# Now You See Me: Robust approach to Partial Occlusions

Karthick Gunasekaran
University of Massachusetts
Amherst, USA
<EMAIL_ADDRESS>Nikita Jaiman
University of Massachusetts
Amherst, USA
<EMAIL_ADDRESS>
###### Abstract

Occlusions of objects is one of the indispensable problems in Computer vision.
While Convolutional Neural Networks (CNNs) provide various state of the art
approaches for regular image classification, they however, prove to be not as
effective for the classification of images with partial occlusions. Partial
occlusion is scenario where an object is occluded partially by some other
object/space. This problem when solved, holds tremendous potential to
facilitate various scenarios. We in particular are interested in autonomous
driving scenario and its implications in the same. Autonomous vehicle research
is one of the hot topics of this decade, there are ample situations of partial
occlusions of a driving sign or a person or other objects at different angles.
Considering its prime importance in situations which can be further extended
to video analytics of traffic data to handle crimes, anticipate income levels
of various groups etc., this holds the potential to be exploited in many ways.
In this paper, we introduce our own synthetically created dataset by utilising
Stanford Car Dataset and adding occlusions of various sizes and nature to it.
On this created dataset, we conducted a comprehensive analysis using various
state of the art CNN models such as VGG-19, ResNet 50/101, GoogleNet, DenseNet
121. We further in depth study the effect of varying occlusion proportions and
nature on the performance of these models by fine tuning and training these
from scratch on dataset and how is it likely to perform when trained in
different scenarios, i.e., performance when training with occluded images and
unoccluded images, which model is more robust to partial occlusions and so on.

##  I Introduction

Classification has proved to be a problem of prime significance in varied
domains be it healthcare, autonomous driving or any other area and undeniably
convolutional neural networks (CNNs) based architectures are the present state
of the art approach in this. When CNNs are reaching leaps and bounds in normal
image classification [6], these are relatively inefficient when classifying
the objects with occlusions[occ]. Occ [Occlusions] have become quite
challenging for the field of computer vision and hold a tremendous potential
to uncover various benefits such as face detection, vehicle detection etc.

Being motivated by it’s scope, in the project, we are restricting to the
problem of occ in the vehicular dataset.The dataset we consider is from
Stanford called the ”Cars Dataset”[3] having approx 16k varied car images with
196 categories of cars . The bigger picture that intrigued us is thinking
about how accurately an autonomous self driving car of the future will be able
to detect a human or a traffic sign or another vehicle with partial occs if
they are using one of the state of the art of model. When an object is on the
road it is very highly likely to be occluded by several objects we therefore,
taking into consideration the impact it can have, try to focus on
understanding and solving the problem of Partial Occs in vehicular datasets.
We investigate how to build models resulting in better classification
accuracies on the occluded dataset. For our purpose, an entire synthetic
dataset is generated from the existing dataset with varied proportions and
types of occ and conduct a detailed analysis by training different state of
the art CNN Models which are either pre trained (on ImageNet [1]) or training
the models from scratch for the classification of partially occ cars.

The dataset is quite diverse and covers different car views from different
angles. Classes are typically at the level of Make, Model, Year, e.g. 2012
Tesla Model S or 2012 BMW M3 coupe. The project is mainly a two step process:
First, like discussed above, creating an artificially occluded dataset with
variants. Second, our goal is to evaluate various state of the art CNN
architectures to observe the performance on occluded data as compared to the
unoccluded and explore the ways to improve the classification accuracies by
conducting various in detail experiments.

Through this paper we aim to investigate and answer several questions like, i)
how occ sizes affects the performance of model architectures; ii) does deeper
networks tend to perform better in the classification of partially occluded
datasets; iii) robustness of model in the classification when trained on
dataset with different occ proportion and tested of different. iv) would
nature of artifact such as constant pixel artifact affect model performance as
compared to random pixel artifact on image of same occ size and many more by
conducting experiments of varied kinds extensively.

##  II Background/Related Work

A good amount of work has been done on fine grained categorization of
different objects such as birds [2], plants etc. already. Fine grained
categorization learns the sub details in one class as opposed to regular
classification. Liu et al. [7] explains the fine grained classification
techniques on the Stanford Car Dataset [3] identifying the make and models of
cars from different angles. Owing to the fact that there is lack of huge
dataset specific to cars, Yang et al. [12] collects a large scale dataset
called ”CompCars” to perform fine grained categorization and verification of
these cars on the dataset of both survellience nature and web nature
containing approximately 200k images of 1.7 k car models suggesting
interesting applications based on CNN models which can be made available on
request.

