# Detecting Golodness via Gröbner Degeneration

Keller VandeBogert

###### Abstract.

In this paper we study the extent to which Golodness may be transferred along
morphisms of DG-algebras. In particular, we show that if $I$ is a so-called
fiber invariant ideal, then Golodness of $I$ is equivalent to Golodness of the
initial ideal of $I$. We use this to transfer Golodness results for monomial
ideals to more general classes of ideals. We also prove that any so-called
rainbow monomial ideal with linear resolution defines a Golod ring; this
result encompasses and generalizes many known Golodness results for classes of
monomial ideals. We then combine the techniques developed to give a concise
proof that maximal minors of (sparse) generic matrices define Golod rings,
independent of characteristic.

##  1\. Introduction

Massey products are higher order (co)homological operations that take as
inputs a tuple of homology classes and output a subset of $H_{\bullet}(A)$,
where $A$ is some DG-algebra. In general, Massey products can be very
difficult to compute explicitly and may only be partially defined as a
function. Despite their complexity, these operations have warranted a close
study due to their description in terms of differentials arising from
associated Eilenberg-Moore spectral sequences (see [GM74]). If all higher
order Massey products are trivial (that is, they are defined and only contain
$0$), it is known that the associated Eilenberg-Moore spectral sequences
degenerate on the first page; this connection was used by Gugenheim and May to
carry out explicit cohomological computations of spaces for which more
elementary methods did not apply.

In [Gol62], Golod proved that triviality of Massey products on the Tor-algebra
of a local ring $(R,\mathfrak{m},k)$ was intimately connected to the Poincaré
series $P^{R}_{k}(t)$. More precisely, Golod noticed that the existence of so-
called _trivial Massey operations_ is equivalent to $P^{R}_{k}(t)$ attaining a
coefficient-wise inequality, originally established by Serre. Such rings are
now called _Golod rings_ , and despite their long history remain relatively
misunderstood even for seemingly simple classes of rings.

Even for rings defined by monomial ideals, it is a difficult problem (with a
rather bumpy history) to prove Golodness. Counterexamples to results of
Berglund and Jöllenbeck [BJ07] due to De Stefani [DS16] and Katthän [Kat17]
indicated that there were gaps in the proofs of some of the results of [BJ07],
and has led to more recent work exploring the extent to which the results of
Berglund and Jöllenbeck remain valid (see [Kat17], [DDS20], and [Van22]). It
is worth noting that understanding the Golodness properties of rings defined
by monomial ideals has connections to understanding cohomological properties
of more geometric objects such as toric varieties, since it is known that the
integral cohomology algebra of the associated _moment-angle_ complex is
isomorphic to the Tor-algebra of certain associated Stanley-Reisner ideals
(for more on this perspective, see [DS07]).

The goal of this paper is to study the extent to which Golodness is preserved
upon deforming a ring. One of the main motivations for this is the observation
that there are many results related to Golodness that are known only for
monomial ideals, implying that there may be cases for which Golodness can be
transferred along an associated Gröbner degeneration. We also prove that a
large class of monomial ideals known as _rainbow monomial ideals_ with linear
resolution are always Golod. This produces a large class of Golod rings
defined by monomial ideals that encompasses and generalizes many known classes
of rings throughout the literature. We combine this result with results
established by Conca and Varbaro on squarefree Gröbner degenerations to prove
that ideals of maximal minors of sparse generic matrices define Golod rings;
this generalizes a result of Srinivasan using techniques that require no
characteristic assumptions on the base field.

The paper is organized as follows. In Section 2, we introduce some necessary
background and establish notation for the rest of the paper. We also prove
some results related to the transfer of trivial Massey operations along
certain types of morphisms of DG-algebras which will be employed in later
sections (see Theorem 2.9 and Proposition 2.11). In Section 3 we study the
case of transferring trivial Massey operations along morphisms $\phi:R\to S$
of polynomial rings whose kernel is generated by variable differences; in
particular, the trivial Massey operation is directly induced by the map $\phi$
in such cases.

In Section 4, we examine the relationship between Golodness of an ideal and
its initial ideal. We show that ideals with _fiber invariant_ Betti numbers
(see Definition 4.6) have the property that they are Golod if and only if the
initial ideal is Golod (see Theorem 4.8). We also use results of Conca and
Varbaro to give simple classes of ideals with fiber invariant Betti numbers
(see Theorem 4.7).

In Section 5, we consider a ubiquitous class of monomial ideals which have
been referred to as _rainbow monomial ideals_ or facet ideals associated to
$n$-partite $n$-uniform clutters. These ideals have already been studied for
their connections to polarizations of Artinian monomial ideals ([AFL22],
[AV21]), arithmetically Cohen-Macaulay sets of points in products of
projective space, and the combinatorial structure of the homological
invariants associated to such ideals ([Nem21], [Van21a]). We prove that any
rainbow monomial ideal with linear resolution is Golod; this result
encompasses many of the Golodness results known for monomial ideals in the
literature, since numerous classes of monomial ideals are obtained by
specializing rainbow monomial ideals with linear resolution (see the statement
of Corollary 5.14 for a list of these classes of monomial ideals).

Finally, in Section 6 we combine many of the results established earlier in
the paper to give a concise proof that any ideal generated by the maximal
minors of a (sparse) generic matrix is Golod. This extends a result due to
Srinivasan, who proved that ideals of maximal minors are Golod in
characteristic $0$. One of the ingredients of the proof is also of independent
interest: more precisely, we show that any power of an ideal of maximal minors
of a (sparse) generic matrix has fiber invariant Betti numbers, generalizing a
result of Boocher [Boo12].

##  2\. Background and the Transfer of Trivial Massey Operations

In this section, we first recall some definitions and establish notation
essential to the remainder of the paper. This includes the definition of a DG-
algebra and Golodness in terms of trivial Massey operations. We also prove two
main results about the transfer of trivial Massey operations along certain
classes of morphisms of DG-algebras which we will use for later sections to
deduce Golodness.

