manifolds satisfying this property are the _highly connected
manifolds_ , that is, those $M$ with $\mathrm{H}^{i}(M;\mathbb{Q})=0$ when
$0<i<\lfloor\dim(M)/2\rfloor$.

Since Gorenstein rings that are not regular or hypersurfaces are never Golod,
the Serre bound Eq. 3.4.1 must be strict for such rings. However, a tighter
bound can be established for Gorenstein local rings, as we show now. The case
of equality below is equivalent (when $d=0$) to the formula for
$\operatorname{P}^{R}_{k}(t)$ given in [RŞ14, Proposition 6.2], and our proof
is essentially equivalent to that of _loc. cit_.

######  Proposition 3.19.

Let $R$ be a local ring having dimension $d$ and embedding dimension $e$, with
residue field $k$ and Koszul complex $K^{R}$. If $R$ is Gorenstein but not
regular or a hypersurface, then there is a coefficientwise inequality

$\frac{\operatorname{P}^{R}_{k}(t)}{(1+t)^{d}-t^{2}\operatorname{P}^{R}_{k}(t)}\preccurlyeq\frac{(1+t)^{e-d}}{1-t^{2}(1+t)^{e-d}+t^{e-d+2}-\sum_{i=1}^{e-d-1}\operatorname{rank}_{k}\mathrm{H}_{i}(K^{R})t^{i+1}}$

and equality holds if and only if $R$ is almost Golod.

While the left-hand side is not equal to $\operatorname{P}^{R}_{k}(t)$, it
increases monotonically with $\operatorname{P}^{R}_{k}(t)$, and so it directly
measures the growth of the resolution of $k$. Therefore within the class of
Gorenstein local rings, almost Golod rings display extremal behavior analogous
to Golod rings.

###### Proof.

As $R$ is Gorenstein, prime avoidance yields a regular sequence
$\bm{x}=x_{1},\ldots,x_{d}$ that is part of a minimal generating set for
$\mathfrak{m}_{R}$, so that $\bar{R}=R/(\bm{x})$ is artinian Gorenstein and of
embedding dimension $e-d$. By Nagata’s theorem [Nag62, Section 27] we have
$\operatorname{P}^{R}_{k}(t)=\operatorname{P}^{\bar{R}}_{k}(t)(1+t)^{d}$ and
so the first equality below holds:

$\displaystyle\frac{\operatorname{P}^{R}_{k}(t)}{(1+t)^{d}-t^{2}\operatorname{P}^{R}_{k}(t)}$
$\displaystyle=\frac{\operatorname{P}^{\bar{R}}_{k}(t)}{1-t^{2}\operatorname{P}^{\bar{R}}_{k}(t)}$
$\displaystyle=\operatorname{P}^{\bar{R}/\operatorname{soc}(\bar{R})}_{k}(t)$
$\displaystyle\preccurlyeq\frac{\operatorname{P}^{Q}_{k}(t)}{1-t(\operatorname{P}^{Q}_{\bar{R}/\operatorname{soc}(\bar{R})}(t)-1)}$
$\displaystyle=\frac{\operatorname{P}^{Q}_{k}(t)}{1-t(\operatorname{P}^{Q}_{\bar{R}}(t)-t^{e-d}+t\operatorname{P}^{Q}_{k}(t)-t^{e-d+1}-1)}\,.$

The second equality holds using [LA78, Theorem 2], the coefficientwise
inequality is the Serre bound Eq. 3.4.1 for
$\bar{R}/\operatorname{soc}(\bar{R})$, and the last equality holds using
[LA78, Theorem 1]. Since $\operatorname{P}^{Q}_{k}(t)=(1+t)^{e-d}$ and
$\operatorname{P}^{Q}_{\bar{R}}(t)=\sum_{i=0}^{i=e-d}\operatorname{rank}_{k}\mathrm{H}_{i}(K^{R})t^{i}$,
we obtain the claimed inequality. It remains to note that $R$ is almost Golod
if and only if $\bar{R}/\operatorname{soc}(\bar{R})$ is Golod, if and only if
equality holds in the third line above. ∎

Later, in Section 8.15, we explicitly construct resolutions over almost Golod
Gorenstein rings that achieve the bound of Proposition 3.19.

###  3.20. Monomial rings

In this subsection we consider rings of the form $R=Q/I$, where
$Q=k[x_{1},\ldots,x_{n}]$ and $I$ is generated by monomials
$m_{1},\ldots,m_{r}$. By Fröberg’s theorem [Frö75], $R$ is Koszul as a
$k$-algebra if and only if each $m_{i}$ is quadratic. However, the condition
that $R$ is Cohen Koszul is more common and more subtle.

######  Example 3.21 (Almost linear resolutions).

Let $d\geqslant 3$ be an integer and let $m$ be a monomial in $Q$ that is
divisible by all monomials of degree $\deg(m)-d+1$. Let $R=Q/I$, where $I$ is
the ideal generated by all monomials of degree $d$ not dividing $m$. Then by
[DE22, Theorem 4.2] the ring $R$ is Gorenstein and has an almost linear
resolution over $Q$, and moreover every zero dimensional monomial Gorenstein
ring having an almost linear resolution is of this form. These rings are
almost Golod Gorenstein, and in particular Cohen Koszul, by Remark 3.17.

An exact combinatorial characterization of which monomial rings are Cohen
Koszul would be very interesting; this seems possible but likely non-trivial.
We describe a special case that produces a large number of explicit examples.

######  Proposition 3.22.

Let $Q=k[x_{1},\ldots,x_{n}]$ be a polynomial ring over a field $k$, and let
$I=(m_{1},\ldots,m_{r})$ a monomial ideal such that each $m_{i}$ contains a
variable not in any other $m_{j}$. Then $R=Q/I$ is Cohen Koszul.

###### Proof.

The Taylor resolution $A$ of $R$ over $Q$ has a basis $\\{e_{I}\\}$ indexed by
subsets $I\subseteq\\{1,\ldots,n\\}$, with $e_{I}$ in homological degree
$|I|$, and the differential is defined by

$\partial(e_{I})=\sum_{i\in
I}\pm\frac{m_{I}}{m_{I\smallsetminus\\{i\\}}}e_{I\smallsetminus\\{i\\}}\,,$

where $m_{I}=\operatorname{lcm}\left\\{m_{i}\,\middle|\,i\in I\right\\}$; see
[Tay66] for details and signs. The hypothesis of the proposition exactly
guarantees that the Taylor resolution is minimal. Gemeda [Gem76] proved that
the Taylor resolution has a dg algebra structure with product

$e_{I}e_{J}=\pm\frac{m_{I}m_{J}}{m_{I\cup J}}e_{I\cup J}$

when $I\cap J=\varnothing$, and with $e_{I}e_{J}=0$ otherwise. By Remark 2.7
it follows that $R\otimes^{\mathsf{L}}_{Q}k=A\otimes_{Q}k$ is formal.

It remains to show that $A\otimes_{Q}k$ is a Koszul $k$-algebra. Let $M$ be
the graph with vertices $\\{1,\ldots,r\\}$ and an edge connecting $i$ and $j$
if and only if $\gcd(m_{i},m_{j})\neq 1$. From the description of $A$ above it
follows that

$A\otimes_{Q}k=\frac{k[e_{I}\mid I\subseteq M\text{
connected}]}{(e_{I}e_{J}\mid\gcd(m_{I},m_{J})\neq 1)}\,,$

where $k[e_{I}]$ is the free graded-commutative algebra on the indicated
$e_{I}$; compare this with [Ber05, 6.2]. Assigning each $e_{I}$ weight $1$, we
are done because quadratic monomial quotients of free graded-commutative
algebras are Koszul by Fröberg’s theorem [Frö75] (such rings belong to class B
in [Frö75, Section 3], and Fröberg constructs linear resolutions of the
residue field for all rings of class B). ∎

######  Remark 3.23.

One can readily exhibit monomial rings satisfying the hypothesis of the
proposition, and not falling into the other classes described above. For
example, $R=k\llbracket a,b,c,d,e,f\rrbracket/(abc,cd,ae,acf)$.

The next examples are $k$-algebras that fail to be Cohen Koszul; the first is
a Koszul $k$-algebra and the second has $\mathrm{H}(K^{R})$ a Koszul
$k$-algebra. Both examples fail to be Cohen Koszul since in each case $K^{R}$
admits a nonzero triple Massey product, and hence is not formal; cf. [May69]
for more details on Massey products.

######  Example 3.24.

The $k$-algebra $R=k\llbracket a,b,c,d\rrbracket/(a^{2},ab,bc,cd,d^{2})$ is
the completion of a Koszul $k$-algebra (in the classical sense) by [Frö75,
Corollary 1]. However, the map $k\llbracket a,b,c,d\rrbracket\to R$ is not
Koszul. Indeed, by [Avr81, Example 5.1.4], $K^{R}$ has a nonzero triple Massey
product, and so $K^{R}$ is not formal.

######  Example 3.25.

Let $Q=k\llbracket a,b,c,d,e\rrbracket$ and consider the quotient map

$\varphi\colon Q\to
R\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}Q/(ab^{2},cd^{2},e^{3},abcd,d^{2}e^{2},b^{2}e^{2},ace,b^{2}d^{2}e)\,.$

In [Kat17, Theorem 3.1], it is shown that $\mathrm{H}(K^{R})$ is a trivial
extension that admits a nonzero triple Massey product; the latter is an
obstruction to the formality of $K^{R}$, while the former justifies that
$\mathrm{H}(K^{R})$ is a Koszul $k$-algebra.

######  Remark 3.26.

To any monomial ideal $I\subseteq Q$ one may associate a square-free monomial
ideal $I^{\circ}$ in a larger polynomial ring $Q^{\circ}$, known as the
_polarization_ of $I$. Fröberg [Frö82] proved that there is a regular sequence
of linear forms $y_{1},\ldots,y_{t}$ in the quotient
$R^{\circ}=Q^{\circ}/I^{\circ}$ such that $R=R^{\circ}/(y_{1},\ldots,y_{t})$.
From Proposition 2.17 it follows that $R$ is Cohen Koszul if and only if
$R^{\circ}$ is Cohen Koszul.

A _simplicial complex_ $\Delta$ on $[n]=\\{1,\ldots,n\\}$ is a nonempty family
of subsets of $[n]$, closed under taking subsets. The _Stanley–Reisner ring_
associated to $\Delta$, denoted $k[\Delta]$, is the quotient of
$k[x_{1},\ldots,x_{n}]$ by the ideal generated by monomials $x_{i_{1}}\cdots
x_{i_{t}}$ such that $\\{i_{1},\ldots,i_{t}\\}\notin\Delta$. Every square free
monomial ring is the Stanley–Reisner ring of some simplical complex, and so by
Remark 3.26 we may restrict to such monomials rings.