With these findings in the varied car dataset domain, there has been some work
done in the domain of partial occ’s as well. One challenging problem in the
domain of vision is the problem of occs. Opitz et al. [8] demonstrate in the
domain of face detection their method outperforms the standard CNN models by
introducing a new loss layers for CNN. How CNNs prove to be less effective in
classification of occluded objects when they provide state of the art results
for classifying an image without it, is talked about in Pepik et al. [10]
where they propose that one key step to improve classification is by making
the architectural changes enhancing the robustness to occs.

Osherov et al. [9] discuss the relative insufficiency of convolutional neural
networks in classification of partially occluded objects as compared to
regular classification suggesting an alternate approach to enhance the
robustness of CNNs to occs. They establish the finding that smaller filter
size prove to be beneficial for the model’s robustness to occs and propose an
algorithm with new special regularisation to design a classifier that results
in robustness towards occs.

##  III Approach

###  III-A Custom Dataset Generation:

One of the key challenges of the project was to create our own partially
occluded dataset, since no such kind is available. Therefore, on the existing
Stanford Car Dataset we artificially augmented partial occs of different
proportions as demonstrated in figure 1 to create the synthetic dataset. We
also demonstrate the behaviour and performances of different model
architectures with not just the occ size but also demonstrate what kind of
difference will it make to have varied artifacts such as random pixel (type
1),constant pixel (type 2) or real images (type 3).

The dataset contained lot of variations in images such as images with
different angles,zoom levels, varied backgrounds etc. Data processing was
carried to standardize the images such that occ size will remain uniform
across images. In data processing step all the bounding boxes of images were
first extracted. The images which had bounding boxes sizes less than 224*224
were discarded so that upscaling doesn’t affect the quality of the image.
Extracting the bounding boxes helps in standardizing the zoom level of objects
in the images so that the occ ratio remains approximately the same across all
the objects in image. Images were then down sampled to the standard size of
224 X 224. Post that artifacts were added to the images according to the size
of occ. Images were then normalized and random horizontal flip was used as a
data augmentation technique before being used by the model.

Data balancing was also the primary requirement,as the dataset has
inconsistent number of images in different classes. It was done to obtain even
number of images across all the classes within threshold range. First, we
divide the training and test data in a 70-30 % split restricting the number of
images in each class within a particular range from the average (of total
images). This reduces the likelihood of skew towards one particular class with
the highest number of images. Next, we divide training data into 80-20 %
train-validation split ensuring each class has nearly equal representation in
the validation dataset.

(a) 13%

(b) 23%

(c) 33%

Figure 1: Different Occlusion Proportion

###  III-B Network Architectures:

A variety of different architectures where studied and six were selected based
on the how different they are from one another to perform our experiments on.
However three variants of ResNet architecture were choosen to perform
experiments on the depth of the network. The network architectures chosen
include VGG-19, GoogleNet [11], DenseNet-121 [5] and variants of ResNet [4];
ResNet-50/101/151 for comparison of the model performance across all the
combinations of occ proportions and artifact types. In total we trained eight
state of the art models. The implementation was done in PyTorch. We consider
two types of optimizers; Adam and Stochastic Gradient Descent (SGD) with
momentum of 0.9. We observed that Adam is currently recommended as the default
algorithm to use which certainly gives better performance for classification,
therefore it was the first choice. However, we noticed that for our partially
occluded dataset VGG-19 produced relatively lower accuracy with Adam hence,
SGD + momentum was used as the optimizer with it.

We begin with two different techniques of weight initialisation i) using pre-
trained weights on the ImageNet Dataset and ii) randomly initialising the
weights to train it from scratch although we understand the repercussion of
this approach owing to the small dataset size. Out of eight models, six of
these we train using are pre-trained weights and other two from scratch.

When CNNs perform exceptionally on image classification, with respect to
partial occ we intend to address following questions and findings such as

  * •

How performance of models is affected when trained on occluded dataset as
compared to training on un-occluded dataset when tested on partially occluded
dataset?

  * •

Whether deeper networks are always able to learn finer image representations
in the presence of occs as compared to simpler networks

  * •

Does the model performance change with varying occ size in training and
testing or it remains invariant?

  * •

Will training the model from scratch, despite of having smaller dataset with
data augmentation techniques perform better than pre-trained?

