# Luminoso at SemEval-2018 Task 10: Distinguishing Attributes Using Text
Corpora and Relational Knowledge

Robyn Speer
Luminoso Technologies, Inc.
675 Massachusetts Avenue
Cambridge, MA 02139
<EMAIL_ADDRESS>
&Joanna Lowry-Duda
Luminoso Technologies, Inc.
675 Massachusetts Avenue
Cambridge, MA 02139
<EMAIL_ADDRESS>

###### Abstract

Luminoso participated in the SemEval 2018 task on “Capturing Discriminative
Attributes” with a system based on ConceptNet, an open knowledge graph focused
on general knowledge. In this paper, we describe how we trained a linear
classifier on a small number of semantically-informed features to achieve an
$F_{1}$ score of 0.7368 on the task, close to the task’s high score of 0.75.

##  1 Introduction

Word embeddings are most effective when they learn from both unstructured text
and a graph of general knowledge Speer and Lowry-Duda (2017). ConceptNet 5
Speer et al. (2017) is an open-data knowledge graph that is well suited for
this purpose. It is accompanied by a pre-built word embedding model known as
ConceptNet Numberbatch111https://github.com/commonsense/conceptnet-
numberbatch, which combines skip-gram embeddings learned from unstructured
text with the relational knowledge in ConceptNet.

A straightforward application of the ConceptNet Numberbatch embeddings took
first place in SemEval 2017 task 2, on semantic word similarity. For SemEval
2018, we built a system with these embeddings as a major component for a
slightly more complex task.

The Capturing Discriminative Attributes task Paperno et al. (2018) emphasizes
the ability of a semantic model to recognize relevant differences between
terms, not just their similarities. As the task description states, “If you
can tell that americano is similar to capuccino and espresso but you can’t
tell the difference between them, you don’t know what americano is.”

The ConceptNet Numberbatch embeddings only measure the similarity of terms,
and we hypothesized that we would need to represent more specific
relationships. For example, the input triple “frog, snail, legs” asks us to
determine whether “legs” is an attribute that distinguishes “frog” from
“snail”. The answer is yes, because a frog _has_ legs while a snail does not.
The _has_ relationship is one example of a specific relationship that is
represented in ConceptNet.

To capture this kind of specific relationship, we built a model that infers
relations between ConceptNet nodes, trained on the existing edges in
ConceptNet and random negative examples. There are many models designed for
this purpose; the one we decided on is based on Semantic Matching Energy (SME)
Bordes et al. (2014).

Our features consisted of direct similarity over ConceptNet Numberbatch
embeddings, the relationships inferred over ConceptNet by SME, features that
compose ConceptNet with other resources (WordNet and Wikipedia), and a purely
corpus-based feature that looks up two-word phrases in the Google Books
dataset.

We combined these features based on ConceptNet with features extracted from a
few other resources in a LinearSVC classifier, using liblinear Fan et al.
(2008) via scikit-learn Pedregosa et al. (2011). The classifier used only 15
features, of which 12 ended up with non-zero weights, from the five sources
described. We aimed to avoid complexity in the classifier in order to prevent
overfitting to the validation set; the power of the classifier should be in
its features.

The classifier produced by this design (submitted late to the contest
leaderboard) successfully avoided overfitting. It performed better on the test
set than on the validation set, with a test $F_{1}$ score of 0.7368, whose
margin of error overlaps with the evaluation’s reported high score of 0.75.

At evaluation time, we accidentally submitted our results on the validation
data, instead of the test data, to the SemEval leaderboard. Our code had
truncated the results to the length of the test data, causing us to not notice
the mismatch. This erroneous submission got a very low score, of course. This
paper presents the corrected test results, which we submitted to the post-
evaluation CodaLab leaderboard immediately after the results appeared. We did
not change the classifier or data; the change was a one-line change to our
code for outputting the classifier’s predictions on the test set instead on
the validation set.

##  2 Features

In detail, these are the five sources of features we used:

#### ConceptNet vector similarity.

Given the triple ($\mathit{term}_{1}$, $\mathit{term}_{2}$, $\mathit{att}$),
we look up the ConceptNet Numberbatch embeddings for the root words of the
three terms (with root words determined using ConceptNet’s built-in
lemmatizer). We determine the cosine similarity of ($\mathit{term}_{1}$,
$\mathit{att}$) and the cosine similarity of ($\mathit{term}_{2}$,
$\mathit{att}$). We then subtract the square roots of the similarity scores
(floored at 0). If this difference is large enough, it indicates a positive
example, a discriminative attribute that applies to $\mathit{term}_{1}$ and
not to $\mathit{term}_{2}$.

#### ConceptNet relational inference.

We train a Semantic Matching Energy model to represent ConceptNet nodes and
relations as vectors, along with a 3-tensor of interactions between them. This
model can then assign a confidence score to any triple (a relation connecting
two terms). We used this model to infer values for each of 11 different
ConceptNet relations. As in the case of vector similarity, each feature value
is the difference between the value inferred for _rel_($\mathit{term}_{1}$,
$\mathit{att}$) and _rel_($\mathit{term}_{2}$, $\mathit{att}$). This model is
described in more detail in the next section.

#### Wikipedia lead sections.

This feature expands on ConceptNet vector similarity: instead of computing the
similarity between the attribute and the term, it computes the maximum of the
similarity between the attribute and any word that appears in the lead section
of the Wikipedia article for the term Wikipedia (2017). This helps to identify
attributes that would be used to define the term, such as “amphibian” as an
attribute for “frog”.

#### WordNet entries.

This feature is similar to the “Wikipedia lead sections” feature. It expands
each term by looking up its synonyms in WordNet Miller et al. (1998), the
synonyms in synsets it is connected to, and the words in its gloss
(definition), and taking the maximum similarity of the attribute to any of
these terms.

#### Google Books 2-grams.

This feature determines if $\mathit{term}_{1}$ forms a significant two-word
phrase with $\mathit{att}$, more than $\mathit{term}_{2}$ does, based on the
Google Books English Fiction data Lin et al. (2012). The “significance” (s) of
a two-word phrase is determined by comparing the smoothed log-likelihood of
the individual unigrams to the smoothed log-likelihood of the phrase:

$\begin{split}\mathrm{s}(\mathit{term},\mathit{att})=10+\log_{10}(\\#(\mathit{term},\mathit{att})+1)\\\
-\log_{10}((\\#(\mathit{term})+10^{5})(\\#(\mathit{att})+10^{5}))\end{split}$

where $\\#$ represents the number of occurrences of a unigram or bigram in the
corpus.

The “ConceptNet relational inference” feature provides 11 entries to the
feature vectors, while the other sources each provide one. In total, there are
15 features that represent each input triple.

Across multiple data sources, we use the square root of cosine similarity to
measure the strength of the match between a term and an attribute. Because
attributes should be at least somewhat related to the terms they describe, and
because weak semantic similarity can be interpreted as relatedness, the square
root helps us emphasize the important part of the scale. The difference
between “somewhat related” and “not related” is more important to the task
than the difference between “very similar” and “somewhat related”, as a
discriminative attribute should ideally be unrelated to the second term.

###  2.1 The Relational Inference Model

To infer truth values for ConceptNet relations, we use a variant of the
Semantic Matching Energy model Bordes et al. (2014), adapted to work well on
ConceptNet’s vocabulary of relations. Instead of embedding relations in the
same space as the terms, this model assigns new 10-dimensional embeddings to
ConceptNet relations, yielding a compact model for ConceptNet’s relatively
small set of relations.

The model is trained to distinguish positive examples of ConceptNet edges from
negative ones. The positive examples are edges directly contained in
ConceptNet, or those that are entailed by changing the relation to a more
general one or switching the directionality of a symmetric relation. The
negative examples come from replacing one of the terms with a random other
term, the relation with a random unentailed relation, or switching the
directionality of an asymmetric relation.

We trained this model for approximately 3 million iterations (about 4 days of
computation on an nVidia Titan Xp) using PyTorch Paszke et al. (2017). The
code of the model is available at
https://github.com/LuminosoInsight/conceptnet-sme.

To extract features for the discriminative attribute task, we focus on a
subset of ConceptNet relations that would plausibly be used as attributes:
_RelatedTo_ , _IsA_ , _HasA_ , _PartOf_ , _CapableOf_ , _UsedFor_ ,
_HasContext_ , _HasProperty_ , and _AtLocation_.

For most of these relations, the first argument is the term, and the second
argument is the attribute. We use two additional features for _PartOf_ and
_AtLocation_ with their arguments swapped, so that the attribute is the first
argument. The generic relation _RelatedTo_ , unlike the others, is intended to
be symmetric, so we add its value to the value of its swapped version and use
it as a single feature.

##  3 The Overfitting-Resistant Classifier

The classifier that we use to make a decision based on these features is
scikit-learn’s LinearSVC, using the default parameters in scikit-learn 0.19.1.
(In Section 4, we discuss other models and parameters that we tried.) This
classifier makes effective use of the features while being simple enough to
avoid some amount of overfitting.

One aspect of the classifier that made a noticeable difference was the scaling
of the features. We tried $L_{1}$ and $L_{2}$-normalizing the columns of the
input matrix, representing the values of each feature, and decided on $L_{2}$
normalization.