######  Notation 2.1.

The notation $(F_{\bullet},d_{\bullet})$ or $(F_{\bullet},d^{F}_{\bullet})$
will denote a complex $F_{\bullet}$ with differentials $d_{\bullet}$ or
$d^{F}_{\bullet}$, respectively (the notation $d^{F}_{\bullet}$ is used to
distinguish differentials when multiple different complexes are present). When
no confusion may occur, the more concise notation $F$ may be written, where
the notation $d^{F}$ is understood to mean the differential of $F$ (in the
appropriate homological degree); that is, $d^{F}(f_{i})=d^{F}_{i}(f_{i})$,
where $f_{i}\in F_{i}$.

Given a complex $F_{\bullet}$ as above, elements of $F_{n}$ will often be
written with a subscript $n$, for example as $f_{n}$, without specifying that
they are in $F_{n}$.

######  Definition 2.2.

A _differential graded algebra_ $(F,d)$ (or _DG-algebra_) over a commutative
Noetherian ring $R$ is a complex of finitely generated free $R$-modules with
differential $d$ and with a unitary, associative multiplication
$F\otimes_{R}F\to F$ satisfying

  1. (a)

$F_{i}F_{j}\subseteq F_{i+j}$,

  2. (b)

$d_{i+j}(f_{i}f_{j})=d_{i}(f_{i})f_{j}+(-1)^{i}f_{i}d_{j}(f_{j})$,

  3. (c)

$f_{i}f_{j}=(-1)^{ij}f_{j}f_{i}$, and

  4. (d)

$f_{i}^{2}=0$ if $i$ is odd,

where $f_{k}\in F_{k}$.

In this paper, all DG-algebras will be assumed to be associative. The next
definition will be essential for defining Golod rings. If $z_{\lambda}$ is a
cycle in some complex $A$, then the notation $[z_{\lambda}]$ denotes the
homology class of $z_{\lambda}$. In the following definition,
$\overline{a}:=(-1)^{|a|+1}a$, where $|a|$ denotes the homological degree of
$a\in A$.

######  Definition 2.3.

Let $A$ be a DG-algebra with $H_{0}(A)\cong k$. Then $A$ admits a _trivial
Massey operation_ if for some $k$-basis
$\mathcal{B}=\\{h_{\lambda}\\}_{\lambda\in\Lambda}$, there exists a function

$\mu:\coprod_{i=1}^{\infty}\mathcal{B}^{i}\to A$

such that

$\displaystyle\mu(h_{\lambda})=z_{\lambda}\quad\textrm{with}\quad[z_{\lambda}]=h_{\lambda},\
\textrm{and}$ $\displaystyle
d\mu(h_{\lambda_{1}},\dots,h_{\lambda_{p}})=\sum_{j=1}^{p-1}\overline{\mu(h_{\lambda_{1}},\dots,h_{\lambda_{j}})}\mu(h_{\lambda_{j+1}},\dots,h_{\lambda_{p}}).$

Observe that taking $p=2$ in the above definition yields that $H_{\geqslant
1}(A)^{2}=0$, so the induced algebra structure on $H(A)$ is trivial for a DG-
algebra admitting a trivial Massey operation.

######  Notation 2.4.

Throughout the paper, if $R$ is a regular local ring or standard graded
$k$-algebra, the notation $K^{R}$ will denote the Koszul complex on the
minimal generating set of the (graded) maximal ideal of $R$.

We can now state the definition of a Golod ring.

######  Definition 2.5.

Let $(R,\mathfrak{m})$ be a local ring and let $K^{R}$ denote the Koszul
complex on the generators of $\mathfrak{m}$. If $K^{R}$ admits a trivial
Massey operation $\mu$, then $R$ is called a _Golod ring_. An ideal $I$ in
some ring $Q$ will be called _Golod_ if the quotient $Q/I$ is Golod.

######  Remark 2.6.

Note that even though trivial Massey operations are defined with respect to
some choice of basis, it is known that _every_ choice of basis for the Koszul
homology of a Golod ring admits a trivial Massey operation [Avr98, Remark
5.2.3].

Equivalently, Golodness may be defined by saying that the multiplication is
trivial and all higher order Massey products are defined and only contain the
element $0$; we will use this perspective more explicitly in Section 5, see
Remark 5.11 in particular.

The following result of Herzog and Maleki gives a large class of ideals that
are known to be Golod; it is worth noting that the same result holds in
characteristic $0$ for arbitrary ideals by a result of Herzog and Huneke (see
[HH13]). This result will be needed later in Section 6.

######  Theorem 2.7 ([HM18, Theorem 3.1(d)]).

Let $I\subset R$ be a proper monomial ideal in a polynomial ring $R$. Then
$I^{n}$ and $I^{(n)}$ are Golod for all $n\geqslant 2$.

For the remainder of this section, we consider the following question: suppose
that $\phi:A\to B$ is a morphism of DG-algebras and either $A$ or $B$ admits a
trivial Massey operation with respect to some basis. What conditions are
needed on $\phi$ in order to transfer this Massey operation to $B$ or $A$,
respectively?

This question is relatively straightforward if $A$ has a trivial Massey
operation, but is a little more subtle when one wants to “pull back” a trivial
Massey operation from $B$. We first introduce the following notation.

######  Notation 2.8.

Given a DG-algebra $A$ with $H_{0}(A)\cong k$, where $k$ is some field, the
notation $\mathcal{B}^{A}$ will denote any subset $\mathcal{B}^{A}\subset
H_{\bullet}(A)$ forming a $k$-basis for the homology algebra $H_{\bullet}(A)$.

If $\phi:A\to B$ is a morphism of DG-algebras, the notation
$\phi(\mathcal{B}^{A})$ will denote the set of all elements of the form
$\phi_{*}(h)$, where $h\in\mathcal{B}^{A}$ and $\phi_{*}:H_{\bullet}(A)\to
H_{\bullet}(B)$ is the induced map on homology.

The following theorem gives conditions for which a trivial Massey operations
can be pulled back along a morphism of DG-algebras.

######  Theorem 2.9.

Let $\phi:A\to B$ be a morphism of DG-algebras inducing a surjection on cycles
and an injection on homology. Assume that there exists a trivial Massey
operation $\mu$ on some basis $\mathcal{B}^{B}$ of $B$ with
$\phi(\mathcal{B}^{A})\subset\mathcal{B}^{B}$. Then there exists a trivial
Massey operation $\mu^{\prime}$ on $A$ such that

$\phi\circ\mu^{\prime}(h_{\lambda_{1}},\dots,h_{\lambda_{p}})=\mu(\phi(h_{\lambda_{1}}),\dots,\phi(h_{\lambda_{p}})),$

for all choices of $\lambda_{1},\dots,\lambda_{p}$.