######  Remark 3.27.

We make some remarks about the connections to toric topology; for precise
definitions and background on this area the reader may consult [BP15].

To a simplical complex $\Delta$ one also associates the moment angle complex
$\mathcal{Z}_{\Delta}$, a finite CW-complex with an action of the torus
$(S^{1})^{n}$. The homotopy quotient
$\mathcal{DJ}_{\Delta}=\mathcal{Z}_{\Delta}/\\!/(S^{1})^{n}$ is known as the
Davis–Januszkiewicz space of $\Delta$. By [DJ91, Theorem 4.8] and [NR05,
Theorem 4.8] the cochain algebra of this space is quasi-isomorphic to the
Stanley–Reisner ring:

$C^{*}(\mathcal{DJ}_{\Delta};k)\simeq k[\Delta]\,,$

where the variables $x_{i}$ are given cohomological degree $2$. From this it
follows that $k[\Delta]$ is a Koszul $k$-algebra if and only if
$\mathcal{DJ}_{\Delta}$ is a Koszul space in the sense of [Ber14]. As remarked
in [Ber14, Example 5.8], this happens exactly when $k[\Delta]$ is a quadratic
algebra, or equivalently if $\Delta$ is a _flag complex_ , that is, the
minimal faces not belonging to $\Delta$ are all edges.

The question of when $\mathcal{Z}_{\Delta}$ is a Koszul space seems to be more
interesting. By [BP15, Lemma 3.1] there is a quasi-isomorphism of dg
$k$-algebras

$C^{*}(\mathcal{Z}_{\Delta};k)\simeq k[\Delta]\otimes^{\mathsf{L}}_{Q}k\,.$

Thus $k[\Delta]$ is Cohen Koszul if and only if $\mathcal{Z}_{\Delta}$ is a
Koszul space. The related condition that $\mathcal{Z}_{\Delta}$ is formal has
been investigated in [DS07, Lim19].

The almost Golod condition is also connected with the minimally non-Golod
condition for simplicial complexes introduced in [BJ07]. Indeed, the proof of
[Ame20, Theorem 1.1] shows that if $M=\mathcal{Z}_{\Delta}$ is a moment angle
manifold, and if $M\smallsetminus\\{{\rm pt}\\}$ is rationally homotopy
equivalent to a wedge of spheres, then $\Delta$ is minimally non-Golod (over
$\mathbb{Q}$).

##  4\. Background on $\mathrm{A}_{\infty}$-algebras and coalgebras

Stasheff introduced $\mathrm{A}_{\infty}$-algebras in topology to characterize
loop spaces [Sta63a, Sta63b], and they have since proven a powerful tool in
algebra as a flexible generalization of dg algebras; for an overview see
[Kel01]. In our context, the minimal $Q$-free resolution of a finite
$Q$-algebra $R$ can be equipped with an $\mathrm{A}_{\infty}$-algebra
structure (see Section 5), and this will be leveraged to characterize Koszul
homomorphisms in terms of presentations similar to the quadratic presentations
for classical Koszul algebras (see Section 7).

From now on $Q$ is always a local ring with maximal ideal $\mathfrak{m}_{Q}$
and residue field $k$, and unadorned tensor products and Hom sets are taken
over $Q$.

######  4.1.

An _$\mathrm{A}_{\infty}$ -algebra_ is a graded $Q$-module $A$ equipped with
$Q$-linear maps

$m_{n}\colon A^{\otimes n}\to A\quad\text{for }n\geqslant 1$

of degree $(n-2)$ satisfying the _Stasheff identities_

(4.1.1) $\sum_{\begin{subarray}{c}r+s+t=n\\\ r,t\geqslant 0,s\geqslant
1\end{subarray}}(-1)^{r+st}m_{r+1+t}\left(\operatorname{id}^{\otimes r}\otimes
m_{s}\otimes\operatorname{id}^{\otimes t}\right)=0\,.$

Taking $n=1$ this says that $m_{1}$ is a degree $-1$ square zero endomorphism
of $A$, so we can (and will) make $A$ a complex with $\partial=m_{1}$. Taking
$n=2$ yields a product satisfying the Leibniz rule $\partial
m_{2}=m_{2}(\partial\otimes\operatorname{id}+\operatorname{id}\otimes\partial)$.
The next Stasheff identity, for $n=3$, can be interpreted as saying that
$m_{2}$ is associative up to a homotopy given by $m_{3}$, that is,

$m_{2}(\operatorname{id}\otimes m_{2}-m_{2}\otimes\operatorname{id})=\partial
m_{3}+m_{3}(\partial\otimes\operatorname{id}\otimes\operatorname{id}+\operatorname{id}\otimes\partial\otimes\operatorname{id}+\operatorname{id}\otimes\operatorname{id}\otimes\partial)\,.$

If for some $n$ the Stasheff identity Eq. 4.1.1 holds for every integer less
than $n$, then the obstruction

(4.1.2)
$\operatorname{obs}_{n}^{A}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\sum_{\begin{subarray}{c}r+s+t=n\\\
r,t\geqslant
0,n>s>1\end{subarray}}(-1)^{r+st}m_{r+1+t}\left(\operatorname{id}^{\otimes
r}\otimes m_{s}\otimes\operatorname{id}^{\otimes t}\right)$

is a chain map $A^{\otimes n}\to A$; see [LH03, Corollaire B.1.2].

A _morphism of $\mathrm{A}_{\infty}$-algebras_ $\varphi\colon A\to B$ consists
of $Q$-linear maps

$\varphi_{n}\colon A^{\otimes n}\to B\quad\text{for }n\geqslant 1$

of degree $(n-1)$ satisfying

(4.1.3) $\displaystyle\sum_{\begin{subarray}{c}r+s+t=n\\\ r,t\geqslant
0,s\geqslant 1\end{subarray}}(-1)^{r+st}\varphi_{r+1+t}$
$\displaystyle\left(\operatorname{id}^{\otimes r}\otimes
m^{A}_{s}\otimes\operatorname{id}^{\otimes t}\right)$
$\displaystyle=\sum_{p=1}^{n}\sum_{\begin{subarray}{c}\bm{\alpha}\in\mathbb{N}^{p}\\\
|\bm{\alpha}|=n\end{subarray}}(-1)^{v(\bm{\alpha})}m^{B}_{p}(\varphi_{\alpha_{1}}\otimes\ldots\otimes\varphi_{\alpha_{p}})$

where $\bm{\alpha}=(\alpha_{1},\ldots,\alpha_{p})$ and
$|\bm{\alpha}|=\sum_{k=1}^{p}\alpha_{k}$, with
$v(\bm{\alpha})=\sum_{k=1}^{p}(p-k)(\alpha_{k}-1)$.

If for some $n$ the Stasheff identity Eq. 4.1.3 holds for every integer less
than $n$, then we define

$\displaystyle\operatorname{obs}_{n}^{\varphi}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\sum_{\begin{subarray}{c}r+s+t=n\\\
r,t\geqslant 0,s\geqslant 2\end{subarray}}(-1)^{r+st}\varphi_{r+1+t}$
$\displaystyle\left(\operatorname{id}^{\otimes r}\otimes
m^{A}_{s}\otimes\operatorname{id}^{\otimes t}\right)$
$\displaystyle-\sum_{p=1}^{n-1}\sum_{\begin{subarray}{c}\bm{\alpha}\in\mathbb{N}^{p}\\\
|\bm{\alpha}|=n\end{subarray}}(-1)^{v(\bm{\alpha})}m^{B}_{p}(\varphi_{\alpha_{1}}\otimes\ldots\otimes\varphi_{\alpha_{p}})\,.$

Then the Stasheff identity Eq. 4.1.3 holds if and only if

$\operatorname{obs}_{n}^{\varphi}=m_{1}\varphi_{n}+(-1)^{n}\varphi_{n}(m_{1}\otimes\operatorname{id}^{\otimes(n-1)}+\cdots+\operatorname{id}^{\otimes(n-1)}\otimes
m_{1})\,.$

A morphism $\varphi$ of $\mathrm{A}_{\infty}$-algebras is a _quasi-
isomorphism_ if the chain map $\varphi_{1}$ is a quasi-isomorphism of
complexes. The morphism $\varphi$ is _strict_ if $\varphi_{n}=0$ for $n>1$.

The composition of morphisms $\varphi\colon A\to B$ and $\psi\colon B\to C$ is
defined by

$(\psi\circ\varphi)_{n}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\sum_{p=1}^{n}\sum_{\begin{subarray}{c}\bm{\alpha}\in\mathbb{N}^{p}\\\
|\bm{\alpha}|=n\end{subarray}}(-1)^{v(\bm{\alpha})}\psi_{p}(\varphi_{\alpha_{1}}\otimes\ldots\otimes\varphi_{\alpha_{p}})\,.$

######  4.2.

An $\mathrm{A}_{\infty}$-algebra $A$ is _strictly unital_ if there exists
$1_{A}\in A_{0}$ such that

(4.2.1) $\begin{gathered}m_{2}(1_{A}\otimes a)=a=m_{2}(a\otimes
1_{A})\quad\text{for all }a\in A\quad\text{and}\\\
m_{n}(a_{1}\otimes\ldots\otimes a_{i-1}\otimes 1_{A}\otimes
a_{i+1}\otimes\ldots\otimes a_{n})=0\quad\text{for all }1\leqslant i\leqslant
n\end{gathered}$

for any $a_{1},\ldots,a_{n}\in A$ and $n>2$. A morphism of strictly unital
$\mathrm{A}_{\infty}$-algebras $\varphi\colon A\to B$ is a morphism of
$\mathrm{A}_{\infty}$-algebras such that

(4.2.2) $\begin{gathered}\varphi_{1}(1_{A})=1_{B}\quad\text{and}\\\
\varphi_{n}(a_{1}\otimes\ldots\otimes a_{i-1}\otimes 1_{A}\otimes
a_{i+1}\otimes\ldots\otimes a_{n})=0\quad\text{for all }1\leqslant i\leqslant
n\end{gathered}$

for any $a_{j}\in A$ and $n>1$.

######  4.3.

An $\mathrm{A}_{\infty}$-algebra is _connective_ if it is concentrated in non-
negative degrees. If $A=Q\oplus\bar{A}$ is a graded module and $1_{A}$ a free
generator of the direct summand $Q$, then $A$ is an _$\mathrm{A}_{\infty}$
-algebra with a split unit_.