We took advantage of the design of our features and the asymmetry of the task
as a way to further mitigate overfitting. All of the features were designed to
identify a property that $\mathit{term}_{1}$ has and $\mathit{term}_{2}$ does
not, as is the case for the discriminative examples, so they should all make a
non-negative contribution to a feature being discriminative. We can inspect
the coefficients of the features in the SVC’s decision boundary. If any
feature gets a negative weight, it is likely a spurious result from
overfitting to the training data. So, after training the classifier, we clip
the coefficients of the decision boundary, setting all negative coefficients
to zero.

If we were to remove these features and re-train, or require non-negative
coefficients as a constraint on the classifier, then other features would
inherently become responsible for overfitting. By neutralizing the features
_after_ training, we keep the features that are working well as they are, and
remove a part of the model that appears to purely represent overfitting.
Indeed, clipping the negative coefficients in this way increased our
performance on the validation set.

Table 1 shows the coeffcients assigned to each feature based on the training
data.

Feature | Coefficient
---|---
ConceptNet vector similarity | 13.82
SME: RelatedTo | 14.01
SME: ($x$ IsA $a$) | 2.13
SME: ($x$ HasA $a$) | 0.00
SME: ($x$ PartOf $a$) | 0.56
SME: ($x$ CapableOf $a$) | 3.72
SME: ($x$ UsedFor $a$) | 0.92
SME: ($x$ HasContext $a$) | 0.88
SME: ($x$ HasProperty $a$) | 0.00
SME: ($x$ AtLocation $a$) | 0.00
SME: ($a$ PartOf $x$) | 3.22
SME: ($a$ AtLocation $x$) | 0.69
Wikipedia lead sections | 12.46
WordNet relatedness | 13.95
Google Ngrams | 28.82
Table 1: Coefficients of each feature in our linear classifier. $x$ represents
a term and $a$ represents the attribute.

##  4 Other experiments

There are other features that we tried and later discarded. We experimented
with a feature similar to the Google Books 2-grams feature, based on the AOL
query logs dataset Pass et al. (2006). It did not add to the performance, most
likely because any information it could provide was also provided by Google
Books 2-grams. Similiarly, we tried extending the Google Books 2-grams data to
include the first and third words of a selection of 3-grams, but this, too,
appeared redundant with the 2-grams.

We also experimented with a feature based on bounding box annotations
available in the OpenImages dataset Krasin et al. (2017). We hoped it would
help us capture attributes such as colors, materials, and shapes. While this
feature did not improve the classifier’s performance on the validation set, it
did slightly improve the performance on the test set.

Before deciding on scikit-learn’s LinearSVC, we experimented with a number of
other classifiers. This included random forests, differentiable models made of
multiple ReLU and sigmoid layers, and SVM with an RBF kernel or a polynomial
kernel.

We also experimented with different parameters to LinearSVC, such as changing
the default value of the penalty parameter $C$ of the error term, changing the
penalty from $L_{2}$ to $L_{1}$, solving the primal optimization problem
instead of the dual problem, and changing the loss from squared hinge to
hinge. These changes either led to lower performance or had no significant
effect, so in the end we used LinearSVC with the default parameters for
scikit-learn version 0.19.1.

##  5 Results

Dataset | F1 |  Error (SEM)
---|---|---
train | .7617 |  $\pm$ .0032
validation | .7281 |  $\pm$ .0085
test | .7368 |  $\pm$ .0091
Table 2: $F_{1}$ scores by dataset. The reported $F_{1}$ score is the
arithmetic mean of the $F_{1}$ scores for both classes.

When trained on the training set, the classifier we describe achieved an
$F_{1}$ score of 0.7617 on the training set, 0.7281 on the validation set, and
0.7368 on the test set. Table 2 shows these scores along with their standard
error of the mean, supposing that these data sets were randomly sampled from
larger sets.

###  5.1 Ablation Analysis

Figure 1: This ablation analysis shows the contributions of subsets of the
five sources of features. Ellipses indicate standard error of the mean,
assuming that the data is sampled from a larger, unseen set.

We performed an ablation analysis to see what the contribution of each of our
five sources of features was. We evaluated classifiers that used all non-empty
subsets of these sources. Figure 1 plots the results of these 31 classifiers
when evaluated on the validation set and the test set.

It is likely that the classifier with all five sources (_ABCDE_) performed the
best overall. It is in a statistical tie ($p>.05$) with _ABDE_ , the
classifier that omits Wikipedia as a source.

Most of the classifiers perfomed better on the test set than on the validation
set, as shown by the dotted line. Some simple classifiers with very few
features performed particularly well on the test set. One surprisingly high-
performing classifier was _A_ (ConceptNet vector similarity), which gets a
test $F_{1}$ score of 0.7355 $\pm$ 0.0091. This is simple enough to be called
a heuristic instead of a classifier, and we can express it in closed form. It
is equivalent to this expression over ConceptNet Numberbatch embeddings:

$\mathrm{sim}(\mathit{term}_{1},\mathit{att})-\mathrm{sim}(\mathit{term}_{2},\mathit{att})>0.0961$

where $\mathrm{sim}(a,b)=\sqrt{\max\left(\frac{a\cdot
b}{||a||\cdot||b||},0\right)}$.

It is interesting to note that source _A_ (ConceptNet vector similarity)
appears to dominate source _B_ (ConceptNet SME) on the test data. SME led to
improvements on the validation set, but on the test set, any classifier
containing _AB_ performs equal to or worse than the same classifier with _B_
removed. This may indicate that the SME features were the most prone to
overfitting, or that the validation set generally required making more
difficult distinctions than the test set.

##  6 Reproducing These Results

The code for our classifier is available on GitHub at
https://github.com/LuminosoInsight/semeval-discriminatt, and its input data is
downloadable from https://zenodo.org/record/1183358.

## References

  * Bordes et al. (2014) Antoine Bordes, Xavier Glorot, Jason Weston, and Yoshua Bengio. 2014.  A semantic matching energy function for learning with multi-relational data.  _Machine Learning_ , 94(2):233–259.
  * Fan et al. (2008) Rong-En Fan, Kai-Wei Chang, Cho-Jui Hsieh, Xiang-Rui Wang, and Chih-Jen Lin. 2008\.  LIBLINEAR: A library for large linear classification.  _Journal of Machine Learning Research_ , 9(Aug):1871–1874.
  * Krasin et al. (2017) Ivan Krasin, Tom Duerig, Neil Alldrin, Vittorio Ferrari, Sami Abu-El-Haija, Alina Kuznetsova, Hassan Rom, Jasper Uijlings, Stefan Popov, Andreas Veit, Serge Belongie, Victor Gomes, Abhinav Gupta, Chen Sun, Gal Chechik, David Cai, Zheyun Feng, Dhyanesh Narayanan, and Kevin Murphy. 2017.  OpenImages: A public dataset for large-scale multi-label and multi-class image classification.
  * Lin et al. (2012) Yuri Lin, Jean-Baptiste Michel, Erez Lieberman Aiden, Jon Orwant, Will Brockman, and Slav Petrov. 2012.  Syntactic annotations for the Google Books Ngram Corpus.  In _Proceedings of the ACL 2012 system demonstrations_ , pages 169–174. Association for Computational Linguistics.
  * Miller et al. (1998) George Miller, Christiane Fellbaum, Randee Tengi, P Wakefield, H Langone, and BR Haskell. 1998.  _WordNet_.  MIT Press Cambridge.
  * Paperno et al. (2018) Denis Paperno, Alessandro Lenci, and Alicia Krebs. 2018.  SemEval-2018 Task 10: Capturing discriminative attributes.  In _Proceedings of the 12th International Workshop on Semantic Evaluation (SemEval-2018)_ , New Orleans, LA, United States. Association for Computational Linguistics.
  * Pass et al. (2006) Greg Pass, Abdur Chowdhury, and Cayley Torgeson. 2006.  A picture of search.  In _Proceedings of the 1st International Conference on Scalable Information Systems_ , InfoScale ’06, New York, NY, USA. ACM.
  * Paszke et al. (2017) Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. 2017.  Automatic differentiation in pytorch.
  * Pedregosa et al. (2011) Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, et al. 2011.  Scikit-learn: Machine learning in Python.  _Journal of machine learning research_ , 12(Oct):2825–2830.
  * Speer et al. (2017) Robyn Speer, Joshua Chin, and Catherine Havasi. 2017.  ConceptNet 5.5: An open multilingual graph of general knowledge.  In _AAAI_ , San Francisco.
  * Speer and Lowry-Duda (2017) Robyn Speer and Joanna Lowry-Duda. 2017.  ConceptNet at SemEval-2017 task 2: Extending word embeddings with multilingual relational knowledge.  In _Proceedings of the 11th International Workshop on Semantic Evaluation (SemEval-2017)_ , pages 85–89, Vancouver, Canada. Association for Computational Linguistics.
  * Wikipedia (2017) Wikipedia. 2017.  Wikipedia, the free encyclopedia — English data export.  (A collaborative project with thousands of authors.) Retrieved from https://dumps.wikimedia.org/enwiki/ on 2017-12-20.