###### Proof.

Proceed by induction on $p$. Let $p=1$ and
$\\{h_{\lambda}\\}_{\lambda\in\Lambda}$ denote the basis $\mathcal{B}^{A}$,
and define $\mu^{\prime}(h_{\lambda_{1}}):=z_{\lambda_{1}}$, where
$\phi(z_{\lambda_{1}})=\mu(\phi(h_{\lambda_{1}}))$ (the element
$z_{\lambda_{1}}$ exists since the induced map $\phi:Z_{\bullet}(A)\to
Z_{\bullet}(B)$ is assumed to be surjective). By construction,
$\phi\circ\mu^{\prime}(h_{\lambda_{1}})=\mu(\phi(h_{\lambda_{1}}))$.

Assume now that $p>1$. For any choice of $\lambda_{1},\dots,\lambda_{p}$,
consider the element

$m:=\sum_{i=1}^{p-1}\overline{\mu^{\prime}(h_{\lambda_{1}},\dots,h_{\lambda_{i}})}\mu^{\prime}(h_{\lambda_{i+1}},\dots,h_{\lambda_{p}}).$

By the inductive hypothesis,
$\phi(m)=d\mu(\phi(h_{\lambda_{1}}),\dots,\phi(h_{\lambda_{p}}))$, and since
$\phi$ induces an injection on homology, $m=d^{A}(a)$ for some $a\in A$. By
selection of $a$, this implies that

$\phi(a)-\mu(\phi(h_{\lambda_{1}}),\dots,\phi(h_{\lambda_{p}}))\in
Z_{\bullet}(B).$

Since $\phi:Z_{\bullet}(A)\to Z_{\bullet}(B)$ is surjective, there exists
$a^{\prime}\in Z_{\bullet}(A)$ such that

$\phi(a-a^{\prime})=\mu(\phi(h_{\lambda_{1}}),\dots,\phi(h_{\lambda_{p}})).$

Defining $\mu^{\prime}(h_{\lambda_{1}},\dots,h_{\lambda_{p}}):=a-a^{\prime}$,
it follows that $\mu^{\prime}$ satisfies the required properties. ∎

The following proposition is stated here to give a more specialized class of
morphisms of DG-algebras for which the hypotheses of Theorem 3.5 are
satisfied.

######  Proposition 2.10.

Let $\phi:A\to B$ be a surjective quasi-isomorphism of complexes. Then the
induced map $\phi:Z_{\bullet}(A)\to Z_{\bullet}(B)$ is a surjection.

###### Proof.

Let $b_{i}\in Z_{i}(B)$ be any cycle. By assumption, $\phi(a_{i})=b_{i}$ for
some $a_{i}\in A_{i}$. It remains to show that $a_{i}$ is in fact a cycle. The
assumption that $\phi$ is a quasi-isomorphism implies that there also exists
$a_{i}^{\prime}\in Z_{i}(A)$ and $b_{i+1}\in B_{i+1}$ such that
$b_{i}=\phi(a_{i}^{\prime})+d^{B}(b_{i+1})$, so that
$\phi(a_{i}-a_{i}^{\prime})=d^{B}(b_{i+1})$. Since $\phi_{*}$ is in particular
an injection, this implies that $a_{i}-a_{i}^{\prime}=d^{A}(a_{i+1})$ for some
$a_{i+1}\in A_{i+1}$. This of course implies that $a_{i}\in Z_{i}(A)$. ∎

The following proposition gives conditions for pushing forward a trivial
Massey operation along a morphism of DG-algebras.

######  Proposition 2.11.

Let $\phi:A\to B$ be a quasi-isomorphism of DG-algebras. If $\mu$ is a trivial
Massey operation with respect to some basis $\mathcal{B}^{A}$ of $A$, then the
function $\mu^{\prime}$ defined by

$\mu^{\prime}(\phi(h_{\lambda_{1}}),\dots,\phi(h_{\lambda_{p}})):=\phi\big{(}\mu(h_{\lambda_{1}},\dots,h_{\lambda_{p}})\big{)}$

is a trivial Massey operation on $B$ with respect to the basis
$\phi(\mathcal{B}^{A})$.

###### Proof.

Since $\phi$ is a quasi-isomorphism, $\phi(\mathcal{B}^{A})$ is a basis for
the homology algebra $H_{\bullet}(B)$. Using the fact that $\phi$ is a
morphism of DG-algebras, one finds:

$\displaystyle
d^{B}\mu^{\prime}(\phi(h_{\lambda_{1}}),\dots,\phi(h_{\lambda_{p}}))$
$\displaystyle=d^{B}\phi(\mu(h_{\lambda_{1}},\dots,h_{\lambda_{p}})$
$\displaystyle=\phi d^{A}\mu(h_{\lambda_{1}},\dots,h_{\lambda_{p}})$
$\displaystyle=\phi\Big{(}\sum_{i=1}^{p-1}\overline{\mu(h_{\lambda_{1}},\dots,h_{\lambda_{i}})}\mu(h_{\lambda_{i+1}},\dots,h_{\lambda_{p}})\Big{)}$
$\displaystyle=\sum_{i=1}^{p-1}\overline{\mu^{\prime}(\phi(h_{\lambda_{1}}),\dots,\phi(h_{\lambda_{i}}))}\mu^{\prime}(\phi(h_{\lambda_{i+1}}),\dots,\phi(h_{\lambda_{p}})).$

Thus, $\mu^{\prime}$ is a trivial Massey operation as desired. ∎

##  3\. Complete Intersections Generated by Variable Differences

In this section, we consider transferring trivial Massey operations along
morphisms of polynomial rings with kernel generated by variable differences.
In particular, the results of Section 2 imply that Golodness can be
transferred in either direction along such a morphism. This result in
particular encompasses polarizations of monomial ideals, which for convenience
we define below.