A split unital $\mathrm{A}_{\infty}$-algebra structure on a graded module
concentrated in non-negative degrees is equivalent to the existence of
$Q$-linear maps

$\bar{m}_{n}\colon\bar{A}^{\otimes n}\to\bar{A}\quad\text{for }n\geqslant 1$

of degree $(n-2)$ and $Q$-linear maps

$h_{1}\colon\bar{A}\to Q\quad\text{and}\quad h_{2}\colon\bar{A}^{\otimes 2}\to
Q$

of degrees $-1$ and $0$, respectively, such that for $n\neq 2,3$ the Stasheff
identities Eq. 4.1.1 hold when replacing $m_{i}$ by $\bar{m}_{i}$, and for
$n=2,3$

$\sum_{\begin{subarray}{c}r+s+t=n\\\ r,t\geqslant 0,s\geqslant
1\end{subarray}}(-1)^{r+st}\bar{m}_{r+1+t}(\operatorname{id}^{\otimes
r}\otimes\bar{m}_{s}\otimes\operatorname{id}^{\otimes
t})+(h_{n-1}\otimes\operatorname{id}-\operatorname{id}\otimes h_{n-1})=0\,,$

and additionally

$\begin{gathered}h_{1}\bar{m}_{1}=0\,,\quad
h_{1}\bar{m}_{2}-h_{2}(\bar{m}_{1}\otimes\operatorname{id}+\operatorname{id}\otimes\bar{m}_{1})=0\quad\text{and}\\\
h_{1}\bar{m}_{3}+h_{2}(\bar{m}_{2}\otimes\operatorname{id}-\operatorname{id}\otimes\bar{m}_{2})=0\,;\end{gathered}$

the former replaces the second and third Stasheff identity and the latter
supplements the first three Stasheff identities. In particular, for $n>2$ the
maps $\bar{m}_{n}$ are the appropriate restrictions of $m_{n}$. For $n=2$, we
obtain $m_{2}$ by $\bar{m}_{2}+h_{2}$ and additionally enforcing Eq. 4.2.1.
For $n=1$, we have $m_{1}=\bar{m}_{1}+h_{1}$. This treatment is similar to
[Bur15, Section 3], but is slightly more general since we allow
$\bar{A}_{0}\neq 0$ and hence need $h_{2}$ as well as $h_{1}$. If $A$ were not
connective then we would also need maps $h_{n}$ for $n\geqslant 3$. Taken
together the $h_{n}$ will correspond to the curvature term on the bar
construction of $A$; see 4.7.

######  Remark 4.4.

Let $A$ be a connective $\mathrm{A}_{\infty}$-algebra with a split unit. Then
the projection $A\to Q$ onto the free summand containing the unit need not be
a morphism of strictly unital $\mathrm{A}_{\infty}$-algebras. In fact, this
happens if and only if $h_{1}=0$ and $h_{2}=0$. Such
$\mathrm{A}_{\infty}$-algebras are called _augmented_.

######  4.5.

Fix a graded coalgebra $(C,\Delta)$. Recall $C$ is _counital_ if there exists
a counit map $\varepsilon\colon C\to Q$ such that

$(\operatorname{id}\otimes\varepsilon)\Delta=\operatorname{id}=(\varepsilon\otimes\operatorname{id})\Delta\,.$

We say $C$ is a _curved dg coalgebra_ it is equipped with a coderivation
$\partial$ of degree $-1$ and a curvature term $h\colon C\to Q$ of degree $-2$
such that

$\partial^{2}=(h\otimes\operatorname{id}-\operatorname{id}\otimes
h)\Delta\quad\text{and}\quad h\partial=0\,.$

A curved dg coalgebra $C$ is _connected_ if it is non-negatively graded,
counital and $C_{0}=Q$. In this setting, we write $C=Q\oplus\bar{C}$ for
$\bar{C}=\ker(\varepsilon)$ and set

$\begin{gathered}\bar{\Delta}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\left(\bar{C}\to
C\xrightarrow{\Delta}C\otimes C\to\bar{C}\otimes\bar{C}\right)\,,\\\
\bar{\partial}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\left(\bar{C}\to
C\xrightarrow{\partial}C\to\bar{C}\right)\quad\text{and}\quad\bar{h}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\left(\bar{C}\to
C\xrightarrow{h}Q\right)\end{gathered}$

for the restrictions to $\bar{C}$. These maps satisfy the same relations as
$\Delta$, $\partial$ and $h$.

######  4.6.

The tensor algebra ${\rm T}^{a}(V)$ on a graded $Q$-module $V$ has underlying
graded module ${\rm
T}(V)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\bigoplus_{n\geqslant
0}V^{\otimes n}$, and the multiplication

$\mu((v_{1}\otimes\cdots\otimes
v_{k})\otimes(v^{\prime}_{1}\otimes\cdots\otimes
v^{\prime}_{\ell}))\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}v_{1}\otimes\cdots\otimes
v_{k}\otimes v^{\prime}_{1}\otimes\cdots\otimes v^{\prime}_{\ell}\,.$

The tensor algebra is bigraded by ${\rm T}_{(n)}^{a}(V)_{i}=(V^{\otimes
n})_{i}$.

The tensor coalgebra ${\rm T}^{c}(V)$ on a graded $Q$-module $V$ has
underlying graded module ${\rm T}(V)$, and the comultiplication

$\Delta(v_{1}\otimes\cdots\otimes
v_{n})\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\sum_{i=0}^{n}(v_{1}\otimes\cdots\otimes
v_{i})\otimes(v_{i+1}\otimes\cdots\otimes v_{n})\,.$

The tensor coalgebra is bigraded by ${\rm T}_{(n)}^{c}(V)_{i}=(V^{\otimes
n})_{i}$. The data of an $\mathrm{A}_{\infty}$-algebra can equivalently be
encoded as a differential on a tensor coalgebra, as we see next.

######  4.7.

Let $A$ be a split unital connective $\mathrm{A}_{\infty}$-algebra. Then the
tensor coalgebra ${\rm T}^{c}(\Sigma\bar{A})$ has an induced curved dg
coalgebra structure. The curvature term has components

$h_{1}{\scriptstyle\mathsf{\Sigma}}^{-1}\colon{\rm
T}_{(1)}^{c}(\Sigma\bar{A})\to Q\quad\text{and}\quad
h_{2}({\scriptstyle\mathsf{\Sigma}}^{-1})^{\otimes 2}\colon{\rm
T}_{(2)}^{c}(\Sigma\bar{A})\to Q$

and zero otherwise. The coderivation $\partial$ has components

$(-1)^{\frac{k(k+1)}{2}}\sum_{\begin{subarray}{c}i+j=n-k\\\ i,j\geqslant
0\end{subarray}}(\operatorname{id}^{\otimes
i}\otimes{\scriptstyle\mathsf{\Sigma}}\bar{m}_{k}({\scriptstyle\mathsf{\Sigma}}^{-1})^{\otimes
k}\otimes\operatorname{id}^{\otimes j})\colon{\rm
T}_{(n)}^{c}(\Sigma\bar{A})\to{\rm T}_{(n-k)}^{c}(\Sigma\bar{A})$

for $k\geqslant 1$, and zero otherwise. The map $\partial$ is well-defined
since $A$ is concentrated in non-negative homological degree. With this
structure ${\rm T}^{c}(\Sigma\bar{A})$ is a connected curved dg coalgebra, and
we call

$\mathsf{B}_{(*)}(A)_{\bullet}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\left({\rm
T}_{(*)}^{c}(\Sigma\bar{A})_{\bullet},h,\partial,\Delta\right)$

the _bar construction of $A$_. For $a_{1},\ldots,a_{n}\in\bar{A}$ we write

$[a_{1}|\ldots|a_{n}]\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}({\scriptstyle\mathsf{\Sigma}}a_{1}\otimes\ldots\otimes{\scriptstyle\mathsf{\Sigma}}a_{n})\in\mathsf{B}_{(n)}(A)\,.$

For a split unital connective $\mathrm{A}_{\infty}$-algebra the canonical
projection and inclusion induce a degree $-1$ map of graded modules

$\mathsf{B}(A)\twoheadrightarrow\Sigma\bar{A}\hookrightarrow\Sigma A\to A\,.$

Let $C$ be a connected curved dg coalgebra. Then the algebra ${\rm
T}^{a}(\Sigma^{-1}\bar{C})$ has an induced dg algebra structure. The
differential $m_{1}$ has components

$\begin{gathered}-\bar{h}{\scriptstyle\mathsf{\Sigma}}\colon{\rm
T}_{(1)}^{a}(\Sigma^{-1}\bar{C})\to{\rm
T}_{(0)}^{a}(\Sigma^{-1}\bar{C})\,,\quad-{\scriptstyle\mathsf{\Sigma}}^{-1}\bar{\partial}{\scriptstyle\mathsf{\Sigma}}\colon{\rm
T}_{(1)}^{a}(\Sigma^{-1}\bar{C})\to{\rm T}_{(1)}^{a}(\Sigma^{-1}\bar{C})\\\
\text{and}\quad({\scriptstyle\mathsf{\Sigma}}^{-1})^{\otimes
2}\bar{\Delta}{\scriptstyle\mathsf{\Sigma}}\colon{\rm
T}_{(1)}^{a}(\Sigma^{-1}\bar{C})\to{\rm
T}_{(2)}^{a}(\Sigma^{-1}\bar{C})\,;\end{gathered}$

and zero otherwise. With this structure ${\rm T}^{a}(\Sigma^{-1}\bar{C})$ is a
split unital connective dg algebra, and we call

$\mathsf{\Omega}_{(*)}(C)_{\bullet}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\left({\rm
T}_{(*)}^{a}(\Sigma^{-1}\bar{C})_{\bullet},m_{1},m_{2}\right)$

the _cobar construction of $C$_. For $c_{1},\ldots,c_{n}\in\bar{C}$ we write

$\langle
c_{1}|\ldots|c_{n}\rangle\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}({\scriptstyle\mathsf{\Sigma}}^{-1}c_{1}\otimes\ldots\otimes{\scriptstyle\mathsf{\Sigma}}^{-1}c_{n})\in\mathsf{\Omega}_{(n)}(C)\,.$

For a connected curved dg coalgebra the canonical inclusion and projection
maps induces a degree $-1$ map of graded modules

$C\to\Sigma^{-1}C\twoheadrightarrow\Sigma^{-1}\bar{C}\hookrightarrow\mathsf{\Omega}(C)\,.$

######  Remark 4.8.

The bar and cobar constructions define an adjoint pair of functors when
restricted to split unital connective dg algebras and connected curved dg
coalgebras; see [Lyu13, Section 3]. It remains an adjunction when restricted
to augmented connective dg algebras and connected dg coalgebras.