First all the models where trained with different hyper parameters for
different no of epochs.Then we experimented with increasing number of epochs
realising that models from scratch would need higher epoch number to learn
finer image representations. However, for pre-tained the fact that they are
trained on large datasets there was no such necessity.

##  IV Experiment

###  IV-A Dataset

####  IV-A1 Cars Dataset

Stanford Cars dataset [3] was used for this task. This dataset was choosen
since one of the needs of any autonomous vehicles will be to identify objects
which are partially occluded and cars will be one of main objects of interests
for autonomous driving. As mentioned previously creating datasets was one of
the main tasks of the project as there weren’t any partially occluded datasets
available before hand.

####  IV-A2 Statistics

For training and validation phase, we used 1̃0.3K images (details provided in
table I) with 177 different classes of cars. Out of 10.3K images, approx 8k
images where used for training and 2k images were used for validation. The
images were evenly distributed across multiple classes with a mean of 59
images per classes and variance of 6 images across classes. Also, we have
divided the dataset in 70%-30% train-test split, respectively. For testing
around 4k images where used.

TABLE I: Dataset Statistics Summary | Count
---|---
Images In’Car Dataset’ | 16k
Total classes | 196
Total Occluded Images | 14.8k
Classes with even distribution of images | 177
Training images (7̃0%) | 10.5k
Train (8̃0%) -Val (2̃0%) Split | 8.3k - 2.1k
Test Images (3̃0%) | 4.3k
Occlusion proportions used | (13%,23%,33%)

####  IV-A3 Artifact types

Three different types of artifacts were generated to create varied datasets
such that experiments can be performed to analyse the behaviour of models with
respect to each artifact since in the real world, there are varying patterns
of artifacts. The different artifact types are shown in figures 3, 4 and 5.

Type 1 Artifact:

This artifact was created by assigning random values between 0-255 for each
pixel in all the images. Figure 3(a) shows one such example of the random
pixel artifact.

Type 2 Artifact:

This artifact was result of using a constant value for all the pixels across
images. Figure 3(b) shows the constant pixel artifact.

Type 3 Artifact:

This was created by adding real life images on top of the images in the
dataset. Figure 3(c) shows the example of one such artifact.

####  IV-A4 Occlusion Sizes

Datasets with varying Occ proportions were created to understand the effect of
Occ sizes on the various model performances. The Occ sizes tested were 0% (Un
occluded), 13%, 23% and 33%as shown below.

Un occluded dataset:

The unoccluded dataset was created to make a comparison with occluded dataset
using various models as baseline is shown in Figure 2.

(a)

(b)

(c)

Figure 2: Unoccluded dataset

13% Occluded dataset:

An artifact was added such that 13% of the pixels in the original data was
occluded. For this occ ratio three different types of artifacts where created
as shown in 3.

(a) Type 1

(b) Type 2

(c) Type 3

Figure 3: Different artifacts in dataset with 13% occlusion on Images

23% Occluded dataset:

Similar to previous one, artifact was created such that 23% of the pixels in
the original data was occluded. Three different types of artifacts were
created for this occ ratio as shown in 4.

(a) Type 1

(b) Type 2

(c) Type 3

Figure 4: Different artifacts in datasets with 23% occlusion

33% Occluded dataset:

Artifact were created in such a fashion that 33% of pixels in the original
image were occluded. Similar to previous data set,three different types of
artifacts were created as shown in Figure 5.

(a) Type 1

(b) Type 2

(c) Type 3

Figure 5: Different artifacts in datasets with 33% occlusion

###  IV-B Evaluation Metric

For the experiments conducted,top-1% and top-5% accuracies were considered as
an evalutation metric for the model’s performance. The top-1% accuracy of the
model is obtained by only considering the result with highest probability and
comparing it with the ground truth label. The top-5% accuracy is taken by
considering 5 highest probabilities in the predicted result and checking if
one of them is the ground truth label.

###  IV-C Results

####  IV-C1 Effect of model trained with lower occluded data on higher
occluded data:

An interesting experiment was conducted to understand the behaviour of models
when they come across higher occluded data but the training had only smaller
sized occ. A model which was trained on 13% Occ dataset was tested with 23%
and also 33% Occ dataset. The baseline results for this case would be the
model to be tested on same Occ size as trained i.e. baseline models trained on
13% and 23% will be tested on 13% and 23% respectively to be compared. The
results can be seen in table II.