######  Definition 3.1.

Let $I$ be a monomial ideal in the polynomial ring $S=k[x_{1},\ldots,x_{n}]$,
such that for every index $i$, $x_{i}$ appears with power at most $m_{i}$ in
any minimal generator of $I$. Let
$\check{X}_{i}=\\{x_{i1},x_{i2},\ldots,x_{im_{i}}\\}$ be a set of variables,
and let $\tilde{S}=k[\check{X}_{1},\ldots\check{X}_{n}]$ be the polynomial
ring in the union of these variables. An ideal $\tilde{I}\subset\tilde{S}$ is
a _polarization_ of $I$ if

$\displaystyle\sigma=\\{x_{11}-x_{12},x_{11}-x_{13},\ldots,x_{11}-x_{1m_{1}}\\}\cup\\{x_{21}-x_{22},\ldots,x_{21}-x_{2m_{2}}\\}\cup\ldots$
$\displaystyle\cup\\{x_{n1}-x_{n2},\ldots,x_{n1}-x_{n,m_{n}}\\}$

is a regular sequence in $\tilde{S}/\tilde{I}$ and
$\tilde{I}\otimes\tilde{S}/\sigma\cong I$.

######  Observation 3.2.

Let $\phi:R\to S$ be a surjective graded morphism of standard-graded
$k$-algebras. Then $\phi$ induces a surjective morphism of DG-algebras
$K^{R}\to K^{S}$.

###### Proof.

Any graded $k$-algebra morphism is equivalently determined by the data of a
morphism of vector spaces $R_{1}\to S_{1}$, and this map can be extended to a
morphism of algebras on the corresponding exterior algebras. ∎

######  Setup 3.3.

Let $R$ and $S$ be standard-graded polynomial rings over a fixed field $k$,
and let $\phi:R\to S$ denote any graded surjective morphism of $k$-algebras
with the property that $\mathfrak{a}:=\operatorname{Ker}\phi$ is a complete
intersection.

######  Remark 3.4.

Notice that the assumption that $\phi:R\to S$ is a graded morphism of
$k$-algebras in Setup 3.3 implies that $\mathfrak{a}$ is generated by linear
forms.

######  Theorem 3.5.

Adopt notation and hypotheses as in Setup 3.3. Let $I\subset R$ be any ideal
such that $\mathfrak{a}$ is regular on $R/I$. Then $I$ is Golod if and only if
$\phi(I)$ is Golod.

######  Remark 3.6.

Theorem 3.5 can also be deduced in the local case by considering the Poincaré
series of the respective rings; see for instance [Avr98, Proposition 5.2.4].
One advantage of the argument given here is that the results of Section 2 tell
one precisely how the Massey product is induced by the map $\phi$, which is
useful for explicit computations.

###### Proof.

$\implies:$ Assume that $I$ is Golod. Since $\mathfrak{a}$ is regular on
$R/I$, the induced map $R/I\otimes_{R}K^{R}\to S/\phi(I)\otimes_{S}K^{S}$ is a
quasi-isomorphism of DG-algebras, so $\phi(I)$ is Golod by Proposition 2.11.

$\impliedby:$ Assume that $\phi(I)$ is Golod. Since $\mathfrak{a}$ is regular
on $R/I$, the induced map $\widehat{\phi}:R/I\otimes_{R}K^{R}\to
S/\phi(I)\otimes_{S}K^{S}$ is a quasi-isomorphism of DG-algebras, and
Observation 3.2 implies that this induced map is also a surjection. Let
$\mathcal{B}^{R/I}$ denote any $k$-basis for the homology algebra. Then
$\phi(\mathcal{B}^{R/I})$ may be extended to a $k$-basis
$\mathcal{B}^{S/\phi(I)}$ for $H_{\bullet}(S/\phi(I)\otimes_{S}K^{S})$, and
the Golodness assumption of $\phi(I)$ implies that there exists a trivial
Massey operation with respect to $\mathcal{B}^{S/\phi(I)}$. Combining
Proposition 2.10 with Theorem 2.9, it follows that $I$ is Golod. ∎

######  Corollary 3.7.

Let $\widetilde{I}$ denote the polarization of any monomial ideal $I$. Then
$I$ is Golod if and only if $\widetilde{I}$ is Golod.

##  4\. Fiber Invariant Gröbner Degenerations

In this section we study one of the main motivating questions of this paper:
to what extent can Golodness be transferred along an associated Gröbner
degeneration? We first give the necessary background on Gröbner degenerations,
and use this material to define the notion of _fiber invariant_ Betti numbers
(with respect to a given term order). We then use results of Conca and Varbaro
from [CV20] to furnish some simple classes of ideals with fiber invariant
Betti numbers. Finally, we prove that for an ideal $I$ with fiber invariant
Betti numbers, Golodness of $I$ is equivalent to Golodness of the initial
ideal $\textrm{in}_{<}(I)$. This result will be particularly useful in Section
6.

The results on Gröbner degenerations are stated here without proof; for a more
complete introduction to this material see [Gre98] and the references therein.

######  Setup 4.1.

Let $R=k[x_{1},\dots,x_{n}]$ and fix a weight vector $w\in\mathbb{N}^{n}$
inducing a term order $<$. Define $R^{h}:=R[t]$. Let $f\in R$ and write

$f=\underset{1\leqslant i\leqslant l}{\sum}c_{i}x^{\mathbf{a}_{i}}$

with $w(\textbf{a}_{1})\geqslant w(\textbf{a}_{2})\geqslant\cdots\geqslant
w(\textbf{a}_{l})$. Define the homogenization of $f$ with respect to $w$ as

$f^{h}:=t^{w(\textbf{a}_{1})}\cdot f(t^{-w_{1}}x_{1},\dots,t^{-w_{n}}x_{n}).$

For any homogenenous ideal $I\subset R$, define the homogenization of $I$ via

$I^{h}:=\\{f^{h}\mid f\in I\\}.$

######  Observation 4.2.

Adopt notation and hypotheses as in Setup 4.1. If $I\subset R$ is a
homogeneous ideal, then $I^{h}$ is homogeneous with respect to the following
gradings:

  1. (1)

$\deg(x_{i})=1$ for $1\leqslant i\leqslant n$ and $\deg(t)=0$, or

  2. (2)

$\deg(x_{i})=w_{i}$ for $1\leqslant i\leqslant n$ and $\deg(t)=1$.

By convention, we will always think of a homogenized ideal $I^{h}$ as being
graded with respect to the grading $(2)$ of Observation 4.2. The following
proposition is a fundamental result in the study of initial ideals; it says
that $I$ and $\textrm{in}_{<}I$ can be connected via a flat family.