######  4.9.

A morphism
$\varphi\colon(C,\Delta,\varepsilon,\partial,h)\to(C^{\prime},\Delta^{\prime},\varepsilon^{\prime},\partial^{\prime},h^{\prime})$
of connected curved dg coalgebras consists of $Q$-linear maps

$\varphi_{0}\colon C\to Q\quad\text{and}\quad\varphi_{1}\colon C\to
C^{\prime}$

of degree $-1$ and 0, respectively, satisfying

$\begin{gathered}\varepsilon=\varepsilon^{\prime}\varphi_{1}\,,\quad
h^{\prime}\varphi_{1}=h-\varphi_{0}\partial+(\varphi_{0}\otimes\varphi_{0})\Delta\,,\\\
\partial^{\prime}\varphi_{1}=\varphi_{1}\partial+(\varphi_{0}\otimes\varphi_{1}-\varphi_{1}\otimes\varphi_{0})\Delta\quad\text{and}\quad\Delta^{\prime}\varphi_{1}=(\varphi_{1}\otimes\varphi_{1})\Delta\,;\end{gathered}$

see [Pos11, Chapter 4]. This induces a map of dg algebras
$\mathsf{\Omega}(\varphi)\colon\mathsf{\Omega}(C)\to\mathsf{\Omega}(C^{\prime})$,
and we say $\varphi$ is a _weak equivalence_ if $\mathsf{\Omega}(\varphi)$ is
a quasi-isomorphism.

######  4.10.

Let $A$ be an $\mathrm{A}_{\infty}$-algebra. An _$\mathrm{A}_{\infty}$ -module
over $A$_ is a graded module $M$ equipped with maps

$m_{n}^{M}\colon A^{\otimes(n-1)}\otimes M\to M\quad\text{for }n\geqslant 1$

of degree $(n-2)$, satisfying

$\sum_{\begin{subarray}{c}r+s+t=n\\\ r\geqslant 0,s,t\geqslant
1\end{subarray}}(-1)^{r+st}m^{M}_{r+1+t}\left(\operatorname{id}^{\otimes
r}\otimes m_{s}\otimes\operatorname{id}^{\otimes
t}\right)+\sum_{\begin{subarray}{c}r+t=n\\\ r\geqslant 0,s\geqslant
1\end{subarray}}(-1)^{r}m^{M}_{r+1}\left(\operatorname{id}^{\otimes r}\otimes
m_{s}^{M}\right)=0\,.$

If $A$ is strictly unital, we say an $\mathrm{A}_{\infty}$-module $M$ over $A$
is _strictly unital_ if

$\begin{gathered}m_{2}(1_{A}\otimes m)=m\quad\text{for all }m\in
M\quad\text{and}\\\ m_{n}(a_{1}\otimes\ldots\otimes a_{i-1}\otimes
1_{A}\otimes a_{i+1}\otimes\ldots\otimes a_{n-1}\otimes m)=0\quad\text{for all
}1\leqslant i\leqslant n-1\end{gathered}$

for any $a_{1},\ldots,a_{n-1}\in A$ and $m\in M$ with $n\neq 2$.

If $A$ is connective and has a split unit, then a strictly unital
$\mathrm{A}_{\infty}$-module structure over $A$ on $M$ is equivalent to the
existence of maps

$\bar{m}^{M}_{n}\colon\bar{A}^{\otimes(n-1)}\otimes M\to M\quad\text{for
}n\geqslant 1$

of degree $(n-2)$ such that for $n\neq 2,3$ the Stasheff identities hold when
replacing $m$ by $\bar{m}$, and for $n=2,3$ there is an extra curvature term
$h_{n-1}\otimes\operatorname{id}$ similar to 4.3.

######  4.11.

Let $A$ be a split unital connective $\mathrm{A}_{\infty}$-algebra. The data
of a strictly unital $\mathrm{A}_{\infty}$-module structure over $A$ is
equivalent to that of a strictly unital dg module structure over
$\mathsf{\Omega}(\mathsf{B}(A))$. Explicitly, if $\\{\bar{m}^{M}_{n}\\}$ is a
strictly unital $\mathrm{A}_{\infty}$-module structure on a graded module $M$,
then the dg module structure on $M$ is given by the same differential
$m^{M}_{1}$, and the multiplication $\mathsf{\Omega}(\mathsf{B}(A))\otimes
M\to M$ induced by

$-(-1)^{\frac{n(n-1)}{2}}m^{M}_{n+1}(({\scriptstyle\mathsf{\Sigma}}^{-1})^{\otimes
n}{\scriptstyle\mathsf{\Sigma}}\otimes\operatorname{id}_{M})\colon\Sigma^{-1}\bar{\mathsf{B}}_{(n)}(A)\otimes
M\to M\,.$

Moreover, this construction is natural in $A$ and $M$, and any quasi-
isomorphism of $\mathrm{A}_{\infty}$-modules over $A$ yields a quasi-
isomorphism of dg modules over $\mathsf{\Omega}(\mathsf{B}(A))$.

##  5\. Transfer of $\mathrm{A}_{\infty}$-algebra structures

In this section, as above, $Q$ is a local ring with maximal ideal
$\mathfrak{m}_{Q}$ and residue field $k$. Let $R$ be an
$\mathrm{A}_{\infty}$-algebra over $Q$, and let $A\to R$ be a quasi-
isomorphism of complexes over $Q$. We would like to know whether the
$\mathrm{A}_{\infty}$-algebra structure on $R$ induces an
$\mathrm{A}_{\infty}$-algebra structure on $A$. This is well-understood in the
case that $Q$ is field, so that $A\to R$ is a homotopy equivalence; the first
result is due to Kadeishvili [Kad82] when $A=\mathrm{H}(R)$. For general
homotopy equivalences this was studied, for example, in [Mar06]. Burke has
shown that if $R$ is a quotient of $Q$ and $A$ is a $Q$-free resolution of
$R$, then the product on $R$ lifts to an $\mathrm{A}_{\infty}$-structure on
$A$ [Bur18, Proposition 3.6]. We give a proof in a more general situation.

######  Proposition 5.1.

Let $R$ be a strictly unital connective $\mathrm{A}_{\infty}$-algebra and
$\varepsilon\colon A\to R$ a surjective quasi-isomorphism of complexes over
$Q$, with $A$ degree-wise free and concentrated in non-negative degrees. Then
there exists an $\mathrm{A}_{\infty}$-algebra structure with a split unit on
$A$ such that $\varepsilon$ is a strict quasi-isomorphism of
$\mathrm{A}_{\infty}$-algebras.

###### Proof.

Since $\varepsilon$ is surjective we may choose a splitting $A=\bar{A}\oplus
Q$ such that $\varepsilon$ maps the free generator of $Q$ to the unit of $R$.
We inductively construct higher multiplication maps $m_{n}$ on $A$ satisfying
the $n$th Stasheff identity. To begin with we set
$m_{1}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\partial$
where $\partial$ is the differential of $A$.

For $n=2$ we consider the diagram

${A\otimes Q+Q\otimes A}$${A}$${A^{\otimes 2}}$${R^{\otimes
2}}$${R\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\simeq}$$\scriptstyle{\varepsilon}$$\scriptstyle{\varepsilon^{\otimes
2}}$$\scriptstyle{m_{2}^{R}}$

Since the right vertical arrow is a surjective quasi-isomorphism and the left
vertical arrow is injective in each degree and the cokernel in each degree is
projective, there exists a lift $m_{2}^{A}\colon A^{\otimes 2}\to A$ such that
the diagram commutes; see for example [DS95, Section 7]. This morphism
$m_{2}^{A}$ satisfies the desired properties by construction.

For $n>2$, the obstruction $\operatorname{obs}_{n}^{A}$ from Eq. 4.1.2 is a
chain map. We have a short exact sequence of complexes

$0\to\sum_{i+j=n-1}A^{\otimes i}\otimes Q\otimes A^{\otimes
j}\xrightarrow{\eta_{n}}A^{\otimes n}\to\bar{A}^{\otimes n}\to 0\,.$

By direct computation we obtain $\operatorname{obs}_{n}^{A}\eta_{n}=0$. So the
obstruction $\operatorname{obs}_{n}^{A}$ factors through $\bar{A}^{\otimes
n}$. We consider the diagram

${\bar{A}^{\otimes
n}}$${\Sigma^{-1}\operatorname{cone}(\varepsilon)}$${A}$${R\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\operatorname{obs}_{n}^{A}}$$\scriptstyle{0}$$\scriptstyle{\alpha}$$\scriptstyle{\pi}$$\scriptstyle{\simeq}$$\scriptstyle{\varepsilon}$

Since $\bar{A}^{\otimes n}$ is, as graded modules, a direct summand of
$A^{\otimes n}$, and the higher multiplications $m_{i}^{A}$ for $i<n$ commute
with $\varepsilon$, the right triangle commutes up to homotopy
$m_{n}^{R}\varepsilon^{\otimes n}$. Then there exists a chain map $\alpha$
such that the left triangle commutes up to homotopy $\sigma$. Since
$\varepsilon$ is surjective, we may assume $m_{n}^{R}\varepsilon^{\otimes
n}=\varepsilon\sigma$. That is

$m^{A}_{n}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\left(A^{\otimes
n}\to\bar{A}^{\otimes n}\xrightarrow{\sigma}A\right)$

satisfies the $n$th Stasheff identity. ∎

######  5.2.

In the setup of Proposition 5.1 we can also transfer
$\mathrm{A}_{\infty}$-module structures: If $M$ is a strictly unital
$\mathrm{A}_{\infty}$-module over $R$ and with semifree resolution
$\gamma\colon G\to M$ over $Q$, in the sense discussed later in 6.2, then
there exists a strictly unital $\mathrm{A}_{\infty}$-module structure on $G$
over $A$ and $\gamma$ is a strict morphism of $\mathrm{A}_{\infty}$-modules;
compare with [Bur18]. When the homology of $M$ is bounded below (for example,
if $M$ is an honest module), one can take $G$ to be a bounded below complex of
free $Q$-modules.

######  Proposition 5.3.

Let $\varepsilon\colon R\to S$ be a surjective strict quasi-isomorphism of
strictly unital $\mathrm{A}_{\infty}$-algebras over $Q$. Further let $A$ be a
split unital, connective, degree-wise free $\mathrm{A}_{\infty}$-algebra and
$\varphi\colon A\to S$ a morphism of strictly unital
$\mathrm{A}_{\infty}$-algebras. Then there exists a morphism of strictly
unital $\mathrm{A}_{\infty}$-algebras $\psi\colon A\to R$ such that
$\varphi=\varepsilon\psi$.

###### Proof.

The unit $Q\to S$ factors through $\varepsilon$ and $\varphi_{1}$, so by
[DS95, Section 7], there is a chain map $\psi_{1}\colon A\to R$ such that
$\varphi_{1}=\varepsilon\psi_{1}$ and $\psi_{1}(1_{A})=1_{R}$.