# Iterated Mapping Cones for Strongly Koszul Algebras

Keller VandeBogert  University of Notre Dame

###### Abstract.

In this paper we extend the well-known iterated mapping cone procedure to
monomial ideals in strongly Koszul algebras. We study properties of ideals
generated by monomials in commutative Koszul algebras and show that the linear
strand of ideals generated by linear forms is obtained as a subcomplex of the
Priddy complex. In the case of strongly Koszul algebras, this shows that the
minimal free resolution of a monomial ideal admitting linear quotients is
obtained as an iterated mapping cone, immediately extending results for such
ideals in polynomial rings to strongly Koszul algebras. We then consider
monomial ideals admitting a so-called regular ordering, a generalization of
regular decomposition functions, and show that the comparison maps in the
iterated mapping cone construction can be computed explicitly. In particular,
this gives a closed form for the minimal free resolution of monomial ideals
admitting a regular ordering over strongly Koszul algebras.

###### Key words and phrases:

Koszul algebras, iterated mapping cones, minimal free resolutions, linear
strands

##  1\. Introduction

A $k$-algebra $A$ is _Koszul_ if its residue field $k$ has a linear minimal
free resolution over $A$. Koszul algebras are one method of generalizing
standard graded polynomial rings and make their appearance in a wide range of
seemingly disconnected settings. Topologically, Koszul duality for quadratic
algebras can be used to translate between facts about equivariant and standard
cohomology (see [10]). Representation theoretically, Koszul algebras are, in
the words of Beilinson, Ginzburg, and Soergel, “as close to semisimple as a
$\mathbb{Z}$-graded ring can possibly be” and arise in the context of mixed
complexes/Hodge modules (see [2]). In the setting of number theory,
Positselski (see [15]) has shown that certain classes of Milnor rings are
Koszul algebras and relates the Milnor-Bloch-Kato conjecture to Koszulness of
certain quotient rings. Fröberg gives an overview in [9] of conditions
equivalent to Koszulness along with many examples in the literature of Koszul
algebras, including extremal Gorenstein rings and Veronese/Segre algebras. For
a more complete exposition/overview of Koszul algebras, see [14].

For non-regular Koszul algebras, the minimal free resolution of the residue
field is necessarily infinite, and this resolution can be explicitly described
by the Priddy complex, introduced by Stewart Priddy in [16]. One interesting
feature of this resolution is the association of the algebra $A$ to its
quadratic dual, $A^{!}$. In the “classical” case of a polynomial ring, $A$ is
the symmetric algebra on some vector space and its quadratic dual is the
exterior algebra; in particular the Priddy complex recovers the well-known
Koszul complex. This leads to the idea that one can try to generalize
constructions involving symmetric and exterior algebras by replacing these
objects with Koszul algebras and their respective quadratic duals.

This is the approach taken by the authors in [8] (and indeed was the original
motivation for this paper), where the $L$-complex construction of Buchsbaum
and Eisenbud (see [3]) was extended to Koszul algebras by replacing certain
Schur modules associated to hook partitions with the image of the natural maps

$A\otimes_{k}(A^{!})^{*}_{i+1}\otimes_{k}A_{d-1}\to
A\otimes_{k}(A^{!})^{*}_{i}\otimes_{k}A_{d}$

induced by multiplication by the trace element. It is shown that the complexes
induced by these building blocks constitute a minimal free resolution of
powers of the maximal ideal of $A$, a direct generalization of the original
case considered in [3].

Powers of the maximal ideal are, in particular, monomial ideals, with the
minimal generating set given by the classes of all monomials of a given degree
(one has to be careful about what they mean by “monomial ideal” in a general
Koszul algebra; see Remark 3.10). Considering the combinatorial nature of many
naturally occurring Koszul algebras (for instance, Hibi rings on posets,
Stanley-Reisner rings of graphs, or (sometimes) Orlik-Solomon algebras
associated to matroids), one is tempted to ask: what techniques relating to
monomial ideals in polynomial rings generalize to arbitrary Koszul algebras?

In this paper, we begin to answer this question from a homological perspective
by generalizing the well-known iterated mapping cone procedure for ideals with
linear quotients (see [13]) to _strongly_ Koszul algebras. The iterated
mapping cone procedure spawns from the observation that the short exact
sequence

$0\to\frac{A}{(I:m)}\xrightarrow[]{m}\frac{A}{I}\xrightarrow[]{}\frac{A}{I+(m)}\to
0$

can be used to inductively compute a free resolution of any quotient ring; in
the case that the ideal has linear quotients, this procedure will yield the
_minimal_ free resolution. This recovers the well-known (squarefree) Eliahou-
Kervaire resolution for (squarefree) stable ideals (see [7] and [1]), and
extends to many more classes of monomial ideals. One hitch to this approach
for general Koszul algebras is the fact that an ideal generated by a subset of
variables may not have linear resolution (see Example 3.4); this is remedied
by restricting to _strongly_ Koszul algebras (see Definition 3.1), originally
introduced by Herzog, Hibi, and Restuccia in [11].

This paper is organized as follows. In Section 2 we introduce/recall some
results on linear strands and Koszul algebras. Most importantly, we recall the
construction of the quadratic dual of a quadratic algebra and use this to
construct the Priddy complex. In Section 3, we begin the process of extending
results on monomial ideals in polynomial rings to Koszul algebras. In
particular, we construct the linear strand of any ideal generated by linear
forms as a subcomplex of the Priddy complex (see Proposition 3.8); in the case
of a strongly Koszul algebra, this yields the minimal free resolution
explicitly. We then generalize the idea of a _decomposition function_ to
general strongly Koszul algebras and show that the minimal free resolutions of
ideals admitting linear quotients can be constructed as an iterated mapping
cone.

In Section 4, we consider cases for which the induced comparison maps in the
iterated mapping cone procedure can be constructed explicitly. This leads to
the definition of monomial ideals admitting a _regular ordering_ (see
Definition 4.4), a direct generalization of _regular_ decomposition functions
as in [13]. This culminates in Theorem 4.9, whose proof includes the original
Theorem $1.12$ of [13], and shows that the minimal free resolution of any
monomial ideal admitting a regular ordering in a strongly Koszul algebra can
be computed explicitly in a manner that directly generalizes the case for
polynomial rings. We conclude with further questions on different choices of
decomposition for computing the comparison maps in the iterated mapping cone
procedure and additional combinatorial structure on the resolution of Theorem
4.9 arising from infinite-dimensional cell complexes.

##  2\. Koszul Algebras and The Priddy Complex

In this section, we introduce some necessary background on linear strands and
Koszul algebras that will be needed for the remainder of the paper. We record
two results of [12] that are useful for proving that a candidate complex does
indeed arise as the linear strand of a module. After defining Kozul algebras,
we introduce some machinery for defining the so-called Priddy complex. We
conclude with the well-known connection between Koszul algebras and acyclicity
of the associated Priddy complex, originally proved in [16].

Throughout the paper, all complexes are assumed to be concentrated only in
nonnegative degrees. All $k$-algebras will be assumed to be finitely generated
and endowed with the standard grading (that is, all variables have degree
$1$), and denoted by $A$.

######  Definition 2.1.

Let $F_{\bullet}$ be a minimal graded $A$-free complex with $F_{0}$ having
initial degree $d$. Then the _linear strand_ of $F_{\bullet}$, denoted
$F_{\bullet}^{\operatorname{lin}}$, is the complex obtained by restricting
$d_{i}^{F}$ to the subcomplex generated by components $(F_{i})_{d+i}$ for each
$i\geqslant 1$.

######  Remark 2.2.

Observe that the minimality assumption in Definition 2.1 ensures that the
linear strand is well defined. Choosing bases, the linear strand can be
obtained by restricting to the columns where only linear entries occur in the
matrix representation of each differential.

######  Theorem 2.3 ([12], Theorem 1.1).

Let $G_{\bullet}$ be a linear complex of free $A$-modules with initial degree
$n$. Then the following are equivalent:

  1. (1)

The complex $G_{\bullet}$ is the linear strand of a finitely generated
$A$-module with initial degree $n$.

  2. (2)

The homology $H_{i}(G_{\bullet})_{i+n+j}=0$ for all $i>0$ and $j=0,\ 1$.

The following Proposition shows that linear strands behave in a manner that is
quite similar to minimal free resolutions.

######  Proposition 2.4 ([12], Corollary 1.2).

Let $G_{\bullet}$ be a linear complex of free $A$-modules with initial degree
$n$ such that $H_{i}(G_{\bullet})_{i+n+j}=0$ for all $i>0$, $j=0,\ 1$.

Let $N$ be a finitely generated $A$-module with minimal graded free resolution
$F_{\bullet}$. Assume that there exist isomorphisms making the following
diagram commute:

$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{G_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{F_{1}^{\textrm{lin}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{0}^{\textrm{lin}}.}$

Then $G_{\bullet}\cong F_{\bullet}^{\textrm{lin}}$.

######  Remark 2.5.

In the original formulation of Theorem 2.3 and Proposition 2.4, it was assumed
that the complex $G_{\bullet}$ had finite length. However, the proofs extend
with no effort to the infinite case.

Next, we introduce our main objects of study for the rest of the paper.