######  Proposition 4.3.

Adopt notation and hypotheses as in Setup 4.1. If $G$ is a $w$-Gröbner basis
of $I$, then:

  1. (a)

$I^{h}=(g^{h}\mid g\in G)$, and

  2. (b)

$R^{h}/I^{h}$ is flat over $k[t]$.

In particular,

$R^{h}/I^{h}\otimes R^{h}/(t)=R/\textrm{in}_{w}(I)\quad\textrm{and}\quad
R^{h}/I^{h}\otimes R^{h}/(t-1)=R/I.$

The flatness of $R^{h}/I^{h}$ implies that free resolutions over $R^{h}$ will
descend to free resolutions over $R$. This fact is made precise in the
following:

######  Corollary 4.4.

Adopt notation and hypotheses as in Setup 4.1. Let

$\displaystyle F:\quad\cdots\to F_{i}^{h}\to\cdots\to F^{h}_{1}\to R^{h}\to
0.$

be a minimal free resolution of $R^{h}/I^{h}$. Then this resolution descends
to:

  1. (1)

A minimal $R$-free resolution of $R/in_{w}(I)$, by applying the functor
$-\otimes_{k[t]}k[t]/(t)$.

  2. (2)

A (possibly non-minimal) $R$-free resolution of $R/I$, by applying the functor
$-\otimes_{k[t]}k[t]/(t-1)$

In particular, one has

$\beta_{ij}^{R^{h}}(I^{h})=\beta_{ij}^{R}(\textrm{in}_{w}(I))\quad\textrm{for
all}\ i,j.$

The next proposition is an evident observation resulting from Corollary 4.4;
we state it explicitly here because it will be an essential ingredient for
later proofs.

######  Proposition 4.5.

Let $G$ denote a reduced Gröbner basis of an ideal $I$, and let $F^{h}$ denote
the homogeneous minimal free resolution of $I^{h}$. Then
$\beta_{ij}(\textrm{in}_{<}I)=\beta_{ij}(I)$ if and only if $F^{h}\otimes
k[t]/(t-1)$ is a minimal complex.

The following definition introduces a key notion that will allow us to
transfer Golodness along associated Gröbner degenerations.

######  Definition 4.6.

Let $R$ be a standard graded polynomial ring over a field $k$ endowed with
some term order $<$. A homogeneous ideal $I$ is said to have _fiber invariant_
Betti numbers (with respect to $<$) if

$\beta_{ij}(R/I)=\beta_{ij}(R/\textrm{in}_{<}I)\quad\textrm{for all}\ i,j.$

It is worth noting that the property
$\beta_{ij}(R/I)=\beta_{ij}(R/\textrm{in}_{<}I)$ for all $i,\ j$ at least
implies that $I$ has a minimal generating set that forms a Gröbner basis,
which is already an uncommon property. That being said, the following theorem
will give us a sufficiently abundant source of examples of ideals with fiber
invariant Betti numbers; in particular, such ideals exist (the fact that such
ideals exist has already been known in the literature, see for example [Boo12]
and [MS14]).

######  Theorem 4.7.

Assume $I$ is a homogeneous ideal with linear minimal free resolution. The
ideal $I$ has fiber invariant Betti numbers in each of the following cases:

  1. (1)

the initial ideal $\textrm{in}_{<}(I)$ has linear resolution, or

  2. (2)

the initial ideal $\textrm{in}_{<}(I)$ is squarefree.

###### Proof.

Proof of $(1):$ If $\textrm{in}_{<}(I)$ has linear resolution, then there are
no nontrivial consecutive cancellations (see [Pee04]) in the Betti table of
$\textrm{in}_{<}(I)$, so the Betti tables must coincide.

Proof of $(2)$: If $\textrm{in}_{<}(I)$ is squarefree, then $I$ also has
linear resolution by [CV20, Theorem 1.2]. The proof then follows by $(1)$. ∎

The following theorem shows that Golodness may be transferred along Gröbner
degenerations associated to ideals with fiber invariant Betti numbers.

######  Theorem 4.8.

Let $I$ be any ideal with fiber invariant Betti numbers with respect to some
term order $<$. Then,

$R/I\ \textrm{is Golod}\quad\iff\quad R/\textrm{in}_{<}(I)\ \textrm{is
Golod}.$

###### Proof.

Let $I^{h}\subset R^{h}$ denote the associated homogenization with respect to
the term order $<$, and let $I^{a}$ denote $I^{h}\otimes_{k[t]}k[t]/(t-a)$.
The assumption of fiber invariance is equivalent to the statement that there
is an isomorphism of $R$ (or $k$)-modules
$\operatorname{Tor}_{\bullet}^{R}(R^{h}/I^{h},k)\cong
F^{h}\otimes_{R}k\cong\operatorname{Tor}_{\bullet}^{R}(R/I^{a},k)$ for all
$a\in k$, and this isomorphism is furthermore an isomorphism of algebras. This
implies that the induced map $R^{h}/I^{h}\otimes_{R}K\to R/I^{a}\otimes_{R}K$
is a surjective quasi-isomorphism of DG-algebras, whence $I^{h}$ is Golod as
an $R$-module if and only if $I^{a}$ is Golod for some $a\in k$ by Proposition
2.10 and Theorem 2.9. Employing Proposition 2.11, this is in fact equivalent
to $I^{a}$ being Golod for _all_ $a\in k$. ∎

The standard inequality
$\beta_{ij}(R/I)\leqslant\beta_{ij}(R/\textrm{in}_{<}(I))$ may tempt one to
believe that trivial homology algebra (or even Golodness) of
$R/\textrm{in}_{<}(I)$ implies trivial homology algebra/Golodness for $I$, but
the following simple example shows that this is not true.

######  Example 4.9.

Let $R=k[x_{1},x_{2},x_{3}]$ and consider the ideal

$I=\left({x}_{1}^{2},{x}_{1}{x}_{3},-{x}_{1}{x}_{2}+{x}_{3}^{2},{x}_{2}{x}_{3},{x}_{2}^{2}\right).$

The ideal $I$ is a grade $3$ Gorenstein ideal and hence is not Golod. With
respect to the standard lexicographic order on the variable order
$x_{1}>x_{2}>x_{3}$, $I$ has initial ideal

$\textrm{in}_{<}(I)=\left({x}_{1}^{2},{x}_{1}{x}_{2},{x}_{2}^{2},{x}_{1}{x}_{3},{x}_{2}{x}_{3},{x}_{3}^{3}\right),$

and one can check using Macaulay2 that $\textrm{in}_{<}(I)$ _is_ Golod. In
particular, having a Golod initial ideal does not imply that the original
ideal is Golod.