TABLE II: Effect of training on Lower Occ Data and Testing on higher Occ Data Trained Occ | Tested Occ | Model
---|---|---
Top 1% | Top 5%
13% | 23% | 80.14 | 94.47
13% | 33% | 70.62 | 91.03
23% | 23% | 81.87 | 95.81
33% | 33% | 72.85 | 92.42
0%(Un-occ) | 23% | 62.87 | 86.50
0%(Un-occ) | 33% | 45.68 | 74.17

From Table II above, we can see that the models trained on lower Occ data gave
similar performance when tested on the higher Occ data as compared to the
models trained on higher Occ data when tested with same Occ size. Also, we
note that the models trained on unoccluded dataset does not perform well. The
Figure 6 clearly shows that across all the models same behaviour can be seen.

Figure 6: Analysis on Effect of model trained with lower occluded data on
higher occluded data

Therefore, we understand that the models are unable to generalize to artifacts
if they have not seen these before during the training time, however, are able
to adapt and generalize well for larger artifacts if they have seen smaller
artifacts previously.

####  IV-C2 Effect of model trained with higher occluded data on unoccluded or
lower occluded data :

In this experiment the performance of models on lower Occ or un-occluded data
when they were trained on higher Occ data was analysed. In the real world
scenario, there will be cases where the training set has an object and testing
set doesn’t contain any occ. This test was carried out to understand if the
model which learnt from lesser number of features because of occ would be able
to perform similar to the model which was trained with all the features
available. The baseline in this case is the model which was trained on un-
occluded dataset and tested on un-occluded dataset since they have all the
features available to them during training and were able to classify pretty
much accurately.

TABLE III: Effect of training on Higher Occ Datasets and Testing on Lower Occ Datasets Trained Occ | Tested Occ | Model
---|---|---
Top 1% | Top 5%
13% | 0% | 89.59 | 98.29
23% | 0% | 88.77 | 97.95
0% | 0% | 89.43 | 98.38
13% | 13% | 85.63 | 97.33
23% | 23% | 81.87 | 95.81

The Table III shows the top-1 and top-5 accuracy of model when tested on lower
occ datasets.

Figure 7: Analysis on Effect of model trained with higher occluded data on
unoccluded or lower occluded data

Figure 7 shows that the model performance is almost similar to the baseline.
Therefore, it can be inferred that having occluded objects in the training set
will be an added advantage since it will be able to classify both occluded and
un-occluded objects in test and better helps in overall generalization of the
model.

####  IV-C3 Effect of different models on different Occlusion sizes

Understanding the model performance with respect to occ sizes was one of the
important aspect of this work. Therefore, models were tested for datasets with
different occ percentages of Type 1 Artifact. All the models listed were
pretrained on Imagenet and then again trained on Stanford Cars un-occluded
dataset [3]. Table IV shows the top-1% and top-5% accuracies of the models
when tested with different sizes of artifacts.

TABLE IV: Model performance across different Occlusions Sizes Model | Unacc | 13% | 23% | 33%
---|---|---|---|---
Top 1% | Top 5% | Top 1% | Top 5% | Top 1% | Top 5% | Top 1% | Top 5%
Vgg-19 | 84.01 | 96.29 | 60.73 | 83.60 | 43.54 | 69.89 | 28.14 | 53.79
ResNet-50 | 89.02 | 98.15 | 75.50 | 93.62 | 62.01 | 86.59 | 44.14 | 73.13
ResNet-101 | 89.43 | 98.38 | 77.75 | 94.10 | 62.88 | 86.49 | 45.68 | 74.17
GoogleLeNet | 73.88 | 90.89 | 49.42 | 74.31 | 32.86 | 58.96 | 20.72 | 42.06
DenseNet-121 | 87.57 | 97.56 | 74.50 | 92.46 | 58.80 | 84.22 | 40.86 | 67.59

From plot 8 it can be seen that the accuracy of all the models decreases in
almost a linear fashion as the occ sizes increase. Resnet-101,ResNet-50 and
DenseNet tend to perform better overall for all types of data. The models are
unable to classify accurately as the occ sizes in the datasets increase.

Figure 8: Accuracy[Top-1%] of Models Vs Occlusion sizes

Further probing with top-5% accuracy as shown in the Figure 9 it can be seen
that Residual Network architecture in general is better compared with
DenseNet, since a sudden decrease of top-5% accuracy is seen in DenseNet which
is not observed in ResNet. No specific effect was observed in models with
respect to Occ, the performance of all the models degrade as the Occ size
increases.