######  Definition 2.6.

A $k$-algebra $A$ is _Koszul_ if the residue field $k$ has a linear graded
minimal free resolution over $A$.

######  Example 2.7.

Let $A=k[x_{1},\dots,x_{n}]$. Then the ideal of variables forms a regular
sequence, so the residue field is resolved by the Koszul complex on
$x_{1},\dots,x_{n}$, which is a linear complex.

The following will be needed for stating Definition 2.11.

######  Definition 2.8.

Let $A$ be a graded $k$-algebra. The algebra $A$ is _quadratic_ if $A=T(V)/Q$,
where $V$ is a $k$-vector space, $T(V)$ is the tensor algebra, and $Q$ is a
quadratic ideal of $T(V)$.

Given a quadratic $k$-algebra $A$, the _quadratic dual_ is the algebra

$A^{!}:=\frac{T(V^{*})}{Q^{\perp}},$

with $V^{*}:=\operatorname{Hom}_{k}(V,k)$, and $Q^{\perp}$ is the quadratic
ideal generated by the orthogonal complement to $Q_{2}$ with respect to the
natural pairing

$\displaystyle(V\otimes V)\otimes(V^{*}\otimes V^{*})$ $\displaystyle\to k,$
$\displaystyle(v_{1}\otimes v_{2})\otimes(u_{1}\otimes u_{2})$
$\displaystyle\mapsto u_{1}(v_{1})\cdot u_{2}(v_{2}).$

######  Example 2.9.

If $A=S(V)$ is the symmetric algebra on some vector space $V$, then
$A^{!}=\bigwedge V^{*}$ is the exterior algebra on $V^{*}$.

######  Notation 2.10.

If $x_{1},\dots,x_{n}$ are the generators of the $k$-algebra $A$, then the
notation $x_{1}^{*},\dots,x_{n}^{*}$ will denote the dual generators of
$A^{!}$.

######  Definition 2.11.

Let $A$ be a quadratic algebra. The _Priddy complex_ is the complex
$P_{\bullet}^{A}$ with

$P_{i}^{A}:=A\otimes_{k}(A^{!})_{i}^{*}$

and differential $P_{i}^{A}\to P_{i-1}^{A}$ induced by multiplication by the
trace element, $\sum_{i}x_{i}\otimes x_{i}^{*}$.

The following theorem was proved by Priddy in [16]; other proofs may be found
in, for instance, [14].

######  Theorem 2.12.

Let $A$ be a quadratic $k$-algebra. Then $A$ is Koszul if and only if the
Priddy complex $P_{\bullet}^{A}$ is acyclic.

##  3\. Iterated Mapping Cones for Strongly Koszul Algebras

In this section, we turn our attention to the iterated mapping cone
construction for Koszul algebras. As it turns out, this construction may not
yield a minimal free resolution for arbitrary Koszul algebras (see Example
3.4), but everything works out nicely if we restrict to the slightly smaller
class known as _strongly Koszul algebras_. These were introduced originally by
Herzog, Hibi, and Restuccia, and have the convenient property that any ideal
generated by variables has linear resolution. We also show that for an
arbitrary Koszul algebra, the linear strand of any ideal generated in degree
$1$ is obtained as a subcomplex of the Priddy complex. In the case of strongly
Koszul algebras, this subcomplex will be the minimal free resolution.

With these results established, we obtain a more general version of Lemma
$1.5$ of [13]. As an immediate consequence, ideals with linear quotients in
strongly Koszul algebras satisfy a nearly identical set of properties to those
in the polynomial ring case. For the remainder of the paper,
$A:=k[x_{1},\dots,x_{n}]/I$ will denote a commutative Koszul algebra.

######  Definition 3.1.

A Koszul algebra $A$ is _strongly Koszul_ if there exists a basis $X$ for
$A_{1}$ such that for every $Y\subset X$ and every $x\in X\backslash Y$, there
exists a subset $Z\subset X$ such that $\big{(}(Y):x\big{)}=(Z)$.

A Koszul algebra $A$ is _universally Koszul_ if every ideal generated by
linear forms has a linear resolution.

######  Remark 3.2.

The above definition of strongly Koszul is a definition introduced by Conca,
De Negri, and Rossi in [5, Definition 3.11], and is slightly stronger than the
original definition used by Herzog, Hibi, and Restuccia [11]. Throughout the
rest of the paper, we will assume by convention that if $A$ is a strongly
Koszul algebra, then the presentation $A=k[x_{1},\dots,x_{n}]/I$ has been
chosen in such a way that $X=\\{x_{1},\dots,x_{n}\\}$.

The following Lemma is an essential component of what makes the iterated
mapping cone construction work well for strongly Koszul algebras.

######  Lemma 3.3 ([11]).

Let $A$ be a strongly Koszul algebra. Then every ideal generated by a subset
of the variables has a linear minimal free resolution.

It is important to note that Lemma 3.3 does not hold for general Koszul
algebras. The following example was coomunicated to the author by Hailong Dao,
and is a well-known example due to Conca of a Koszul algebra that is _not_ LG-
quadratic (see [4, Example 1.20]).

######  Example 3.4.

Let

$A:=\frac{k[a,b,c,d]}{(ac,ad,ab-bd,a^{2}+bc,b^{2})}.$

Then $A$ is a Koszul algebra, but the ideal $(b)$ does not have linear
resolution since the relation $c^{2}\cdot b=0$ is minimal.

The following is an exercise in linear algebra which gives a straightforward
relation between generators of the Koszul algebra and its quadratic dual.

######  Observation 3.5.

Let $A$ be a commutative Koszul algebra and let $\\{x_{s}x_{t}\mid(s,t)\in
S\\}$ be a basis for $A_{2}$. For all $(u,v)\notin S$, there exist
coefficients $f^{u,v}_{s,t}\in k$ such that

$x_{u}x_{v}=\sum_{(s,t)\in S}f^{u,v}_{s,t}x_{s}x_{t}.$

Given this notation, $A^{!}_{2}$ has basis
$\\{x^{*}_{u}x^{*}_{v}\mid(u,v)\notin S\\}$ with relations

$x^{*}_{s}x^{*}_{t}=-\sum_{(u,v)\notin S}f^{u,v}_{s,t}x^{*}_{u}x^{*}_{v}.$

In the following construction, we introduce the subcomplex of the Priddy
complex which will serve as the linear strand/minimal free resolution of the
ideals of interest.

######  Construction 3.6.

Let $A$ be a Koszul algebra and $(\ell_{1},\dots,\ell_{k})$ an ideal minimally
generated by linear forms, and let $L:=\sum_{i\notin E}A^{!}x_{i}^{*}$ be the
left ideal generated by the variables with indices not appearing in $E$.
Define

$B:=A^{!}/L.$

Observe that $B^{*}$ inherits the structure of a right $A^{!}$-module via the
action $(\phi\cdot a)(b):=\phi(ab)$, for $\phi\in B^{*}$, $a\in A^{!}$, and
$b\in B$. Consider the induced complex

$\cdots\to A\otimes B^{*}_{d}\to A\otimes B^{*}_{d-1}\to\cdots\to A\otimes
B^{*}_{1}\to A\to 0,$

where each differential is right multiplication by the trace element
$\sum_{i=1}^{n}x_{i}\otimes x_{i}^{*}$.

######  Proposition 3.7.

Let $A$ be a Koszul algebra and $A\otimes B^{*}_{\bullet}$ denote the complex
of Construction 3.6. Then

$(H_{i}(A\otimes B^{*}_{\bullet}))_{i+j}=0\quad\textrm{for}\ i>0\
\textrm{and}\ j=0,1.$

###### Proof.

Making an appropriate change of variable, it is of no loss of generality to
assume that each $\ell_{j}=x_{i_{j}}$ is a variable. Moreover, this is
compatible with the Priddy differential since this differential is induced by
multiplication by the image of $1$ under the isomorphism
$\operatorname{Hom}(V,V)\cong V\otimes V^{*}$, which is independent of the
choice of basis for $V$.

Notice that $H_{i}(A\otimes B_{\bullet}^{*})_{i}=0$ trivially, since there is
a natural inclusion $A\otimes B_{\bullet}^{*}\hookrightarrow
A\otimes(A^{!})^{*}_{\bullet}$ (obtained by dualizing the natural surjection
$A^{!}\to B$), and there are no degree $i$ cycles in homological degree $i$ in
the Priddy complex.