Let $n\geqslant 2$ and assume that for $i<n$ the chain maps $\psi_{i}\colon
A^{\otimes i}\to R$ exist, the $i$th Stasheff identities Eq. 4.1.3 and Eq.
4.2.2 hold, and $\varepsilon\psi_{i}=\varphi_{i}$. A computation shows that
$\operatorname{obs}_{n}^{\psi}$ and $\operatorname{obs}_{n}^{\varphi}$ vanish
when any of its inputs is $1_{A}$. Hence we can view
$\operatorname{obs}_{n}^{\psi}$ and $\operatorname{obs}_{n}^{\varphi}$ as maps
on $\bar{A}^{\otimes n}$. Taking homology classes in
$\operatorname{Hom}(\bar{A}^{\otimes n},S)$ we have

$\varepsilon[\operatorname{obs}_{n}^{\psi}]=[\operatorname{obs}_{n}^{\varphi}]=0\,,$

since
$\varepsilon\operatorname{obs}_{n}^{\psi}=\operatorname{obs}_{n}^{\varphi}$
and $\varphi$ is a morphism of strictly unital $\mathrm{A}_{\infty}$-algebras;
cf. 4.1. Since $\varepsilon$ is a surjective quasi-isomorphism, and using the
assumptions on $A$, the induced map $\operatorname{Hom}(\bar{A}^{\otimes
n},R)\to\operatorname{Hom}(\bar{A}^{\otimes n},S)$ is a quasi-isomorphism and
hence $\operatorname{obs}_{n}^{\psi}$ is a boundary in
$\operatorname{Hom}(\bar{A}^{\otimes n},R)$. That is, there is
$\bar{\psi}_{n}\colon\bar{A}^{\otimes n}\to R$ such that

$\operatorname{obs}_{n}^{\psi}=m_{1}^{R}\bar{\psi}_{n}+(-1)^{n}\bar{\psi}_{n}(\bar{m}_{1}^{A}\otimes\operatorname{id}^{\otimes
n}+\cdots+\operatorname{id}^{\otimes n}\otimes\bar{m}_{1}^{A})\,.$

Setting
$\psi_{n}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}(A^{\otimes
n}\to\bar{A}^{\otimes n}\xrightarrow{\bar{\psi}_{n}}R)$, we now have
$\psi_{1},\ldots,\psi_{n}$ satisfying the required identities Eq. 4.1.3 and
Eq. 4.2.2, completing the induction. ∎

It is well-known that $\mathrm{A}_{\infty}$-algebras can be used to
characterize formality of dg algebras over fields [Kad82]. We record the
following generalization in local algebra.

######  Proposition 5.4.

Let $\varphi\colon Q\to R$ be a finite local homomorphism. The fiber
$R\otimes^{\mathsf{L}}_{Q}k$ is formal as a dg $k$-algebra if and only if the
minimal $Q$-free resolution $\varepsilon\colon A\to R$ admits an
$\mathrm{A}_{\infty}$-structure $\\{m_{n}\\}$ making $\varepsilon$ a strict
quasi-isomorphism of $\mathrm{A}_{\infty}$-algebras, and satisfying
$m_{n}\otimes_{Q}k=0$ for $n\geqslant 3$.

###### Proof.

Suppose that the minimal $Q$-free resolution $A$ of $R$ has an
$\mathrm{A}_{\infty}$-structure $\\{m_{n}\\}$ with the stated property. If
$A^{\prime}$ is a $Q$-free dg algebra resolution of $R$, then it follows from
Proposition 5.3 that $A$ and $A^{\prime}$ are quasi-isomorphic as
$\mathrm{A}_{\infty}$-algebras over $Q$. Therefore $A\otimes_{Q}k$ and
$R\otimes^{\mathsf{L}}_{Q}k=A^{\prime}\otimes_{Q}k$ are quasi-isomorphic as
$\mathrm{A}_{\infty}$-algebras over $k$. By assumption $m_{n}\otimes_{Q}k=0$
for $n\geqslant 3$, so $A\otimes_{Q}k$ is a graded algebra, canonically
isomorphic to $\operatorname{Tor}^{Q}(R,k)$. Two dg $k$-algebras are quasi-
isomorphic as dg algebras if and only if they are quasi-isomorphic as
$\mathrm{A}_{\infty}$-algebras [Kad82], and we can conclude that
$R\otimes^{\mathsf{L}}_{Q}k$ is formal.

Suppose conversely that $R\otimes^{\mathsf{L}}_{Q}k$ is formal. By Proposition
5.1 the minimal $Q$-free resolution $A$ of $R$ admits an
$\mathrm{A}_{\infty}$-structure $\\{m_{n}^{\prime}\\}$. Using the same
reasoning as above, since $R\otimes^{\mathsf{L}}_{Q}k$ is formal
$A\otimes_{Q}k$ and $\operatorname{Tor}^{Q}(R,k)$ are quasi-isomorphic as
$\mathrm{A}_{\infty}$-algebras over $k$. By the uniqueness of minimal models
(that is, $\mathrm{A}_{\infty}$-algebras over a field having zero
differential; see [Kad82]) there is an isomorphism of
$\mathrm{A}_{\infty}$-algebras

$\psi\colon(\operatorname{Tor}^{Q}(R,k),0,\mu,0,\ldots)\xrightarrow{\cong}(A\otimes_{Q}k,0,m_{2}^{\prime}\otimes
k,m_{3}^{\prime}\otimes k,\ldots)\,,$

where $\mu$ is the ordinary product on $\operatorname{Tor}^{Q}(R,k)$. We may
make the identification $A\otimes_{Q}k=\operatorname{Tor}^{Q}(R,k)$ and choose
lifts $\Psi_{i}\colon A^{\otimes i}\to A$ with
$\Psi_{i}\otimes_{Q}k=\psi_{i}$. By Nakayama’s lemma $\Psi_{1}$ is an
isomorphism and we can inductively define operations $m_{n}\colon A^{\otimes
n}\to A$ by the formula
$m_{n}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}$

$\Psi_{1}^{-1}\Big{(}-\sum_{\begin{subarray}{c}r+s+t=n\\\ r,t\geqslant
0,s\geqslant
1\end{subarray}}(-1)^{r+st}{\Psi}_{r+1+t}\left(\operatorname{id}^{\otimes
r}\otimes{m}_{s}\otimes\operatorname{id}^{\otimes
t}\right)+\sum_{\begin{subarray}{c}p,\bm{\alpha}\in\mathbb{N}^{p}\\\
|\bm{\alpha}|=n\end{subarray}}(-1)^{v(\bm{\alpha})}{m}^{\prime}_{p}{\bf\Psi}^{\otimes\bm{\alpha}}\Big{)}\,.$

By construction the map $\Psi\colon(A,\\{m_{n}^{\prime}\\})\to(A,\\{m_{n}\\})$
now satisfies the Stasheff morphism identities (4.1.3), and it follows that
$(A,\\{m_{n}\\})$ is an $\mathrm{A}_{\infty}$-algebra, isomorphic to
$(A,\\{m_{n}^{\prime}\\})$. Finally, from $\Psi\otimes_{Q}k=\psi$ it follows
that $m_{2}\otimes_{Q}k=\mu$ and $m_{n}\otimes_{Q}k=0$ for $n\geqslant 3$, as
stated in the proposition. ∎

The following technical lemma will be used later to help generate examples, by
showing that certain $\mathrm{A}_{\infty}$-operations are minimal.

######  Lemma 5.5.

Let $\varphi\colon A\to T$ be a map of connective split unital
$\mathrm{A}_{\infty}$-algebras, where $T$ is a trivial algebra. If for some
$N$ the map $(\varphi_{1})_{<N}\colon A_{<N}\to T_{<N}$ is injective, then the
$\mathrm{A}_{\infty}$-structure of $A$ vanishes in degrees less than $N$, in
the sense that $(\bar{m}_{n}(\bar{A}^{\otimes n}))_{i}=0$ for all $n\geqslant
1$ and $i<N$.

###### Proof.

We prove this by induction on $n$. It is clear for $n=1$ since $\varphi_{1}$
is a chain map. For $n\geqslant 2$, since $\bar{m}_{s}^{T}=0$ for all $s$, we
can rearrange the Stasheff morphism identities (4.1.3):

$\bar{\varphi}_{1}\bar{m}_{n}=-\sum_{\begin{subarray}{c}r+s+t=n\\\
r,t\geqslant 0,s\geqslant
1\end{subarray}}(-1)^{r+st}\bar{\varphi}_{r+1+t}\left(\operatorname{id}^{\otimes
r}\otimes\bar{m}_{s}\otimes\operatorname{id}^{\otimes t}\right)\,.$

We can assume by induction that $(\bar{m}_{s}(\bar{A}^{\otimes s}))_{<N}=0$
for $s<n$. Since each $\varphi_{r}$ increases degree by $r-1$, this implies
that the the right-hand side above is zero in degrees $i<N$. Since
$(\bar{\varphi}_{1})_{<N}$ is injective, it follows that
$\bar{m}_{n}(\bar{A}^{\otimes n}))_{<N}=0$. ∎

###  5.6. Cyclic $\mathrm{A}_{\infty}$-algebras

For Gorenstein algebras the minimal resolution satisfies a Poincaré duality
property that allows us, in favorable situations, to construct
$\mathrm{A}_{\infty}$-resolutions with additional duality properties.

A _cyclic $\mathrm{A}_{\infty}$-algebra_ of degree $d$ over $Q$ is a complex
$A$ of finitely generated free $Q$-modules with a perfect, $Q$-bilinear
pairing

$\langle-,-\rangle\colon A\otimes A\to\Sigma^{d}Q\,,$

and an $\mathrm{A}_{\infty}$-structure $\\{m_{n}\\}$ on $A$ such that for each
$n$

$\langle
m_{n}(a_{1},\ldots,a_{n}),a_{n+1}\rangle=(-1)^{n+|a_{1}|(|a_{2}|+\cdots+|a_{n+1}|)}\langle
m_{n}(a_{2},\ldots,a_{n+1}),a_{1}\rangle\,;$

see [Kon94].

There is for each $n$ an isomorphism of complexes

$\operatorname{cyc}\colon\operatorname{Hom}(A^{\otimes
n},A)\xrightarrow{\cong}\operatorname{Hom}(A^{\otimes(n+1)},\Sigma^{d}Q)\,,\quad\operatorname{cyc}(f)=\langle
f(-),-\rangle\,.$

We give $A^{\otimes(n+1)}$ the action of the cyclic group $C_{n+1}=\langle
c\rangle$ with generator acting by $c\cdot(a_{1}\otimes\cdots\otimes
a_{n+1})=(-1)^{|a_{1}|(|a_{2}|+\cdots+|a_{n+1}|)}(a_{2}\otimes\cdots\otimes
a_{n+1}\otimes a_{1})$. From this perspective, an
$\mathrm{A}_{\infty}$-structure $\\{m_{n}\\}$ is cyclic if and only if

$\operatorname{cyc}(m_{n})\cdot
c=(-1)^{n}\operatorname{cyc}(m_{n})\quad\text{for all }n\,.$

Let $\varphi\colon Q\to R$ be a surjective local Gorenstein homomorphism of
projective dimension $d$, and let $A$ be the minimal resolution of $R$ over
$Q$. Let $\mu\colon A^{\otimes 2}\to A$ be a chain map lifting the product on
$R$; we can assume that $\mu$ is unital and graded-commutative by [BH98,
3.4.3]. The Gorenstein condition Eq. 3.11.1 guarantees that
$A\otimes_{Q}k=\operatorname{Tor}^{Q}(R,k)$ is a Poincaré duality algebra with
the product induced from $\mu$. It follows from Nakayama’s lemma that
$A_{d}\cong Q$ and we obtain a perfect pairing

(5.6.1) $\langle-,-\rangle\colon A\otimes
A\xrightarrow{\mu}A\twoheadrightarrow\Sigma^{d}A_{d}=\Sigma^{d}Q\,.$

######  Theorem 5.7.

Let $Q\to R$ be a surjective local Gorenstein homomorphism of odd projective
dimension $d$. Assume that $Q$ contains a field of characteristic zero. The
minimal resolution $A$ of $R$ over $Q$ admits the structure of a split unital,
cyclic $\mathrm{A}_{\infty}$-algebra of degree $d$, making the map $A\to R$ a
strict $\mathrm{A}_{\infty}$-algebra quasi-isomorphism.