Figure 9:  Accuracy[Top-5%] of Models Vs Occlusion sizes

####  IV-C4 Effect of Training on occluded dataset and Training on un-occluded
dataset and testing Occluded data

A comparison was made to understand the effect of training the model with
occluded training images. In this, a pretrained model was trained on occluded
dataset and separately on un-occluded datasets. Now both the models were
tested on the occluded dataset to determine how much better the model trained
on occluded will perform. Results can be seen in the table V. From the bar
plot 10 it can be noticed that as the occ size of the objects increases the
performance of the model with Occ training performed much better than the one
trained on un-occluded data. The difference between the performance of the
model trained on occluded dataset and on un-occluded dataset was initially
very low at 13% occ size. In 23% occ size, the difference between them
increased further however at 33% it increased to a very high value of 27%
difference. The trend can be seen that model tend to perform better with
training on occluded data. Similar kind of relationship can be seen with other
models as well.

TABLE V: Model performance across different Occlusions Sizes Occ Size | Unocc | Occ
---|---|---
Top 1% | Top 5% | Top 1% | Top 5%
0% | 89.43 | 98.38 | - | -
13% | 77.75 | 94.10 | 85.63 | 97.34
23% | 62.88 | 86.49 | 81.87 | 95.81
33% | 45.68 | 74.17 | 72.85 | 92.42
Figure 10: Effect of Training on Occlusion Dataset and Training on un-occluded
dataset for different Occlusions

####  IV-C5 Effect of Network depth on Occlusion

This experiment was aimed at understanding the hypothesis how the network
depth impacts the classification with partial occs. We consider different
number of layers (50,101,151) of the same network architecture; Residual
Network (ResNet) and investigate if increasing the number of layers in the
network actually helps in understanding the finer image representations better
with respect to occ than the one with lesser number of layers. For the car
dataset with 13% occ(details mentioned in table VI), we create a bar plot of
for artifacts type 1,2 and 3 and plot the corresponding accuracies obtained by
training the ResNet-50, ResNet-101 and ResNet-151 models as shown in figure
11. The ResNet architecture was chosen because it performed better
comparatively than others.

As seen in the figure, it’s quite clear that in case of partial occs
increasing network depth doesn’t play significant role in increasing or
decreasing the accuracies. We conclude in case of partial occs, depth of the
network doesn’t seem to have impact on model performances in classification.

Figure 11: Accuracies(Top 1%) with variation in network depth for artifact types 1,2 and 3 TABLE VI: Effect of network depth on model performance for different artifacts with 13 % occ Type | ResNet-50 | ResNet-101 | ResNet-151
---|---|---|---
Top 1% | Top 5% | Top 1% | Top 5% | Top 1% | Top 5%
1 | 86.31 | 97.11 | 85.63 | 97.34 | 85.89 | 97.49
2 | 84.13 | 96.90 | 87.54 | 97.79 | 86.35 | 96.63
3 | 85.56 | 97.06 | 84.35 | 96.95 | 84.69 | 96.45

####  IV-C6 Comparison of un-pretrained model and pre-trained model
performance on Occlusion

As discussed above, there are two ways to train a model; one is to pretrain
the model on Imagenet dataset and use the pre-trained weights to retrain on
Stanford cars dataset [3] and second take randomly initialized weights and
train the model from scratch on Stanford cars dataset. For the car dataset
with 13% occs and artifact type 1 we considered two models to be trained by
both using the pre-trained weights and from scratch. Table VII gives the
details of top 1% and 5% accuracies corresponding to these values. One of the
primary reason that lead to poor performance of the models trained from
scratch is due to the limited size of the dataset and large number of classes
(177) even though we used few data augmentation techniques they proved to be
insufficient here. So, the effect of Occ could not be determined in this case.

TABLE VII: Pretrained (PT) vs Not pre-trained(NPT) models for 13% occlusion and artifact type 1 Model | PT | N-PT
---|---|---
Top 1% | Top 5% | Top 1% | Top 5%
VGG-19 | 74.68 | 92.55 | 08.22 | 26.44
ResNet-50 | 86.31 | 97.10 | 20.61 | 45.82

####  IV-C7 Effect of different artifact types

In this experiment, we investigate the impact of varying the artifact type and
examine the model’s performance. We notice that on an average all the models
tend to result in little higher accuracy for artifact type 2. However, overall
it can be seen that there is not much significant difference in model
performances across different artifact types. Therefore it can be assumed that
all the artifacts bring in almost same amount of noise in the image no matter
how uniform or random the artifact is. Figure 12 shows a bar plot of
accuracies for different models.