##  5\. Golodness of Rainbow Monomial Ideals with Linear Resolution

In this section, we prove that so-called _rainbow_ monomial ideals with linear
resolution define Golod rings. This result generalizes and extends many pre-
existing Golodness results on monomial ideals in the literature (see Corollary
5.14). In order to prove this result, we will need to recall a fair amount of
background on the structure of the linear strands of rainbow monomial ideals
along with higher Massey products on Koszul homology.

######  Definition 5.1.

A _rainbow monomial ideal_ on $n$ colors is any monomial ideal generated by
monomials of the form

$x_{1a_{1}}x_{2a_{2}}\cdots x_{na_{n}}$

for some choices of $a_{1},\dots,a_{n}$. Variables of the form $x_{ia_{i}}$
are said to come from the $i$th _color class_. If there are $d_{i}$ possible
values for $a_{i}$ in the $i$th color class, then the $i$th color class will
be said to have $d_{i}$ variables.

######  Remark 5.2.

A rainbow monomial ideal on $n$ colors can be equivalently formulated as a
facet ideal of an $n$-partite $n$-uniform clutter. This is the perspective
taken in [Nem21].

Our main interest in this section will be rainbow monomial ideals having
linear resolution. We first adopt the following setup:

######  Setup 5.3.

Let $I$ be a rainbow monomial ideal on $n$ colors with $d_{i}$ variables in
each color class. Given such an ideal and any subset $L\subset[n]$, let
$p_{L}$ denote the ranked projection onto the colors contained in $L$; that
is, the ideal obtained by setting all variables of colors classes in
$[n]\backslash L$ equal to $1$.

If $A=\\{\textbf{a}^{1},\dots,\textbf{a}^{n}\\}$, where
$\textbf{a}^{i}=\\{a_{1}^{i}<\cdots<a_{j_{i}}^{i}\\}$, then the following
notation will be used:

$x_{[n],A}:=x_{1\textbf{a}^{1}}\cdot x_{2\textbf{a}^{2}}\cdots
x_{n\textbf{a}^{n}}.$

Similarly, for basis elements of the Koszul complex the following notation
will be used:

$e_{[n],A}:=e_{1\textbf{a}^{1}}\wedge e_{2\textbf{a}^{2}}\wedge\cdots\wedge
e_{n\textbf{a}^{n}}.$

A basis element in the Koszul complex will be said to have _valid multidegree_
if for all choices of $b_{i}\in\textbf{a}^{i}$, one has that
$x_{1b_{1}}x_{2b_{2}}\cdots x_{nb_{n}}\in G(I)$. Given a basis element
$e_{[n],A}$ with valid multidegree, define

$\eta_{n-i}(e_{[n],A}):=e_{1\textbf{a}^{1}}\wedge
e_{2\textbf{a}^{2}}\wedge\cdots\wedge
d(e_{n-i+1,\textbf{a}^{n-i+1}})\wedge\cdots\wedge d(e_{n\textbf{a}^{n}}),$

where $d$ denotes the differential of the associated Koszul complex. Notice
that $\eta_{n-i}(e_{[n],A})$ is a cycle in $R/I\otimes_{R}K_{\bullet}$ by
definition of being a valid multidegree.

######  Theorem 5.4 ([Nem21, Corollary 5.3]).

Let $I$ be a rainbow monomial ideal with linear resolution. Then any ranked
projection of $I$ also has linear resolution. Likewise, $I$ has linear
resolution if and only if its complementary ideal $I^{c}$ has linear
resolution.

######  Remark 5.5.

The notation $I^{c}$ for the complementary ideal denotes the ideal with
$G(I^{c})=G(I_{1}\cdots I_{n})\backslash G(I)$, where
$I_{i}=(x_{i1},\dots,x_{id_{i}})$.

In order to study Golodness of rainbow monomial ideals with linear resolution,
we will first need an explicit basis for the associated Koszul homology.

######  Observation 5.6.

Let $I$ be a rainbow monomial ideal with linear resolution. Then a basis for
the Koszul homology $H_{\bullet}(R/I\otimes_{R}K^{R})$ is represented by all

$\eta_{n-1}(e_{[n],A}),\quad\textrm{where}\ e_{[n],A}\ \textrm{has valid
multidegree.}$

###### Proof.

Suppose that $I$ has $n$ colors and $d_{i}$ variables in each color class. Let
$I_{i}=(x_{i1},\dots,x_{id_{i}})$ for each $1\leqslant i\leqslant n$. Then the
minimal free resolution of $I$ is a the subcomplex of the multigraded minimal
free resolution of $I_{1}I_{2}\cdots I_{n}$ obtained by restricting to all
valid multidegrees. Using the explicit form for the Koszul homology given in,
for instance, [Van21b] the result follows. ∎

######  Notation 5.7.

Adopt notation and hypotheses as in Setup 5.3. The notation $h_{[n],A}$ will
be used to denote the homology class of the element $\eta_{n-1}(e_{[n],A})$.
Use the notation $\mathcal{B}^{R/I}$ to denote the set of all such basis
elements.

The following lemma shows that there exists a partially defined trivial Massey
operation with respect to certain tuples of basis elements as in Observation
5.6. We will see that higher Massey products of all other tuples must vanish
for multidegree reasons, which will yield the desired Golodness statement.