We first need a lemma about projective resolutions.

######  Lemma 5.8.

Let $M$ be a finitely generated $Q$-module of projective dimension $d>0$, with
a projective resolution $A\to M$, and set $V=A_{<d}/A_{0}$. Then for any $n$
we have $\mathrm{H}_{i}(V^{\otimes n+1})=0$ whenever $i>n(d-1)+1$ and
$i\neq(n+1)(d-1)$.

###### Proof.

We show this by inducing on $n$. Since $A$ is a projective resolution of $R$,
the homology of $V$ is concentrated in degrees $1$ and $d-1$, and there is an
exact triangle

$\Sigma^{d-1}Q^{s}\longrightarrow V\longrightarrow\Sigma N\,,$

of complexes of $Q$-modules, where $N=\ker(A_{0}\to M)$ and $A_{d}=Q^{\oplus
s}$. This justifies the case $n=0$, and for each $n\geqslant 1$ yields another
exact triangle

$\Sigma^{d-1}(V^{\otimes n})^{\oplus s}\longrightarrow V^{\otimes
n+1}\longrightarrow\Sigma V^{\otimes n}\otimes N\,.$

Clearly $\mathrm{H}_{i}(\Sigma V^{\otimes n}\otimes N)=0$ for $i>n(d-1)+1$, so
by the long exact sequence in homology the map
$\mathrm{H}_{i}(\Sigma^{d-1}V^{\otimes(n-1)})^{\oplus
s}\to\mathrm{H}_{i}(V^{\otimes n})$ is surjective for $i>n(d-1)+1$. By the
induction hypothesis $\mathrm{H}_{i}(\Sigma^{d-1}V^{\otimes(n-1)})=0$ if
$i\neq n(d-1)$ and $i>(n-1)(d-1)+1$. From this we conclude that the lemma
holds for $n$. ∎

###### Proof of Theorem 5.7.

Recall from Eq. 5.6.1 that the pairing on $A$ was defined from a unital and
graded-commutative product $\mu\colon A^{\otimes 2}\to A$. This restricts to a
perfect pairing on $V=A_{<d}/A_{0}$, and we start by constructing operations
$m_{n}^{V}\colon V^{\otimes n}\to V$.

If $|a|+|b|=d+1$ then $\mu(a\otimes b)=0$ in $A$, so

$\mu(\partial(a)\otimes
b)+(-1)^{|a|}\mu(a\otimes\partial(b))=\partial(\mu(a\otimes b))=0\,.$

Using commutativity of $\mu$, this is equivalent to the cyclic identity

$\langle m_{1}^{V}(a),b\rangle=(-1)^{1+|a||b|}\langle
m_{1}^{V}(b),a\rangle\,,$

where
$m_{1}^{V}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\partial$.

Next, we truncate $\mu$ to obtain $\mu^{V}\colon V^{\otimes 2}\to V$, and we
define $m_{2}^{V}$ by symmetrizing $\mu^{V}$ with respect to the
$C_{3}$-action:

$\operatorname{cyc}(m_{2}^{V})\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\operatorname{cyc}(\mu^{V})\cdot\frac{1}{3}(1+c+c^{2})\,.$

The obtained $m_{2}^{V}$ satisfies the required cyclic property by
construction. However, the Stasheff identity Eq. 4.1.1 does not hold for
$n=2$, and instead

(5.8.1)
$\partial(m^{V}_{2}(a,b))-m^{V}_{2}(\partial(a),b)-(-1)^{|a|}m^{V}_{2}(a,\partial(b))=\langle
a,b\rangle\partial(\omega)\,,$

where $\omega\in A_{d}$ is the generator with $\langle\omega,1\rangle=1$.
Nonetheless, since $\langle-,-\rangle$ is a chain map, the same computation as
in Eq. 4.1.2 shows that the obstruction $\operatorname{obs}_{3}^{V}$ is a
chain map, that is, a cycle in $\operatorname{Hom}(V^{\otimes 3},V)$.

We proceed to construct $m_{n}^{V}$ for $n\geqslant 3$ by induction,
satisfying the Stasheff identities Eq. 4.1.1 for $n\geqslant 3$, and all
satisfying $\operatorname{cyc}(m_{n}^{V})\cdot
c=(-1)^{n}\operatorname{cyc}(m_{n}^{V})$. The argument is similar to the proof
of Proposition 5.1. If $m_{i}^{V}$ have been constructed for $i<n$ with
required cyclic symmetry, a computation shows that the obstruction
$\operatorname{obs}_{n}^{V}$ from Eq. 4.1.2 is cyclic as well:

$\operatorname{cyc}(\operatorname{obs}_{n}^{V})=(-1)^{n}\operatorname{cyc}(\operatorname{obs}_{n}^{V})\cdot
c\,.$

Since $\operatorname{Hom}(V^{\otimes n},V)\cong\Sigma^{-nd}V^{\otimes(n+1)}$
we can use Lemma 5.8 with $M=R$ to conclude that

$\mathrm{H}_{i}\big{(}\operatorname{Hom}(V^{\otimes n},V)\big{)}=0\ \text{ for
}i>1-n\text{ and }i\neq d-n-1\,.$

Since $d$ is odd, it is impossible to have
$|\operatorname{obs}_{n}^{V}|=n-3=d-n-1$, hence the complex
$\operatorname{Hom}(V^{\otimes n},V)$ is acyclic in degree $n-3$, and the
class $[\operatorname{obs}_{n}^{V}]$ vanishes. This shows that there is an
operation $\tilde{m}^{V}_{n}$ in $\operatorname{Hom}(V^{\otimes n},V)_{n-2}$
such that $\partial(\tilde{m}^{V}_{n})=\operatorname{obs}_{n}^{V}$, and we
symmetrize this to define $m_{n}$:

$\operatorname{cyc}(m_{n}^{V})\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\operatorname{cyc}(\tilde{m}^{V}_{n})\cdot\textstyle{\sum\limits_{i=0}^{n}\frac{(-1)^{in}c^{i}}{n+1}}\,.$

By construction $m_{n}^{V}$ has the required cyclic symmetry. We note that

$\partial(\operatorname{cyc}(m_{n}^{V}))=\operatorname{cyc}(\operatorname{obs}_{n}^{V})\cdot\textstyle{\sum\limits_{i=0}^{n}\frac{(-1)^{in}c^{i}}{n+1}}=\operatorname{cyc}(\operatorname{obs}_{n}^{V})\,.$

Therefore $\partial(m_{n}^{V})=\operatorname{obs}_{n}^{V}$ and the operations
$\\{m_{n}^{V}\\}$ satisfy the $n$th Stasheff identity. This concludes the
induction.

To finish the proof we define the following operations on $A$:

$m_{2}(a_{1},a_{2})\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\begin{cases}m^{V}_{2}(a_{1},a_{2})&\text{if
}|a_{1}|,|a_{2}|>0\text{ and }|a_{1}|+|a_{2}|<d,\\\ \langle
a_{1},a_{2}\rangle\omega&\text{if }|a_{1}|+|a_{2}|=d,\\\ a_{1}a_{2}&\text{if
}|a_{1}|=0\text{ or }|a_{2}|=0,\end{cases}$

and for $n\geqslant 3$

$m_{n}(a_{1},\ldots,a_{n})\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\begin{cases}m^{V}_{n}(a_{1},\ldots,a_{n})&\text{if
all }|a_{i}|>0\text{ and }|a_{1}|+\cdots+|a_{2}|<d,\\\ 0&\text{if
}|a_{1}|+\cdots+|a_{2}|=d\text{ or any }|a_{i}|=0.\end{cases}$

The $n=2$ Stasheff identity for $A$ is equivalent to the identity Eq. 5.8.1
above.

To verify the $n$th Stasheff identity, with $n\geqslant 3$, we need to divide
into cases depending on the inputs: when any of the inputs have degree zero;
when the output has degree less than $d$; and when the output has degree $d$.
The first two of these cases follow easily from the Stasheff identities for
$\\{m_{n}^{V}\\}$. To check the third case, we suppose that
$|a_{1}|+\cdots+|a_{n}|+n-3=d$, and we compute

$\displaystyle\sum_{r+s+t=n}(-1)^{r+st}m_{r+1+t}\left(\operatorname{id}^{\otimes
r}\otimes m_{s}\otimes\operatorname{id}^{\otimes
t}\right)(a_{1},\ldots,a_{n})=$ $\displaystyle(-1)^{|a_{1}|(n-1)+1}\langle
a_{1},m^{V}_{n-1}(a_{2},\ldots,a_{n})\rangle+(-1)^{n-1}\langle
m^{V}_{n-1}(a_{1},\ldots,a_{n-1}),a_{n}\rangle\,;$

and this vanishes by the cyclic symmetry condition for $m_{n-1}^{V}$.

It follows that $A$ is an $\mathrm{A}_{\infty}$-algebra with the operations
$\\{m_{n}\\}$. Finally, the cyclic symmetry condition on $\\{m_{n}^{V}\\}$
implies $A$ is a cyclic $\mathrm{A}_{\infty}$-algebra. ∎

######  Remark 5.9.

The construction in the proof yields a bijection between unital cyclic
$\mathrm{A}_{\infty}$-algebra structures on $A$ and nonunital cyclic
$\mathrm{A}_{\infty}$-algebra structures on $V$, but with a modified version
of the second Stasheff identity in the latter case. The cyclic condition is
necessary to make this correspondence work.

######  Remark 5.10.

We suspect that the restriction to odd $d$ is not necessary in Theorem 5.7.
More cases can be established by analyzing the proof; for example the case
$d\equiv 2\text{ modulo }4$.