Figure 12: Effect of different artifact types on model’s performance.

##  V Conclusion

The experiments carried out have provided in depth insights on the occ of
objects in the data sets and helped a great deal in understanding how the
modelsṕerformances differ with variants of occs. Following are the findings,
and facts we were able to establish based on our analysis:

  1. 1.

The type of occ or artifact added to object doesn’t seem to have a significant
effect on accuracy of the model. CNN based models tend to behave the same way
with either random, constant or images based occ’s.

  2. 2.

Based on our experiment with of the current dataset varied depth of the same
network architecture(ResNet-50/101/151) we could conclude that the depth of
the networks have no effect on the predictions with occ’s in anyway.

  3. 3.

The effect of Non pretrained model (NPT) was unable to be identified as the
model was not able to generalize well due to the smaller nature of dataset.

  4. 4.

Training the models with occluded dataset helps the these perform better when
tested with datasets containing occ’s. As the occ size increases the accuracy
gap between with and without occ trained models becomes wider.

  5. 5.

Though it can be understood that some models performed better than other for
both the type of datasets un-occluded and occluded. However, there is not much
evidence to show that if some model was performing better with regards to occ.

  6. 6.

Other interesting finding was that models trained on lower occ datasets
perform better on higher occ datasets as compared to the model trained on
unoccluded dataset. This indicates that the model is better able to generalize
when it has already seen some occ in the dataset even though smaller.

  7. 7.

On the contrary, CNN models trained on higher occ datasets perform similar to
models trained on un-occluded dataset when tested on either lower occluded
datasets. This also holds true when the testing is done for the un-occluded
dataset. Therefore, having occ’s in the training set seems to have no adverse
effect on the model (which intuitively we were expecting) even though there
are no occ’s seen in the original image.

One of the future works will be to establish the findings on a real world
occluded dataset to prove out theory from the artificially created dataset.
Also, as we have identified earlier all CNN based models tend to behave the
same when subjected to occ, the possible extension of the work can be with
regards to building a model which is robust to occ’s using the properties
which were identified and established earlier about occ’s.

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# A NOTE ON MINIMAL RESOLUTIONS OF VECTOR-SPREAD BOREL IDEALS

Marilena Crupi, Antonino Ficarra

###### Abstract

We consider vector–spread Borel ideals. We show that these ideals have linear
quotients and thereby we determine the graded Betti numbers and the bigraded
Poincaré series. A characterization of the extremal Betti numbers of such a
class of ideals is given. Finally, we classify all Cohen–Macaulay
vector–spread Borel ideals. †† _Keywords:_ monomial ideals, vector–spread
Borel ideals, Cohen–Macaulay ideals, extremal Betti numbers. _2020 Mathematics
Subject Classification:_ 05B35, 05E40, 13B25, 13D02, 68W30.

Department of Mathematics and Computer Sciences, Physics and Earth Sciences,
University of Messina, Viale Ferdinando Stagno d’Alcontres 31, 98166 Messina,
Italy