######  Lemma 5.8.

Adopt notation and hypotheses as in Setup 5.3. Let
$\mathcal{V}\subset\coprod_{i=1}^{\infty}\Big{(}\mathcal{B}^{R/I}\Big{)}^{i}$
be the set of all tuples $(h_{[n],A_{1}},\dots,h_{[n],A_{p}})$ such that
$x_{[n],A_{1}}\cdots x_{[n],A_{p}}$ has valid multidegree. Then there exists a
function

$\mu:\mathcal{V}\to R/I\otimes_{R}K^{R}$

such that $\mu(h_{[n],A})=\eta_{n-1}(e_{[n],A})$ and

$d\mu(h_{[n],A_{1}},\dots,h_{[n],A_{p}})=\sum_{j=1}^{p-1}\overline{\mu(h_{[n],A_{1}},\dots,h_{[n],A_{i}}})\mu(h_{[n],A_{i+1}},\dots,h_{[n],A_{p}}).$

###### Proof.

Define $\mu$ as follows:

$\mu(h_{[n],A_{1}},\dots,h_{[n],A_{p}}):=\eta_{n-2}(e_{[n],A_{1}})\wedge\eta_{n-1}(e_{[n],A_{2}})\wedge\cdots\wedge\eta_{n-1}(e_{[n],A_{p}}).$

Then a computation identical to that given in [Van22, Lemma 4.1] yields the
result. ∎

Since we will use terminology related to Massey products for the remainder of
this section, we give the definition here for convenience.

######  Definition 5.9.

Let $A$ be a DG-algebra and let $a_{1},\dots,a_{n}\in H_{\bullet}(A)$.

  1. (1)

The _$n$ -ary Massey product_ $\mu_{n}(a_{1},\dots,a_{n})\subset
H_{\bullet}(A)$ is a partially defined set-valued function. It is defined if
there exist $a_{ij}\in A$ for $1\leqslant i\leqslant j\leqslant n$ such that

    1. (a)

the homology class of $a_{ii}$ is equal to $a_{i}$, and

    2. (b)

there is an equality

$da_{ij}=\sum_{k=i}^{j}\overline{a_{ik}}a_{kj}.$

Then the Massey product is defined to be the set of all homology classes of
the sums $\sum_{k=1}^{n}\overline{a_{1k}}a_{kn}$.

  2. (2)

The DG-algebra $A$ satisfies $(B_{r})$ if for all $k\leqslant r$ and elements
$a_{1},\dots,a_{k}\in H_{\bullet}(A)$, one has:

the Massey product $\mu_{k}(a_{1},\dots,a_{k})$ exists and only contains $0$.

######  Remark 5.10.

Observe that if $A$ has an internal grading, then any Massey product may be
chosen to respect the internal grading; in particular, any Massey product on
the Koszul homology of a multigraded algebra may be chosen to respect this
multigrading.

######  Remark 5.11.

Notice that if a DG-algebra $A$ satisfies $B_{r}$ for all $r\geqslant 2$, then
by definition for all elements $a_{1},\dots,a_{r}\in H_{\bullet}(A)$ there
exist $a_{ij}\in A$ such that

$da_{1r}=\sum_{i=1}^{r}\overline{a_{1i}}a_{ir}.$

Defining $\mu(a_{1},\dots,a_{i}):=a_{1i}$ for $i\leqslant r$ and
$\mu(a_{i+1},\dots,a_{r}):=a_{ir}$ for $i<r$ recovers the definition of a
trivial Massey operation, whence Golodness of a ring $R/I$ is equivalent to
the algebra $R/I\otimes_{R}K^{R}$ satisfying $(B_{r})$ for all $r\geqslant 2$.

######  Lemma 5.12 ([May69, Proposition 2.3] or [Kra66, Lemma 20]).

Let $A$ be a DG-algebra satisfying $(B_{r-1})$ for some $r\geqslant 3$. Then
all $r$-ary Massey products exist and contain only a single element.

The following theorem is the main result of this section.

######  Theorem 5.13.

Let $I$ be a rainbow monomial ideal with linear resolution. Then $I$ is Golod.

###### Proof.

The result will follow by proving that all higher order Massey products are
defined and are trivial; that is, all higher order Massey products contain
only the element $0$. To this end, we proceed inductively on the order of the
higher Massey product. When $p=2$, the assumption that $I$ has linear
resolution implies that the homology algebra
$H_{\bullet}(R/I\otimes_{R}K^{R})$ is trivial.

Assume now that $p>2$. By the inductive hypothesis, the algebra
$R/I\otimes_{R}K^{R}$ satisfies $(B_{p-1})$, so by Lemma 5.12 all $p$-ary
Massey products are defined and contain only a single element. Let
$h_{[n],A_{1}},\dots,h_{[n],A_{p}}$ be a collection of basis elements in
$\mathcal{B}^{R/I}$ with valid multidegree (with respect to the ideal $I$).
There are two cases to consider:

Case 1: The element $x_{[n],A_{1}}\cdots x_{[n],A_{p}}$ does not have valid
multidegree. In this case, since the Massey product
$\mu(h_{[n],A_{1}}),\dots,h_{[n],A_{p}})$ does not have valid multidegree, it
represents the trivial homology class.

Case 2: The element $x_{[n],A_{1}}\cdots x_{[n],A_{p}}$ does have valid
multidegree. In this case, there is a trivial Massey operation on the set of
all elements whose product has valid multidegree by Lemma 5.8. This implies
that the Massey operation must only contain the $0$ element. ∎

As previously mentioned, rainbow monomial ideals with linear resolution are
ubiquitous in the literature; the following corllary gives a non-exhaustive
list of monomial ideals obtained as (specializations of) rainbow monomial
ideals with linear resolution (see Section $6$ of [FGH17] for more on co-
letterplace ideals and their specializations).

######  Corollary 5.14.

The following classes of monomial ideals are Golod:

  1. (1)

Edge ideals of Ferrers graphs and hypergraphs.

  2. (2)

Any restricted power of a strongly stable hypergraph ideal.

  3. (3)

Co-letterplace ideals (and by specialization, edge ideals of cointerval
$d$-hypergraphs and uniform face ideals).

  4. (4)

Clutters associated to arithmetically Cohen-Macaulay sets of points in
$(\mathbb{P}^{1})^{\times d}$.

  5. (5)

Alexander duals of polarizations of Artinian monomial ideals.

More generally, any monomial ideal with linear resolution that has a
polarization that can be relabelled as a rainbow monomial ideal is Golod.

The question of when a given monomial ideal with linear resolution has a
polarization that can be relabelled as a rainbow monomial ideal seems to be
subtle; the following example shows that there exist monomial ideals with no
such polarization.