##  6\. Twisted tensor products

Twisted tensor products are an important tool in homological algebra,
especially in the construction of resolutions. In this section we develop
their theory over a commutative ring, producing universal resolutions that
will be applied to Koszul homomorphisms in later sections. However, our method
of construction is new even in when the base ring $Q$ is a field. Similar
results have been obtained using different methods in unpublished work of
Burke [Bur]. We do not explicitly use the language of twisting cochains, but
these objects are present implicitly; and the reader may consult [LV12,
Section 2.1] for more information on twisted tensor products and twisting
cochains.

Let $C$ be a connected curved differential graded (dg) coalgebra over $Q$ that
is free as a graded module. For a right dg module $M$ and a left dg module $N$
over $\mathsf{\Omega}(C)$, we construct a complex
$M\otimes^{\tau}C\otimes^{\tau}N$, with a “twisted” differential; we call this
complex a _twisted tensor product_. First, we define a dg bimodule
$\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}\mathsf{\Omega}(C)$ over
$\mathsf{\Omega}(C)$.

######  Construction 6.1.

Ignoring differentials for now, there is a well-known exact sequence of graded
$\mathsf{\Omega}(C)$-bimodules

(6.1.1)
$0\to\mathsf{\Omega}(C)\otimes\Sigma^{-1}\bar{C}\otimes\mathsf{\Omega}(C)\xrightarrow{\iota}\mathsf{\Omega}(C)\otimes
Q\otimes\mathsf{\Omega}(C)\xrightarrow{\mu}\mathsf{\Omega}(C)\to 0\,,$

where $\iota(x\otimes\langle c\rangle\otimes y)=\mu(x,\langle c\rangle)\otimes
1\otimes y-x\otimes 1\otimes\mu(\langle c\rangle,y)$ for $c\in\bar{C}$ and
$x,y\in\mathsf{\Omega}(C)$, and $\mu$ the multiplication map.

We give $\mathsf{\Omega}(C)\otimes\Sigma^{-1}\bar{C}\otimes\mathsf{\Omega}(C)$
the unique differential $\partial^{\iota}$ making $\iota$ a chain map, and we
set

$\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}\mathsf{\Omega}(C)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\operatorname{cone}(\mathsf{\Omega}(C)\otimes\Sigma^{-1}\bar{C}\otimes\mathsf{\Omega}(C)\xrightarrow{\iota}\mathsf{\Omega}(C)\otimes
Q\otimes\mathsf{\Omega}(C))\,.$

We write $\partial^{\tau}$ for the differential on
$\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}\mathsf{\Omega}(C)$, this is a
dg $\mathsf{\Omega}(C)$-bimodule whose underlying graded bimodule is
$\mathsf{\Omega}(C)\otimes C\otimes\mathsf{\Omega}(C)$, using the evident
multiplication by $\mathsf{\Omega}(C)$ on either side.

For a right dg $\mathsf{\Omega}(C)$-module $F$ and a left dg
$\mathsf{\Omega}(C)$-module $G$, we define

$F\otimes^{\tau}C\otimes^{\tau}G\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}F\otimes_{\mathsf{\Omega}(C)}\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}\mathsf{\Omega}(C)\otimes_{\mathsf{\Omega}(C)}G\,.$

Explicitly its underlying graded module is $F\otimes C\otimes G$ and the
differential is

$\displaystyle\partial^{\tau}$
$\displaystyle=\partial^{F}\otimes\operatorname{id}_{C}\otimes\operatorname{id}_{G}+\operatorname{id}_{F}\otimes\partial^{C}\otimes\operatorname{id}_{G}+\operatorname{id}_{F}\otimes\operatorname{id}_{C}\otimes\partial^{G}$
$\displaystyle\quad+\left(\mu(\operatorname{id}_{F}\otimes{\scriptstyle\mathsf{\Sigma}}^{-1}p)\otimes\operatorname{id}_{C}\otimes\operatorname{id}_{G}-\operatorname{id}_{F}\otimes\operatorname{id}_{C}\otimes\mu({\scriptstyle\mathsf{\Sigma}}^{-1}p\otimes\operatorname{id}_{G})\right)(\operatorname{id}_{F}\otimes\Delta\otimes\operatorname{id}_{G})$

where $p\colon C\to\bar{C}$ is the natural projection and we use $\mu$ for the
right action of $\mathsf{\Omega}(C)$ on $F$ and the left action of
$\mathsf{\Omega}(C)$ on $G$.

######  6.2.

Given a dg algebra $A$, recall that a dg $A$-module $F$ is _semifree_ if it
admits an exhaustive filtration

$0=F(-1)\subseteq F(0)\subseteq F(1)\subseteq\ldots\subseteq F$

where each subquotient $F(i)/F(i-1)$ is a sum of shifts of $A$. As a matter of
terminology, a semifree dg $A$-bimodule is a semifree dg module over $A\otimes
A^{\mathrm{op}}$.

Every dg $A$-module $M$ admits a semifree resolution in the sense that there
exists a surjective quasi-isomorphism $F\xrightarrow{\simeq}M$, with $F$ a
semifree dg $A$-module. Such resolutions are unique up to homotopy; see
[FHT01, Chapter 6] for this fact, as well as other details regarding semifree
dg modules.

######  Lemma 6.3.

The map
$\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}\mathsf{\Omega}(C)\longrightarrow\mathsf{\Omega}(C)$
induced from Eq. 6.1.1 is a semifree resolution of $\mathsf{\Omega}(C)$ as a
dg $\mathsf{\Omega}(C)$-bimodule.

###### Proof.

By construction, Eq. 6.1.1 is a short exact sequence of dg
$\mathsf{\Omega}(C)$-bimodules, and so it induces an exact triangle in the
derived category of dg $\mathsf{\Omega}(C)$-bimodules. We obtain the quasi-
isomorphism by comparing this triangle to the triangle associated to the cone
construction for
$\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}\mathsf{\Omega}(C)$.

The dg module
$\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}\mathsf{\Omega}(C)$ is
semifree as a dg $\mathsf{\Omega}(C)$-bimodule since $C$ is free as a module
over $Q$ and non-negatively graded. ∎

######  6.4.

Let $\varphi\colon Q\to R$ be a finite local homomorphism with $A\to R$ a free
resolution of $R$ over $Q$. Fix an $R$-complex $M$ and a semifree resolution
$\gamma\colon G\to M$ over $Q$. By Proposition 5.1, there exists a split
unital $\mathrm{A}_{\infty}$-algebra structure $\\{m_{n}\\}$ on $A$ and a
strictly unital $\mathrm{A}_{\infty}$-module structure $\\{m_{n}^{G}\\}$ over
$A$ on $G$. Then by 4.11, this induces a dg module structure over
$\mathsf{\Omega}(\mathsf{B}(A))$ on $G$.

Suppose further that $C$ is a connected curved dg coalgebra with counit
$\varepsilon\colon C\to Q$, equipped with a weak equivalence of connected
curved dg coalgebras $C\to\mathsf{B}(A)$; cf. 4.9. Then $R$ and $G$ each have
an induced dg module structure over $\mathsf{\Omega}(C)$.

######  Theorem 6.5.

In the setting of 6.4, the map

$R\otimes^{\tau}C\otimes^{\tau}G\longrightarrow M\quad\text{given by}\quad
r\otimes c\otimes g\mapsto r\varepsilon(c)\gamma(g)$

is a semifree resolution of $M$ over $R$.

###### Proof.

The map $G\to M$ is a quasi-isomorphism of $\mathrm{A}_{\infty}$-modules over
$A$, and by 4.11, it is a quasi-isomorphism of dg modules over
$\mathsf{\Omega}(\mathsf{B}(A))$, and thus over $\mathsf{\Omega}(C)$. From
Lemma 6.3, we obtain the quasi-isomorphism of (left) dg
$\mathsf{\Omega}(C)$-modules

$\beta\colon\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}G=\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}\mathsf{\Omega}(C)\otimes_{\mathsf{\Omega}(C)}G\xrightarrow{\simeq}\mathsf{\Omega}(C)\otimes_{\mathsf{\Omega}(C)}G\cong
G\,.$

Since $G$ is semifree over $Q$, it follows that
$\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}G$ is semifree as a left dg
module over $\mathsf{\Omega}(C)$.

The claimed quasi-isomorphism fits into the commutative diagram below

${\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}G}$${R\otimes^{\tau}C\otimes^{\tau}G}$${M\,,}$$\scriptstyle{\alpha}$$\scriptstyle{\gamma\beta}$

where $\alpha$ is the map induced by the quasi-isomorphism of dg algebras

$\mathsf{\Omega}(C)\xrightarrow{\simeq}\mathsf{\Omega}(\mathsf{B}(A))\xrightarrow{\simeq}R\,.$

Since the latter is a quasi-isomorphism and
$\mathsf{\Omega}(C)\otimes^{\tau}C\otimes^{\tau}G$ is a semifree resolution of
$M$ over $\mathsf{\Omega}(C)$, it follows that $\alpha$ is a quasi-
isomorphism. We have already justified that $\beta$ and $\gamma$ are quasi-
isomorphisms, accounting for the downward arrow on the right. Therefore the
horizontal map is a quasi-isomorphism, as claimed. ∎

##  7\. $\mathrm{A}_{\infty}$-algebra presentations for Koszul homomorphisms

We now have the machinery to show that Koszul homomorphisms admit
presentations analogous to those of classical Koszul $k$-algebras. The next
result lifts these classical quadratic presentations to local algebra, and
explains how one may think of Koszul homomorphisms as
$\mathrm{A}_{\infty}$-deformations of Koszul algebras over fields.

######  Theorem 7.1.

A finite local homomorphism $\varphi\colon Q\to R$ is Koszul if and only if
there is

  1. (1)

a non-negatively graded, degreewise finite rank free $Q$-module $V$, and a
direct summand $W\subseteq V\otimes V$,

  2. (2)

an $\mathrm{A}_{\infty}$-structure $\\{m_{n}\\}$ on the $Q$-module ${\rm
T}(V)/(W)$ with grading induced from the grading of $V$, and

  3. (3)

a $Q$-linear map $V_{0}\to R$,

such that

  1. (i)

the $k$-algebra ${\rm T}^{a}(V\otimes k)/(W\otimes k)$ is Koszul with respect
to the tensor algebra weight grading,

  2. (ii)

$\\{m_{n}\\}$ agrees with the algebra structure on ${\rm T}^{a}(V)/(W)$ modulo
$\mathfrak{m}_{Q}$; that is

$m_{2}\otimes k=\mu\otimes k\quad\text{and}\quad m_{n}\otimes k=0\ \text{ for
}\ n\neq 2\,,$

where $\mu$ is the usual product on the quotient of a tensor algebra, and

  3. (iii)

with the structure $\\{m_{n}\\}$, the induced map ${\rm T}(V)/(W)\to R$ is a
strict $\mathrm{A}_{\infty}$-algebra quasi-isomorphism.