E-mail addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>
## Introduction

In this note we study the class of vector–spread Borel ideals introduced in
[15] as a generalization of the class of $t$–spread ideals, where $t$ is a non
negative integer [13] (see, also, [4, 5, 12] and the reference therein). Let
$S=K[x_{1},\dots,x_{n}]$ be the standard graded polynomial ring over a fixed
field $K$ and let ${\bf t}=(t_{1},t_{2},\dots,t_{d-1})\in{\mathbb{Z}}_{\geq
0}^{d-1}$, with $d\geq 2$, a $(d-1)$–tuple whose entries are non negative
integers. A monomial $u=x_{j_{1}}x_{j_{2}}\cdots x_{j_{\ell}}$ ($1\leq
j_{1}\leq j_{2}\leq\cdots\leq j_{\ell}\leq n$) of degree $\ell\leq d$ of $S$
is called a vector–spread monomial of type ${\bf t}$ or simply a ${\bf
t}$–spread monomial if $j_{i+1}-j_{i}\geq t_{i}$, for $i=0,\dots,\ell-1$. A
${\bf t}$–spread monomial ideal is a monomial ideal generated by ${\bf
t}$–spread monomials. For instance,
$I=(x_{1}x_{3}^{2}x_{5},x_{1}x_{3}^{2}x_{6},x_{1}x_{4}x_{5})$ is a
$(2,0,1)$–spread monomial ideal of the polynomial ring
$S=K[x_{1},\ldots,x_{5}]$, but it is not $(2,1,1)$–spread as
$x_{1}x_{3}^{2}x_{5}\in G(I)$ is not $(2,1,1)$–spread. One can note that any
monomial (ideal) is ${\bf 0}$–spread, where ${\bf 0}=(0,0,\dots,0)$. If
$t_{i}\geq 1$, for all $i$, a ${\bf t}$–spread monomial (ideal) is a
squarefree monomial (ideal). A ${\bf t}$–spread monomial ideal $I$ of $S$ is
said _vector–spread Borel ideal of type_ ${\bf t}$ or simply a ${\bf
t}$–spread strongly stable ideal if for any ${\bf t}$–spread monomial $u\in
I$, and all $j<i$ such that $x_{i}$ divides $u$ and $x_{j}(u/x_{i})$ is ${\bf
t}$–spread, then $x_{j}(u/x_{i})\in I$. For ${\bf t}={\bf 0}=(0,0,\dots,0)$
(${\bf t}={\bf 1}=(1,1,\dots,1)$) one obtains the classical notion of strongly
stable (squarefree strongly stable) ideal [16].
We study the graded Betti numbers of a ${\bf t}$–spread strongly stable ideal
$I$ by combinatorial tools and therefore we obtain a formula for their
computation. As a consequence we are able to give a characterization of the
extremal Betti numbers of such an ideal $I$. Finally, the Cohen–Macaulay ${\bf
t}$–spread strongly stable ideals are classified. Our approach is similar to
the one of [13].
In this note we discuss a unified concept to deal with the classes of monomial
ideals which are among the most important in Combinatorics (stable, strongly
stable ideals) in order to get more general results. Section 1 contains some
preliminaries and notions that will be used in the article. In Section 2, we
review a result on the graded Betti numbers of a ${\bf t}$–spread strongly
stable ideal stated in [15]. We propose a simpler method for determining a
formula for the graded Betti numbers of such a class of monomial ideals
(Corollary 2.4). A key result is Theorem 2.2 which states that a ${\bf
t}$–spread strongly stable ideal has linear quotients. The notion of
vector–spread support (Subsection 1.2) plays an essential role in this
context. Indeed, the linear quotients of vector–spread Borel ideals can be
expressed in terms of vector–spread supports (Corollary 2.3). Moreover, the
bigraded Poincaré series of such ideals is also determined (Corollary 2.5). In
Section 3, we obtain a characterization of the extremal Betti numbers of ${\bf
t}$–spread strongly stable ideals (Proposition 3.3) by the results in Section
2. Finally, Section 4 contains one of the main result in the article. We
analyze the Cohen–Macauleyness of ${\bf t}$–spread strongly stable ideals via
the formula of the graded Betti numbers described in Section 2. Indeed, we
obtain a classification (Theorem 4.3) by investigating the height and the
projective dimension of such ideals.

## 1\. Preliminaries

Throughout the article, we denote by $S=K[x_{1},\dots,x_{n}]$ the standard
graded polynomial ring over a field $K$ and by ${\operatorname{Mon}}(S)$
(${\operatorname{Mon}}_{d}(S)$) the set of all monomials (of degree $d$) in
$S$.

### 1.1. A glimpse to graded Betti numbers.

Given a monomial $u=x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}$ of $S$,
the support of $u$ is the set
${\operatorname{supp}}(u)=\\{i:a_{i}>0\\}=\\{i:x_{i}\ \textup{divides}\ u\\}$.
We denote by $\max(u)$ ($\min(u)$) the maximal (minimal) integer
$i\in{\operatorname{supp}}(u)$.
For a monomial ideal $I$ of $S$, $G(I)$ denotes the unique minimal set of
monomial generators of $I$ and we set $G(I)_{j}=\\{u\in G(I):\deg(u)=j\\}$. It
is known that a monomial ideal $I$ of $S$ has a unique minimal graded free
$S$–resolution

${\mathbb{F}}:0\rightarrow F_{p}\xrightarrow{\ d_{p}\ }F_{p-1}\xrightarrow{\
d_{p-1}\ }\cdots\xrightarrow{\ d_{1}\ }F_{0}\xrightarrow{\ d_{0}\
}I\rightarrow 0,$

where $F_{i}=\bigoplus_{j}S(-j)^{\beta_{i,j}(I)}$. The numbers
$\beta_{i,j}(I)=\dim_{K}\operatorname{Tor}(K,I)_{j}$ are called the graded
Betti numbers of $I$. For all $i$, $\beta_{i}(I)=\sum_{j}\beta_{i,j}(I)$ is
the $i$th total Betti number of $I$. One defines the projective dimension and
the regularity of $I$ as follows