######  Example 5.15 ([RW01]).

Consider the following ideal:

$I:=(x_{3}x_{4}x_{5}x_{6},x_{2}x_{4}x_{5}x_{6},x_{1}x_{4}x_{5}x_{6},$
$x_{3}x_{4}x_{5}x_{7},x_{2}x_{4}x_{5}x_{v},x_{1}x_{3}x_{5}x_{7},x_{1}x_{2}x_{4}x_{7},x_{1}x_{4}x_{6}x_{7},$
$x_{1}x_{5}x_{6}x_{7},x_{3}x_{4}x_{6}x_{7},x_{2}x_{5}x_{6}x_{7},x_{2}x_{3}x_{6}x_{7},x_{1}x_{2}x_{3}x_{7})\subset
k[x_{1},\dots,x_{7}].$

Then $I$ has linear resolution, but if $\operatorname{char}k\neq 2,\ 3$,
Reiner and Welker show that there is no choice of basis in the minimal free
resolution for which the differentials have coefficients $0$ or $\pm 1$. Thus,
there does not exist a polarization of $I$ that can be relabelled as a rainbow
monomial ideal since the minimal free resolution of a rainbow monomial ideal
has coefficients $0$ or $\pm 1$, independent of characteristic.

##  6\. Golodness of Maximal Minors of Sparse Matrices

In this section, we combine many of the results established in this paper thus
far to prove that (any power of) maximal minors of (sparse) generic matrices
define Golod rings. The proof of this result will use some results already
established in the literature which we recall for convenience. Using these
results we will deduce that these ideals have fiber invariant Betti numbers
and hence Golodness may be transferred along to associated Gröbner
degeneration. The initial ideals in these cases will be evidently Golod by
Theorems 5.13 and 2.7, whence the result will follow immediately.

We begin this section by adopting the following setup:

######  Setup 6.1.

Let $X$ be a generic $n\times m$ matrix ($n\leqslant m$) with coordinate ring
$R=k[X]=k[x_{ij}]_{\begin{subarray}{c}1\leqslant i\leqslant n\\\ 1\leqslant
j\leqslant m\end{subarray}}$. Let $X^{\prime}$ be any matrix obtained by
setting entries of $X$ equal to $0$ in such a way that the nonzero entries
form a 2-sided ladder (see, for instance, [CDF+21, Remark 4.11]). The notation
$I_{n}(X^{\prime})$ will denote the ideal of $n\times n$ (maximal) minors of
$X^{\prime}$.

The following theorem combines numerous results from across the literature on
initial ideals of ideals of maximal minors; we will outline the proofs and
references for these implications in Remark 6.3.

######  Theorem 6.2.

Adopt notation and hypotheses as in Setup 6.1 and define
$I:=I_{n}(X^{\prime})$. Then:

  1. (1)

For any term order $<$, the set of maximal minors of $X^{\prime}$ forms a
Gröbner basis for $I$ (that is, the set of maximal minors is a universal
Gröbner basis).

  2. (2)

For any diagonal term order $<$ and all $t\geqslant 1$, there is the equality

$\textrm{in}_{<}(I^{t})=\textrm{in}_{<}(I)^{t}.$

  3. (3)

For any term order $<$, the ideal $\textrm{in}_{<}(I)^{t}$ has linear
resolution for all $t$.

######  Remark 6.3.

Part $(1)$ of Theorem 6.2 is a well-known result due to Sturmfels and
Zelevinsky, see [SZ93] and [BZ93]. In the case that $X^{\prime}$ is a fully
generic matrix, part $(2)$ was proved originally by Conca in [Con97, Theorem
2.1]. Finally, part $(3)$ is immediate from the results of [CDF+21], since
their results show that the defining ideal of the Rees algebra has a quadratic
Gröbner basis and is hence Koszul, so the linear powers property follows from
[HHZ04, Theorem 1.1].

######  Corollary 6.4.

Adopt notation and hypotheses as in Setup 6.1. Then $I_{n}(X^{\prime})$ has
fiber invariant Betti numbers for any term order $<$.

######  Remark 6.5.

In the case that $X^{\prime}$ is a fully generic matrix, Corollary 6.4 was
proved by Boocher in [Boo12, Theorem 3.1].

###### Proof.

By part $(1)$ of Theorem 6.2, all initial ideals of $I_{n}(X^{\prime})$ are
squarefree. Moreover, the pruning technique given by Boocher in [Boo12] shows
that $I_{n}(X^{\prime})$ has linear resolution, so the result follows from
Theorem 4.7. ∎

The following result is the main result of this section and employs the
results many of the earlier sections of this paper.

######  Theorem 6.6.

Adopt notation and hypotheses as in Setup 6.1. Then $I_{n}(X^{\prime})^{t}$ is
Golod for all $t\geqslant 1$.

######  Remark 6.7.

It is worth mentioning that in the case that $X^{\prime}$ is fully generic and
$k$ has characteristic $0$, the fact that $I_{n}(X^{\prime})$ is Golod was
originally proved by Srinivasan in [Sri89]; this was a corollary of the
explicit DG-algebra structure constructed on the Eagon-Northcott complex. This
algebra structure was heavily reliant on the characteristic $0$ assumption,
however, so it is not clear how similar techniques could be used to prove
Theorem 6.6 in arbitrary characteristic. Likewise, in the case that $t>1$, the
fact that $I_{n}(X^{\prime})^{t}$ is Golod in characteristic $0$ follows from
the results of [HH13].

###### Proof.

Let $I:=I_{n}(X^{\prime})$. There are two cases to consider: the first case is
for $t=1$ and the second is for $t>1$.

Case 1: $t=1$. In this case, recall that $I$ has fiber invariant Betti numbers
with respect to any term order. Thus, Golodness of $I$ is equivalent to
Golodness of $\textrm{in}_{<}(I)$. However, $\textrm{in}_{<}(I)$ is a rainbow
monomial ideal with linear resolution and hence Golod by Theorem 5.13.

Case 2: $t>1$. Let $<$ denote the standard diagonal term order. By Theorem
6.2, the initial ideal of $I^{t}$ has linear resolution for all $t>1$. By
Theorem 4.7 the ideal $I^{t}$ has fiber invariant Betti numbers, and by
Theorem 2.7 the ideal $\textrm{in}_{<}(I)^{t}$ is Golod. Employing Theorem
4.8, it follows that $I^{t}$ is Golod for all $t>1$. ∎

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