Moreover, $V$ can be taken to be finite rank whenever $R$ has finite
projective dimension over $Q$.

###### Proof.

If the stated conditions hold then $R\otimes^{\mathsf{L}}_{Q}k$ is formal by
Proposition 5.4, and $\operatorname{Tor}^{Q}(R,k)\cong{\rm T}^{a}(V\otimes
k)/(W\otimes k)$ is Koszul, therefore $\varphi$ is Koszul by definition.

Assume, conversely, that $\varphi$ is Koszul. Since
$\operatorname{Tor}^{Q}(R,k)$ is Koszul, it admits a compatible weight grading
making it quadratic. That is, we have an isomorphism

$\operatorname{Tor}_{i}^{Q}(R,k)_{(w)}\cong\big{(}{\rm
T}_{(w)}^{a}(\bar{V})/(\bar{W})\big{)}_{i}$

identifying the product on Tor with the product on the quotient of the tensor
algebra, where $\bar{V}=\operatorname{Tor}^{Q}(R,k)_{(1)}$ and
$\bar{W}\subseteq\bar{V}\otimes_{k}\bar{V}$ is a graded subspace.

Let $V$ be a free graded $Q$-module such that $V\otimes k=\bar{V}$, and choose
a direct summand $W\subseteq V\otimes V$ such that $W\otimes k=\bar{W}$. If we
define
$A\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}{\rm
T}(V)/(W)$, then $A$ is a free, bigraded $Q$-module and

$A\otimes_{Q}k={\rm T}(V)/(W)\otimes k={\rm
T}(\bar{V})/(\bar{W})\cong\operatorname{Tor}^{Q}(R,k)\,.$

Therefore we may equip $A$ with a differential making it the minimal
$Q$-resolution of $R$. We have constructed Item 1 and Item 3 satisfying
condition Item i. Since $R\otimes^{\mathsf{L}}_{Q}k$ is formal, by Proposition
5.4 there is an $\mathrm{A}_{\infty}$-structure on $A$ as required for Item 2,
inducing the algebra structure on $\operatorname{Tor}^{Q}(R,k)$ and satisfying
the conditions Item ii and Item iii. ∎

In Section 8 we illustrate Theorem 7.1 in detail using the examples in Section
3.

###  7.2. Strictly Koszul presentations

For a local homomorphism $\varphi\colon Q\to R$, Theorem 6.5 allowed us to
obtain free resolutions over $R$ starting from free resolutions over $Q$. The
main input to this theorem was a curved dg coalgebra $C$ over $Q$ with quasi-
isomorphism $\mathsf{\Omega}(C)\to R$. Our philosophy is that when $\varphi$
is Koszul $C$ should have a simple description. In this section we introduce
additional technical assumptions that will allow us to explicitly construct
$C$, mimicking a classical construction of Priddy.

######  Definition 7.3.

Let $\varphi\colon Q\to R$ be Koszul. Recall from Theorem 7.1 that $R$ admits
an $\mathrm{A}_{\infty}$-algebra resolution $A$ over $Q$ with a quadratic
presentation $A\cong{\rm T}(V)/(W)$ satisfying the conditions Items i, ii and
iii. The data $(A,V,W)$ is called a _strictly Koszul presentation_ for
$\varphi$ if, in addition to these conditions,

(7.3.1) $\bar{m}_{1}(V)\subseteq
V\quad\text{and}\quad\bar{m}_{n}\big{(}\bigcap_{i+2+j=n}V^{\otimes i}\otimes
W\otimes V^{\otimes j}\big{)}\subseteq V\quad\text{for }n\geqslant 2\,,$

where we have used the inclusion $\bigcap_{i+2+j=n}V^{\otimes i}\otimes
W\otimes V^{\otimes j}\subseteq V^{\otimes n}\subseteq\bar{A}^{\otimes n}$ to
apply the $\mathrm{A}_{\infty}$-operations $\bar{m}_{n}$ of $\bar{A}$. If the
homomorphism $\varphi$ admits a strictly Koszul presentation, then $\varphi$
is called _strictly Koszul_.

In this setting, we define

$\mathsf{C}_{(n)}(V,W)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\bigcap_{i+2+j=n}(\Sigma
V)^{\otimes i}\otimes\Sigma^{2}W\otimes(\Sigma V)^{\otimes
j}\subseteq\mathsf{B}_{(n)}(A)\,.$

By definition, the curvature term, the coderivation and the comultiplication
on $\mathsf{B}(A)$ restrict to maps on $\mathsf{C}(V,W)$, and hence
$\mathsf{C}(V,W)$ is a counital curved dg coalgebra. We call $\mathsf{C}(V,W)$
the _Priddy coalgebra associated to $(A,V,W)$_. If the presentation is clear
from the context, we say it is the _Priddy coalgebra of $\varphi$_, and we
write

$\mathsf{C}(\varphi)\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\mathsf{C}(V,W)\,.$

What we call the Priddy coalgebra first appeared, for algebras over a field,
in the work of Priddy [Pri70, Section 3], where it is called the Koszul
complex. See also [BGS96, Section 2.6] and [LV12, Chapter 3] (where our
notation is taken from).

In Section 8 we show that complete intersection and Golod homomorphisms are
strictly Koszul, as well as Cohen presentations of almost Golod Gorenstein
local rings. In fact, we are not able to construct surjective Koszul
homomorphisms that are not strictly Koszul, therefore we ask:

######  Question 7.4.

For a surjective Koszul homomorphism $\varphi\colon Q\to R$, is it always
possible to construct a strictly Koszul presentation $(A,V,W)$ as in
Definition 7.3?

We think of $R$ and $\mathsf{C}(\varphi)$ as being Koszul dual to each other
relative to $Q$. The next result justifies this, and in particular it says
that this specializes, at the maximal ideal of $Q$, to classical Koszul
duality over $k$.

######  Theorem 7.5.

Let $\varphi\colon Q\to R$ be a strictly Koszul homomorphism. Then
$\mathsf{C}(\varphi)$ is minimal in the sense that
$\partial(\mathsf{C}(\varphi))\subseteq\mathfrak{m}_{Q}\mathsf{C}(\varphi)$,
and the inclusion $\mathsf{C}(\varphi)\to\mathsf{B}(A)$ is a weak equivalence
of connected curved dg coalgebras. Moreover, both
$T=\operatorname{Tor}^{Q}(R,k)$ and
$E=\operatorname{Hom}(\mathsf{C}(\varphi),k)$ are Koszul $k$-algebras with

$\operatorname{Ext}_{T}(k,k)\cong
E\quad\text{and}\quad\operatorname{Ext}_{E}(k,k)\cong T\,.$

###### Proof.

We fix a strictly Koszul presentation, so that
$\mathsf{C}(\varphi)=\mathsf{C}(V,W)$.

By Theorem 7.1 the $\mathrm{A}_{\infty}$-structure on $A$ satisfies
$m_{n}\otimes k=0$ for $n\neq 2$, and $A\otimes k={\rm T}^{a}(V\otimes
k)/(W\otimes k)$ is a quadratic algebra. It then follows from [LV12,
Proposition 3.3.2] that the differential of $\mathsf{C}(V,W)\otimes
k=\mathsf{C}(V\otimes k,W\otimes k)\subseteq\mathsf{B}(A\otimes k)$ is zero.
Therefore $\mathsf{C}(V,W)$ is minimal.

Since $T=A\otimes k$ is Koszul by assumption, $\mathsf{C}(V\otimes k,W\otimes
k)\to\mathsf{B}(A\otimes k)$ is a weak equivalence by [LV12, Theorem 3.4.6].
Since $V$ is free and $W\subseteq V\otimes V$ is a summand,
$\mathsf{\Omega}(\mathsf{C}(V\otimes k,W\otimes
k))=\mathsf{\Omega}(\mathsf{C}(V,W))\otimes k$ and
$\mathsf{\Omega}(\mathsf{B}(A\otimes k))=\mathsf{\Omega}(\mathsf{B}(A))\otimes
k$, so it follows from the the derived version of Nakayama’s lemma that
$\mathsf{C}(V,W)\to\mathsf{B}(A)$ is a weak equivalence as well.

Since $\mathsf{C}(\varphi)$ is minimal, the coproduct induces the structure of
a graded $k$-algebra on $\operatorname{Hom}(\mathsf{C}(\varphi),k)$, with zero
differential. Writing $(-)^{\vee}=\operatorname{Hom}(-,Q)$, we can directly
compute from the definition of the Priddy coalgebra:

(7.5.1) $\mathsf{C}(\varphi)^{\vee}={\rm
T}^{a}(\Sigma^{-1}V^{\vee})/(\Sigma^{-2}W^{\perp})$

where $W^{\perp}=\left\\{f\in(V\otimes V)^{\vee}\,\middle|\,f(W)=0\right\\}$
(this uses that $V$ is free and $W\subseteq V\otimes V$ is a summand). It
follows that

$E=\mathsf{C}(\varphi)^{\vee}\otimes k={\rm T}^{a}(\Sigma^{-1}V^{\vee}\otimes
k)/(\Sigma^{-2}W^{\perp}\otimes k)\,.$

Therefore $E$ is the quadratic dual of $T={\rm T}^{a}(V\otimes k)/(W\otimes
k)$, and the final statement follows from [BGS96, 2.10]. ∎

###  7.6. The Priddy resolution

We have arrived at one of the main applications of our techniques. The next
result provides explicit “universal resolutions” for modules over the target
of a strictly Koszul homomorphism. It recovers the Shamash resolution in the
case of complete intersection homomorphisms, and the bar resolution of Iyengar
and Burke in the case of Golod homomorphisms. We present these and other
examples in the next section.

######  Theorem 7.7.

Let $\varphi\colon Q\to R$ be a Koszul homomorphism with a strictly Koszul
presentation $(A,V,W)$. Assume that $M$ is an $R$-complex with a semifree
resolution $G\to M$ over $Q$ and that $G$ has a strictly unital
$\mathrm{A}_{\infty}$-module structure over $A$. Then

$R\otimes^{\tau}\mathsf{C}(V,W)\otimes^{\tau}G\longrightarrow M$

is a semifree resolution over $R$, with differential given by

$\displaystyle\partial^{\tau}$
$\displaystyle=\sum_{\begin{subarray}{c}r+s+t=n\\\ r,t\geqslant 0,s\geqslant
1\end{subarray}}(-1)^{\frac{s(s+1)}{2}}\operatorname{id}_{R}\otimes(\operatorname{id}^{\otimes
r}\otimes{\scriptstyle\mathsf{\Sigma}}\bar{m}_{s}({\scriptstyle\mathsf{\Sigma}}^{-1})^{\otimes
s}\otimes\operatorname{id}^{\otimes t})\otimes\operatorname{id}_{G}$
$\displaystyle\quad+\sum_{\begin{subarray}{c}i+j=n+1\\\ i\geqslant
0,j\geqslant