Next, let $y=\sum_{j}\lambda_{j}x_{j}\otimes f_{j}$ be a cycle of degree $i+1$
in homological degree $i$. Since the Priddy complex is acyclic,
$y=d(\lambda\otimes f)$ for some $f\in(A^{!})^{*}_{i+1}$. It remains to show
that if $f\cdot x_{j}^{*}\in B^{*}_{i}$ for all $j$, then $f\in B^{*}_{i+1}$.
By definition, there is a short exact sequence

$0\to B^{*}\to(A^{!})^{*}\to L^{*}\to 0,$

whence $f\in B^{*}$ if and only if $f(L)=0$. Let $h\in L$ be any homogeneous
element of degree $r$; if $r=1$, then by assumption $f\cdot x_{s}^{*}=0$ for
every $s\notin E$. If $r>1$, then $h=\sum_{s\notin E}h_{s}x_{s}^{*}$; since
$\deg(h_{s})\geqslant 1$, one can write $h_{s}=\sum_{j=1}^{k}h_{j,s}x_{s}^{*}$
and hence

$\sum_{j=1}^{k}x_{j}^{*}\Big{(}\sum_{s\notin E}h_{j,s}x_{s}^{*}\Big{)}.$

Since $fx_{j}^{*}=0$ and $\sum_{s\notin E}h_{j,s}x_{s}^{*}\in L$ for each
$1\leqslant j\leqslant k$, one finds that $f(h)=0$ and hence $f\in B^{*}_{i}$
as desired. ∎

######  Proposition 3.8.

Let $A$ be a Koszul algebra. Then the ideal $(\ell_{1},\dots,\ell_{k})$ has
linear strand given by the complex of Construction 3.6. If $A$ is universally
Koszul, then this complex is the minimal free resolution of
$A/(\ell_{1},\dots,\ell_{k})$.

###### Proof.

Assume without loss of generality that each $\ell_{i}$ is a variable. Let
$L:=(\ell_{1},\dots,\ell_{k})$ and $F_{\bullet}$ denote the minimal free
resolution of $L$. In view of Propositions 3.7 and 2.4, it suffices to show
that there is an isomorphism $F_{1}^{\operatorname{lin}}\to
F_{0}^{\operatorname{lin}}\cong A\otimes B_{2}^{*}\to A\otimes B_{1}^{*}$.
From the natural inclusion $A\otimes B_{\bullet}^{*}\hookrightarrow
A\otimes(A^{!})^{*}_{\bullet}$ it is immediate that the map $A\otimes
B_{1}^{*}\to A$ is a minimal presentation of $A/L$ and that multiplication by
the trace element induces a map $A\otimes B_{2}^{*}\to Z_{2}$, where
$Z:=\operatorname{Ker}(A\otimes B_{1}^{*}\to L)$.

Let $\sum_{i}a_{i}\ell_{i}=0$ be any linear relation on the generators of $L$.
This relation is evidently the image of $\sum_{i}a_{i}\otimes\ell_{i}$ under
the Priddy differential. Moreover, by Proposition 3.7,
$\sum_{i}a_{i}\otimes\ell_{i}$ must lie in the image of $A\otimes B_{2}^{*}\to
A\otimes B_{1}^{*}$, whence the result follows. ∎

With Proposition 3.8 established, we introduce some necessary notation for
monomial ideals in strongly Koszul algebras.

######  Notation 3.9.

Let $A$ be a strongly Koszul algebra and $J=(m_{1},\dots,m_{r})$ be a monomial
ideal of $A$. The following notation will be used:

$M(J):=\textrm{the set of all monomials of}\ J,$ $G(J):=\textrm{the unique
minimal generating set of}\ J,\ \textrm{consisting of monic monomials.}$

With the generators of $J$ ordered as above, the notation $J_{i}$ will denote
the ideal $(m_{1},\dots,m_{i})$, where by convention $J_{0}=(0)$. Given any
monomial $m\in A$, define

$\operatorname{Supp}(m):=\\{x_{i}\mid m\in(x_{i})\\}.$

Given any monomial ideal $J$, define

$\operatorname{Supp}(J):=\bigcup_{m\in G(J)}\operatorname{Supp}(m).$

######  Remark 3.10.

It will be understood that an ideal will be referred to as a monomial ideal
if, after a $k$-basis of $A$ has been chosen, all minimal generators can be
written as monomials with respect to this basis. It is important to note that
ideals that are monomial ideals with respect to one choice of basis may _not_
be monomial ideals with respect to a different choice.

As a very simple example of this, the ideal $(xy)\subset\frac{k[x,y,z]}{(xy-
yz-xz)}=:A$ is a monomial ideal as long as the element $xy$ is considered as
part of a $k$-basis for $A$. This is important since otherwise the notation
$G(J)$ as in Notation 3.9 may not be well defined.

######  Definition 3.11.

Let $A$ be a strongly Koszul algebra and $J=(m_{1},\dots,m_{r})$ a monomial
ideal of $A$. Then $J$ admits _linear quotients_ with respect to the given
ordering if for all $2\leqslant i\leqslant n$, the ideal $(J_{i-1}:m_{i})$ is
generated by a subset of the variables.

######  Observation 3.12.

Let $A$ be a strongly Koszul algebra. Then for any monomial $m\in A$, the
ideal $(0:m)$ is generated by a subset of variables. In particular, $(0:m)$
has linear resolution.

###### Proof.

Proceed by induction on the degree of $m$. In the case that $\deg m=1$, $m$ is
a variable and the statement holds by the assumption that $A$ is strongly
Koszul. Now, let $\deg(m)>1$ and write $m=m^{\prime}x_{i}$ for some
$x_{i}\in\operatorname{Supp}(m)$ with $\deg(m^{\prime})<\deg(m)$. By
induction, $(0:m^{\prime})$ is generated by a subset of variables and, since
$(0:m)=((0:m^{\prime}):x_{i})$, the ideal $(0:m)$ is also generated by
variables by the definition of strongly Koszul.

∎

The following proposition is the generalized version of “decomposition
functions” as in [13]. In a general Koszul algebra, if a monomial $m$ is
contained in some other monomial ideal, this does not imply that $m$ must be a
multiple of one of the minimal generators. For example, let $J=(xy,yz)\subset
k[x,y,z]/(xy+xz+yz)$. By construction, $xz\in J$, but $xz$ is not a constant
multiple of either $xy$ or $yz$. To account for this, we have:

######  Proposition 3.13.

Let $A$ be a strongly Koszul algebra and $J$ a monomial ideal in $A$. For any
$v\in M(J)$, there exists a decomposition

$v=\sum_{i}m_{i}^{*}(v)\cdot m_{i},$

where, if $m_{i}^{*}(v)\neq 0$, then $m_{i}^{*}(v)\cdot m_{i}\notin J_{i-1}$
for all $1\leqslant i\leqslant n$.

###### Proof.

The fact that such a decomposition exists is clear, since if
$m_{i}^{*}(v)\cdot m_{i}\in J_{i-1}$ then we can rewrite $m_{i}^{*}(v)m_{i}$
in terms of the generators of $J_{i-1}$ and rechoose $m_{i}^{*}(v)$ to be $0$.
∎

By convention, if $m_{j}\in G(J)$ and $x_{s}m_{j}\notin J_{j-1}$, then we will
set $m_{i}(x_{s}m_{j})=0$ for $i<j$ and $m_{j}(x_{s}m_{j})=x_{s}$.

######  Notation 3.14.

Let $J=(m_{1},\dots,m_{r})$ be an ideal admitting linear quotients with the
given ordering. For each $i\geqslant 1$, write $(J_{i-1}:m_{i})=(x_{j}\mid
j\in E_{i})$, where $E_{i}\subset[n]$. Define

$B^{i}:=A^{!}/L^{i},$

where $L^{i}:=\sum_{j\notin E_{i}}A^{!}x_{j}^{*}$.

Finally, we arrive at the analogue of iterated mapping cones for strongly
Koszul algebras:

######  Lemma 3.15.

Let $A$ be a strongly Koszul algebra and $J=(m_{1},\dots,m_{r})$ a monomial
ideal admitting linear quotients with respect to the given order, where
$\deg(m_{1})\leqslant\cdots\leqslant\deg(m_{r})$. Then the iterated mapping
cone derived from the sequence $m_{1},\dots,m_{r}$ is a minimal graded free
resolution of $A/J$ with basis elements in homological degree $\ell\geqslant
1$ of the form

$\\{m_{i}\otimes f\mid 1\leqslant i\leqslant r,\ f\ \textrm{is a basis element
of}\ \big{(}B^{i}_{\ell-1}\big{)}^{*}\\}.$

###### Proof.

The proof is by induction on the minimal number of generators of $J$, denoted
$r$. If $r=1$, then by Observation 3.12 and Proposition 3.8, the quotient ring
$A/(0:m)$ has minimal free resolution obtained by Construction 3.6. Thus, let
$F_{\bullet}$ denote the minimal free resolution of $A/(0:m)$, as in
Construction 3.6. By the tautological short exact sequence

$0\to\frac{A}{(0:m)}\xrightarrow[]{\cdot m}A\to\frac{A}{(m)}\to 0,$

one immediately finds that the minimal free resolution of $A/(m)$ is obtained
as the augmentation of $F_{\bullet}$ by the homothety $A\xrightarrow[]{\cdot
m}A$. This concludes the base case of the induction.

Let $r>1$ and $J=(m_{1},\dots,m_{r})$. To conclude the proof, apply the
inductive hypothesis to the short exact sequence

$0\to\frac{A}{(m_{1},\dots,m_{r-1}:m_{r})}\xrightarrow[]{\cdot
m_{r}}\frac{A}{(m_{1},\dots,m_{r-1})}\to\frac{A}{(m_{1},\dots,m_{r})}\to 0.$

The mapping cone associated to the above short exact sequence will be minimal
by Proposition 3.8 combined with that assumption that the degrees of the
minimal generators are increasing. ∎

######  Corollary 3.16.

Let $A$ be a strongly Koszul algebra and $J=(m_{1},\dots,m_{r})$ a monomial
ideal admitting linear quotients with respect to the given order, where
$\deg(m_{1})\leqslant\cdots\leqslant\deg(m_{r})$. Then

$\beta_{\ell}(A/J)=\sum_{m_{i}\in G(J)}\operatorname{rank}B_{\ell-1}^{i},$
$\beta_{\ell,\ell+q}(A/J)=\sum_{\begin{subarray}{c}m_{i}\in G(J),\\\
\deg(m_{i})=q\end{subarray}}\operatorname{rank}B_{\ell-1}^{i},\quad\textrm{and}$
$\operatorname{reg}(J)=\max\\{\deg(m_{i})\mid m_{i}\in G(J)\\}.$

In particular, $J$ has a linear minimal free resolution if and only if $J$ is
equigenerated.

To conclude this section, we end with an example applying some of our
techniques to a case considered in [8, Example 6.4]. It is worth mentioning
that, although we obtain the Betti number formula more easily in this case,
the construction of [8] gives the minimal free resolution explicitly.

######  Example 3.17.

Let $A:=\frac{k[x_{1},\dots,x_{n}]}{(x_{1}^{2},\dots,x_{n}^{2})}$, so that
$A^{!}=\frac{k\langle
x_{1}^{*},\dots,x_{n}^{*}\rangle}{(x_{i}^{*}x_{j}^{*}+x_{j}^{*}x_{i}^{*}\mid
i<j)}$. Consider the ideal $\mathfrak{m}^{d}$, where $d\leqslant n$; this
ideal is minimally generated by all squarefree monomials of degree $d$, and
hence can be thought of as parametrized by all subsets $\sigma\subset[n]$ of
size $d$. Ordering the generators of $\mathfrak{m}^{d}$ lexicographically, one
finds

$(\mathfrak{m}^{d})_{<\sigma}:x_{\sigma}=(x_{i}\mid i\leqslant\max(\sigma)),$

whence $\mathfrak{m}^{d}$ admits linear quotient with respect to the
lexicographic ordering. Moreover,

$\operatorname{rank}B_{i}^{\sigma}=\binom{i+\max(\sigma)-1}{\max(\sigma)-1},$

where $B_{i}^{\sigma}$ denotes the degree $i$ component of the quotient of
$A^{!}$ defined by the left ideal $\sum_{i=\max(\sigma)+1}^{n}A^{!}x_{i}^{*}$.
One then computes:

$\displaystyle\beta_{i,i+d}(\mathfrak{m}^{d})$
$\displaystyle=\sum_{j=d}^{n}\sum_{\begin{subarray}{c}\sigma\subset[j-1]\\\
|\sigma|=d-1\\\ \end{subarray}}\binom{i+j-1}{j-1}$
$\displaystyle=\sum_{j=d}^{n}\binom{j-1}{d-1}\binom{i+j-1}{j-1}$
$\displaystyle=\frac{n-d+1}{d+i}\binom{n}{d-1}\binom{n+i}{n}.$

This recovers a simpler form of the formula computed in [8].

##  4\. Minimal Free Resolutions for Ideals Admitting Regular Orderings

In this section, we consider the “correct” generalization of a regular
decomposition function for ideals admitting linear quotients in a polynomial
ring. It turns out that this condition, which we refer to as a _regular
ordering_ , is a little bit more subtle to state in the case of strongly
Koszul algebras. However, once formulated, we obtain the direct analogue of
the explicit differentials obtained in Theorem $1.12$ of [13]; see Theorem
4.9. The proof of this theorem is analogous to the original proof, with the
computations being slightly more delicate. We conclude with questions about
how far the original material of iterated mapping cones can be generalized for
strongly Koszul algebras.

To begin this section, we recall the original definitions given for (regular)
decomposition functions, as given in [13].

######  Definition 4.1.

Let $A$ be a standard graded polynomial ring and $J=(m_{1},\dots,m_{r})$ a
monomial ideal of $A$. The _decomposition function_ $g:M(J)\to G(J)$ is the
function defined by

$g(u)=m_{j},$

where $1\leqslant j\leqslant r$ is the smallest index such that $u\in J_{j}$.
For any $1\leqslant i\leqslant r$, define

$\operatorname{set}(m_{i}):=\\{k\in[n]\mid x_{k}m_{i}\in J_{i-1}\\}.$

######  Definition 4.2.

Let $g:M(J)\to G(J)$ be a decomposition function. Then $g$ is _regular_ if
$\operatorname{set}(g(x_{s}u))\subset\operatorname{set}(u)$ for all
$s\in\operatorname{set}(u)$ and $u\in G(J)$.

For the more general formulation, it turns out that the relations imposed by
the quadratic ideal defining $A$ will be needed. We adopt the following:

######  Setup 4.3.

Let $A=k[x_{1},\dots,x_{n}]/I$ be a strongly Koszul algebra. Let $S$ be a set
of pairs $(s,t)\in S$ such that $x_{s}x_{t}$ is a basis element of $A_{2}$ and
there exist coefficients $f^{u,v}_{s,t}\in k$ for all $(u,v)\notin S$ such
that

$x_{u}x_{v}=\sum_{(s,t)\in S}f^{u,v}_{s,t}x_{s}x_{t}.$

Recall that the relations on $A^{!}_{2}$ are given by Observation 3.5.

In the following definition, recall that the definition of $L^{i}$ is given in
Notation 3.14.

######  Definition 4.4.

Adopt notation and hypotheses as in Setup 4.3. A monomial ideal
$J=(m_{1},\dots,m_{r})$ admits a _regular ordering_ if:

  1. (1)

For all $j\leqslant k$,

$x_{u}m_{j}^{*}(x_{v}m_{k})+x_{v}m_{j}^{*}(x_{u}m_{k})=\sum_{(s,t)\in
S}f_{s,t}^{u,v}\big{(}x_{s}m_{j}^{*}(x_{t}m_{k})+x_{t}m_{j}^{*}(x_{s}m_{j})\big{)}$

  2. (2)

For all $j<k$ and $x_{t}^{*}\notin L^{k}$, if $m_{j}^{*}(x_{t}m_{k})\neq 0$,
then:

    1. (a)

$L^{k}\subset L^{j}$, and

    2. (b)

if $x_{t}^{*}L^{j}\not\subset L^{k}$ and $x_{t}^{*}x_{s}^{*}L^{j}\subset
L^{k}$ for some $s$, then $x_{s}^{*}\in L^{k}$.

  3. (3)

for all $i<k$ and $x_{s},\ x_{t}\in(J_{k-1}:m_{k})$, the following equality
holds:

$m_{i}^{*}(x_{s}x_{t}m_{k})=\sum_{i\leqslant
j<k}m_{i}^{*}(x_{s}m_{j})m_{j}^{*}(x_{t}m_{k}).$

Some comments on the intuition of conditions $(1)-(3)$ in the above proof are
in order. The condition $(1)$ is a compatibility condition between the
decomposition functions $m_{i}^{*}(-)$ and the relations defining the Koszul
algebra. We shall see that in the case of a polynomial ring, $(1)$ is
trivially satisfied, and hence only plays a role in more general Koszul
algebras. The condition $(2)(a)$ is a translation of the regularity condition
in Definition 4.2 for the functions $m_{i}^{*}$, and $(2)(b)$ is a condition
that we will see implies well-definedness of the generalized iterated mapping
cone construction. Finally, condition $(3)$ in the case of a polynomial ring
reduces to the statement that $g(x_{s}x_{t}m_{k})=g(x_{s}g(x_{t}m_{k}))$ for
every $1\leqslant k\leqslant r$, and so we see that $(2)(a)$ implies $(3)$
when $A$ is a polynomial ring.

As a quick sanity check, we should probably check that Definition 4.4 is
actually a generalization of Definition 4.2. Indeed:

######  Observation 4.5.

Let $A$ be a standard graded polynomial ring and $J$ a monomial ideal with
linear quotients admitting a regular decomposition function $g:M(J)\to G(J)$.
Then $J$ admits a regular ordering.

###### Proof.

Notice that condition $(1)$ of Definition 4.4 in the case of a commutative
polynomial ring reduces to:

$x_{u}m_{j}^{*}(x_{v}m_{k})+x_{v}m_{j}^{*}(x_{u}m_{k})=x_{v}m_{j}^{*}(x_{u}m_{k})+x_{u}m_{j}^{*}(x_{v}m_{k})$

for any $u>v$. Thus, the condition $(1)$ holds trivially. It is then a quick
exercise to verify that condition $(2)(a)$ is a simple translation of
Definition 4.2. For $(2)(b)$, notice that if $x_{t}^{*}L^{j}\not\subset
L^{k}$, then there exists some $x_{u}^{*}\in L^{j}$ that is not contained in
$L^{k}$. Since each $L^{j}$ and $L^{k}$ are ideals in an exterior algebra, the
element $x_{t}^{*}x_{s}^{*}x_{u}^{*}=-x_{t}^{*}x_{u}^{*}x_{s}^{*}$ is
contained in $L^{k}$ if and only if $x_{s}^{*}\in L^{k}$.

Finally, as previously mentioned, condition $(3)$ is a retranslation of the
fact that $g(x_{s}x_{t}u)=g(x_{s}g(x_{t}u))$ for regular decomposition
functions. ∎

Observe that if $A$ is a multigraded Koszul algebra (that is, $A$ is the
quotient by a quadratic monomial ideal), then $A$ is strongly Koszul and the
decomposition $g$ of Definition 4.1 is still well-defined. However, in this
case a regular decomposition function is _not_ enough to guarantee that the
ideal admits a regular ordering.

######  Proposition 4.6.

Let $A=k[x_{1},\dots,x_{n}]/I$ be a multigraded Koszul algebra and
$J=(m_{1},\dots,m_{r})$ be a monomial ideal with linear quotients admitting a
regular decomposition function. Then $J$ admits a regular ordering if:

  1. (*)

for all $1\leqslant i\leqslant r$ and all $y\in\operatorname{Supp}(I)$, either
$ym_{i}\notin J_{i-1}$ or $ym_{i}=0$.

###### Proof.

The proof is broken up in line with each condition in Definition 4.4.
Proof of (1): Condition $(1)$ of Definition 4.4 in the case that $A$ is a
multigraded Koszul algebra is implied by the following:

$x_{u}m_{j}^{*}(x_{v}m_{k})=0\ \textrm{for all}\ x_{u}x_{v}\in I\
\textrm{and}\ 1\leqslant j\leqslant k\leqslant r.$

Assume that condition $(*)$ holds. If $x_{v}m_{k}=0$, then evidently
$x_{u}m_{j}^{*}(x_{v}m_{k})=0$, so assume that $x_{v}m_{k}\notin J_{k-1}$. By
definition, one has $m_{j}^{*}(x_{v}m_{k})=\delta_{jk}x_{v}$, whence
$x_{u}m_{j}^{*}(x_{v}m_{k})=\delta_{jk}x_{u}x_{v}=0$.
Proof of (2)(a)-(b): Observe first that if $y\notin\operatorname{Supp}(I)$,
then $y^{*}$ anticommutes with every other variable in $A^{!}$. Thus the proof
of $(2)(a)$ and $(2)(b)$ is identical to the polynomial ring case when
$y\notin\operatorname{Supp}(I)$. If $y\in\operatorname{Supp}(I)$, then
condition $(*)$ implies that $m_{j}^{*}(ym_{k})=0$ for all $j<k$, so there is
nothing to check.
Proof of (3): Let $x_{s},x_{t}\in(J_{k-1}:m_{k})$ for some $1\leqslant
k\leqslant r$. There are three cases to check:
Case 1: Neither $x_{s}$ nor $x_{t}\in\operatorname{Supp}(I)$. In this case,
notice that $x_{s}x_{t}m_{k}\neq 0$ since otherwise $m_{k}=0$. The proof then
reduces to the same proof given in [13, Lemma 1.11].
Case 2: $x_{t}\in\operatorname{Supp}(I)$. Since $x_{t}\in(J_{k-1}:m_{k})$ by
assumption, condition $(*)$ forces $x_{t}m_{k}=0$. Thus both sides of the
equality in $(3)$ are $0$.
Case 3: $x_{s}\in\operatorname{Supp}(I)$ and
$x_{t}\notin\operatorname{Supp}(I)$. It suffices to show that
$x_{s}g(x_{t}m_{k})=0$. However, if $x_{s}g(x_{t}m_{k})\neq 0$, then $(*)$
implies that $x_{s}\notin(J_{j-1}:m_{j})$, where $m_{j}:=g(x_{t}m_{k})$ and
$j<k$. Definition 4.2 then implies that $x_{s}\notin(J_{k-1}:m_{k})$, a clear
contradiction. ∎

######  Example 4.7.

Let $J:=(x_{1}x_{2},x_{2}x_{3})$ in the Koszul algebra
$k[x_{1},x_{2},x_{3}]/(x_{1}x_{3},x_{3}^{2})$. Then with the given ordering,
$J$ has linear quotients and admits a regular ordering.

The following lemma will be essential for showing that the differentials
appearing in Theorem 4.9 are well-defined.

######  Lemma 4.8.

Adopt notation and hypotheses as in Setup 4.3 and assume that the monomial
ideal $J=(m_{1},\dots,m_{r})$ admits a regular ordering. Let
$f\in(B_{\ell}^{k})^{*}$ and assume that for any $j$ with
$m_{j}^{*}(x_{t}m_{k})\neq 0$, one has $x_{t}^{*}L^{j}\not\subset L^{k}$. Then
the following statements hold:

  1. (1)

If $x_{t}^{*}x_{s}^{*}L^{j}\not\subset L^{k}$ for some $s$, then
$x_{t}^{*}x_{s}^{*}L^{i}\not\subset L^{k}$ for every $i<j$ such that
$m_{i}^{*}(x_{s}m_{j})\neq 0$.

  2. (2)

If $x_{t}^{*}x_{s}^{*}L^{j}\subset L^{k}$ for some $s$, then $x_{s}^{*}\in
L^{j}$. In particular,

$m_{i}^{*}(x_{s}m_{j})=\begin{cases}x_{s}&\textrm{if}\ i=j,\\\
0&\textrm{otherwise}.\end{cases}$

###### Proof.

Proof of (1): Suppose for sake of contradiction that there exists some $i<j$
with $m_{i}^{*}(x_{s}m_{j})\neq 0$, but $x_{t}^{*}x_{s}^{*}L^{i}\subset
L^{k}$. Then, by $(2)(a)$ of Definition 4.4, one has that $L^{j}\subset
L^{i}$, implying

$x_{t}^{*}x_{s}^{*}L^{j}\subset x_{t}^{*}x_{s}^{*}L^{i}\subset L^{k},$

which is a contradiction.
Proof of (2): A priori, condition $(2)(b)$ of Definition 4.4 implies that
$x_{s}^{*}\in L^{k}$. But $(2)(a)$ of Definition 4.4 implies that
$L^{k}\subset L^{j}$, so $x_{s}^{*}\in L^{j}$. This is equivalent to saying
$x_{s}m_{j}\notin J_{j-1}$, whence the latter statement follows immediately. ∎

We now arrive at the main result of this section. As previously mentioned,
this gives the generalized version of the explicit minimal free resolution of
ideals with regular decomposition functions for strongly Koszul algebras.

######  Theorem 4.9.

Adopt notation and hypotheses as in Setup 4.3. Let $J=(m_{1},\dots,m_{r})$ be
a monomial ideal with linear quotients admitting a regular ordering, where
$\deg(m_{1})\leqslant\cdots\leqslant\deg(m_{r})$. Then the minimal free
resolution $F_{\bullet}$ of $A/J$ has differentials of the form

$\partial(m_{k}\otimes f)=-\sum_{s}x_{s}(m_{k}\otimes fx_{s}^{*})+\sum_{s;\
j\leqslant k}m^{*}_{j}(x_{s}m_{k})(m_{j}\otimes fx_{s}^{*}),\quad\textrm{for}\
\deg(f)>0,$ $\partial(m_{k}\otimes 1)=m_{k},$

where each $m_{i}\otimes f$ is as in the statement of Lemma 3.15.

###### Proof.

Throughout the proof, right multiplication will be used to denote the opposite
action of the left action. More precisely: $x_{t}^{*}f:=fx_{t}^{*}$ (so that
$x_{s}^{*}x_{t}^{*}f=fx_{t}^{*}x_{s}^{*}$). The proof follows by induction on
$r$. When $r=1$, the ideal $J=(m)$ is principal and the result follows from
the base case of the proof of Lemma 3.15. Assume now that $r>1$ and consider
the short exact sequence:

$0\to\frac{A}{(J_{r-1}:m_{r})}\xrightarrow[]{\cdot
m_{r}}\frac{R}{J_{r-1}}\to\frac{R}{J}\to 0.$

By the inductive hypothesis, the differentials of the minimal free resolution
$F_{\bullet}$ of $R/J_{r-1}$ are as in the statement of the Theorem; let
$K_{\bullet}$ denote the minimal free resolution of $R/(J_{r-1}:m_{r})$
provided by Construction 3.6. The proof follows by showing that the following
map is a morphism of complexes:

$\displaystyle\psi_{i}:K_{i}$ $\displaystyle\to F_{i}$ $\displaystyle
m_{k}\otimes f$ $\displaystyle\mapsto\sum_{t;\
j<k}m_{j}^{*}(x_{t}m_{k})(m_{j}\otimes x_{t}^{*}f).$

That is, the following diagram commutes:

(4.1)
$\textstyle{K_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{i}}$$\textstyle{F_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{i-1}}$$\textstyle{F_{i-1}.}$

For $i=1$ this is evident, so assume that $i\geqslant 2$. Moving clockwise
around diagram (4.1), one obtains:

$\displaystyle m_{k}\otimes f$ $\displaystyle\mapsto\sum_{t;\
j<k}m_{j}^{*}(x_{t}m_{k})(m_{j}\otimes x_{t}^{*}f)$
$\displaystyle\mapsto-\sum_{s}\sum_{t;\
j<k}x_{s}m_{j}^{*}(x_{t}m_{k})(m_{j}\otimes x_{s}^{*}x_{t}^{*}f)$
$\displaystyle+\sum_{t,\ j<k}\sum_{s;\ i\leqslant
j}m_{i}^{*}(x_{s}m_{j})m_{j}^{*}(x_{t}m_{k})(m_{i}\otimes
x_{s}^{*}x_{t}^{*}f).$

Observe that

$\displaystyle\sum_{s}\sum_{t;\ j<k}x_{s}m_{j}^{*}(x_{t}m_{k})(m_{j}\otimes
x_{s}^{*}x_{t}^{*}f)$ $\displaystyle=\sum_{(s,t)\in
S}\sum_{j<k}x_{s}m_{j}^{*}(x_{t}m_{k})(m_{j}\otimes x_{s}^{*}x_{t}^{*}f)$
$\displaystyle+\sum_{(u,v)\notin
S}\sum_{j<k}x_{u}m_{j}^{*}(x_{v}m_{k})(m_{j}\otimes x_{u}^{*}x_{v}^{*}f)$
$\displaystyle=\sum_{(u,v)\notin S}\sum_{j<k}$
$\displaystyle\Big{(}x_{u}m_{j}^{*}(x_{v}m_{k})-\sum_{(s,t)\in
S}f^{u,v}_{s,t}x_{s}m_{j}^{*}(x_{t}m_{k})\Big{)}(m_{j}\otimes
x_{u}^{*}x_{v}^{*}f)$ $\displaystyle=-\sum_{(u,v)\notin S}\sum_{j<k}$
$\displaystyle\Big{(}x_{v}m_{j}^{*}(x_{u}m_{k})-\sum_{(s,t)\in
S}f^{u,v}_{s,t}x_{t}m_{j}^{*}(x_{s}m_{k})\Big{)}(m_{j}\otimes
x_{u}^{*}x_{v}^{*}f),$

where the final equality follows by condition $(1)$ of Definition 4.4. Next,
recall the notation $L^{i}$ for $1\leqslant i\leqslant r$ from Notation 3.14.
It still remains to show that this differential is well defined; that is, if
$m_{j}^{*}(x_{t}m_{k})\neq 0$ and $m_{j}\otimes x_{t}^{*}f$ is not a valid
basis element (ie, $fx_{t}^{*}(L^{j})\neq 0$), then the image of this basis
element under the differential should remain $0$. By definition, notice that
$fx_{t}^{*}(L^{j})\neq 0\iff x_{t}^{*}L^{j}\not\subset L^{k}$. There are two
cases to consider:

Case 1: $m_{j}\otimes x_{s}^{*}x_{t}^{*}f$ is _not_ a valid basis element.
This is equivalent to saying $x_{t}^{*}x_{s}^{*}L^{j}\not\subset L^{k}$. By
$(1)$ of Lemma 4.8, it follows that $m_{i}\otimes fx_{t}^{*}x_{s}^{*}$ is not
a valid basis element for any $i$ with $m_{i}^{*}(x_{s}m_{j})\neq 0$.

Case 2: $m_{j}\otimes x_{s}^{*}x_{t}^{*}f$ is a valid basis element. This is
equivalent to saying $x_{t}^{*}x_{s}^{*}L^{j}\subset L^{k}$, and by $(2)$ of
Lemma 4.8, the following equality holds:

$m_{i}^{*}(x_{s}m_{j})=\begin{cases}x_{s}&\textrm{if}\ i=j,\\\
0&\textrm{otherwise}.\end{cases}$

One then computes:

$m_{j}\otimes x_{t}^{*}f\mapsto-\sum_{s}x_{s}(m_{j}\otimes
x_{s}^{*}x_{t}^{*}f)+\sum_{i\leqslant j}m_{i}^{*}(x_{s}m_{j})(m_{i}\otimes
x_{s}^{*}x_{t}^{*}f).$

Combining both of the above cases, the expression on the right must also be
$0$. Finally, it remains to consider the following:

$\displaystyle\sum_{t;\ j<k}\sum_{s;\ i\leqslant
j}m_{i}^{*}(x_{s}m_{j})m_{j}^{*}(x_{t}m_{k})(m_{i}\otimes
x_{s}^{*}x_{t}^{*}f)$ $\displaystyle=\sum_{(s,t)\in S}\sum_{i\leqslant
j<k}m_{i}^{*}(x_{s}m_{j})m_{j}^{*}(x_{t}m_{k})(m_{i}\otimes
x_{s}^{*}x_{t}^{*}f)$ $\displaystyle+\sum_{(u,v)\notin S}\sum_{i\leqslant
j<k}m_{i}^{*}(x_{u}m_{j})m_{j}^{*}(x_{v}m_{k})(m_{i}\otimes
x_{u}^{*}x_{v}^{*}f).$

Focusing on the bottom term, one computes:

$\displaystyle\sum_{(u,v)\notin S}\sum_{i\leqslant
j<k}m_{i}^{*}(x_{u}m_{j})m_{j}^{*}(x_{v}m_{k})(m_{i}\otimes
x_{u}^{*}x_{v}^{*}f)$ $\displaystyle=$ $\displaystyle\sum_{(u,v)\notin
S}\sum_{i<k}m_{i}^{*}(x_{u}x_{v}m_{k})(m_{i}\otimes
x_{u}^{*}x_{v}^{*}f)\quad(\textrm{by (3) in Definition
\ref{def:regulrityDef}})$ $\displaystyle=$ $\displaystyle\sum_{(u,v)\notin
S}\sum_{i<k}\sum_{(s,t)\in
S}f_{s,t}^{u,v}m_{i}^{*}(x_{s}x_{t}m_{k})(m_{i}\otimes x_{u}^{*}x_{v}^{*}f)$
$\displaystyle=$ $\displaystyle\sum_{(s,t)\in
S)}\sum_{i<k}m_{i}^{*}(x_{s}x_{t}m_{k})(m_{i}\otimes\sum_{(u,v)\notin
S}f^{u,v}_{s,t}x_{u}^{*}x_{v}^{*}f)$ $\displaystyle=$
$\displaystyle-\sum_{(s,t)\in
S)}\sum_{i<k}m_{i}^{*}(x_{s}x_{t}m_{k})(m_{i}\otimes x_{s}^{*}x_{t}^{*}f).$

Thus, the term $\sum_{t,\ j<k}\sum_{s,\ i\leqslant
j}m_{i}^{*}(x_{s}m_{j})m_{j}^{*}(x_{t}m_{k})(m_{i}\otimes
x_{s}^{*}x_{t}^{*}f)$ is identically $0$.

Moving counterclockwise around diagram (4.1), one has:

$\displaystyle m_{k}\otimes f$ $\displaystyle\mapsto\sum_{t}x_{t}(m_{k}\otimes
x_{t}^{*}f)$ $\displaystyle\mapsto\sum_{t}\sum_{s,\
j<k}x_{t}m_{j}^{*}(x_{s}m_{k})(m_{j}\otimes x_{s}^{*}x_{t}^{*}f)$
$\displaystyle=\sum_{(s,t)\in
S}\sum_{j<k}x_{t}m_{j}^{*}(x_{s}m_{k})(m_{j}\otimes x_{s}^{*}x_{t}^{*}f)$
$\displaystyle+\sum_{(u,v)\notin
S}\sum_{j<k}x_{v}m_{j}^{*}(x_{u}m_{k})(m_{j}\otimes x_{u}^{*}x_{v}^{*}f)$
$\displaystyle=\sum_{(u,v)\notin S}\sum_{j<k}$
$\displaystyle\Big{(}x_{v}m_{j}^{*}(x_{u}m_{k})-\sum_{(s,t)\in
S}f^{u,v}_{s,t}x_{t}m_{j}^{*}(x_{s}m_{k})\Big{)}(m_{j}\otimes
x_{u}^{*}x_{v}^{*}f).$

Comparing this with the expression obtained after moving clockwise, one finds
that these terms are equal, whence the result. ∎

Finally, we conclude this section with some discussion about further results
for ideals admitting linear quotients in strongly Koszul algebras. Firstly, it
is worth noting that the original definition of decomposition function given
in [13] is not the only choice that one can make. For instance, for
_cointerval ideals_ (that is, edge ideals corresponding to cointerval graphs),
one can formulate a different definition of a decomposition function to
construct the morphism of complexes appearing in the iterated mapping cone
construction; see Section $4$ of work by Dochtermann and Mohammadi [6]. The
choice of a different decomposition function is, in our notation, a different
choice of the unique decomposition of Proposition 3.13. This then begs the
question:

######  Question 4.10.

What other choices of decompositions as in Proposition 3.13 can be used to
construct a morphism of complexes as in the proof of Theorem 4.9?

Continuing with the work of [6], we can also ask whether it makes sense to try
to put a cell complex structure on the resolution of Theorem 4.9. This
question seems to be less well-posed, since we have the obvious initial
problem of needing an infinite-dimensional cell complex for general strongly
Koszul algebras. Moreover, the differentials of the Priddy complex are more
difficult to deal with than the classical Koszul complex, so it seems unlikely
that the resolution of Theorem 4.9 will be cellular in any sort of generality.

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