$\displaystyle{\operatorname{pd}}(I)$
$\displaystyle=\max\\{i:\beta_{i,j}(I)\neq 0,\ \textup{for some}\
j\\}=\max\\{i:\beta_{i}(I)\neq 0\\},$ $\displaystyle{\operatorname{reg}}(I)$
$\displaystyle=\max\\{j-i:\beta_{i,j}(I)\neq
0\\}=\max\\{j:\beta_{i,i+j}(I)\neq 0\ \textup{for some}\ i\\}.$

These algebraic invariants have been refined in [7] by the notion of _extremal
Betti number_. A graded Betti number $\beta_{k,k+\ell}(I)\neq 0$ of a monomial
ideal $I$ of $S$ is called extremal if $\beta_{i,i+j}(I)=0$ for all $i\geq k$,
$j\geq\ell$ such that $(i,j)\neq(k,\ell)$.

Let $I$ be a monomial ideal. It is known that
$\textup{depth}(S/I)\leq\dim(S/I)$. If the equality holds, we say that $S/I$
is a Cohen–Macaulay ring and $I$ is a Cohen–Macaulay ideal.

### 1.2. A glimpse to vector–spread monomial ideals.

Let ${\bf t}=(t_{1},t_{2},\dots,t_{d-1})\in{\mathbb{Z}}_{\geq 0}^{d-1}$,
$d\geq 2$, a $(d-1)$–tuple whose entries are non negative integers. Let us
write a monomial $u\in S$ as $u=x_{j_{1}}x_{j_{2}}\cdots x_{j_{\ell}}$, $1\leq
j_{1}\leq j_{2}\leq\cdots\leq j_{\ell}\leq n$. We will maintain this
convention throughout the article.

Let $T=K[x_{1},x_{2},\dots,x_{n},\dots]$ be the polynomial ring in infinitely
many variables. Denote by ${\operatorname{Mon}}(T;{\bf t})$ the set of all
${\bf t}$–spread monomials of $T$ and by ${\operatorname{Mon}}(S;{\bf t})$ the
set of all ${\bf t}$–spread monomials of $S$. Furthermore, for all
$0\leq\ell\leq d$, we define the following sets

$\displaystyle{\operatorname{Mon}}_{\ell}(T;{\bf t})$
$\displaystyle=\big{\\{}u\in{\operatorname{Mon}}(T;{\bf
t}):\deg(u)=\ell\big{\\}},$ $\displaystyle{\operatorname{Mon}}_{\ell}(S;{\bf
t})$ $\displaystyle=\big{\\{}u\in{\operatorname{Mon}}(S;{\bf
t}):\deg(u)=\ell\big{\\}}.$

Let us denote by $M_{n,\ell,{\bf t}}$ the set of all ${\bf t}$–spread
monomials of $S$ having degree $\ell$. One may observe that $M_{n,\ell,{\bf
t}}=\varnothing$ for $\ell>d$. In order to compute the cardinality of the set
$M_{n,\ell,{\bf t}}$, one can consider a special shifting operator (see [6,
13]). More in detail, one defines the map $\sigma_{{\bf 0},{\bf
t}}:{\operatorname{Mon}}(T;{\bf 0})\rightarrow{\operatorname{Mon}}(T;{\bf
t})$, by setting $\sigma_{{\bf 0},{\bf t}}(1)=1$, $\sigma_{{\bf 0},{\bf
t}}(x_{i})=x_{i}$ and

$\displaystyle\sigma_{{\bf 0},{\bf t}}(x_{j_{1}}x_{j_{2}}\cdots
x_{j_{\ell}})=\prod_{k=1}^{\ell}x_{j_{k}+\sum_{s=1}^{k-1}t_{s}},$

for all monomials $u=x_{j_{1}}x_{j_{2}}\cdots
x_{j_{\ell}}\in{\operatorname{Mon}}(T;{\bf 0})$ with $2\leq\ell\leq d$. The
map $\sigma_{{\bf 0},{\bf t}}$ is bijective and its inverse is the map
$\sigma_{{\bf t},{\bf 0}}:{\operatorname{Mon}}(T;{\bf
t})\rightarrow{\operatorname{Mon}}(T;{\bf 0})$ defined as follows:
$\sigma_{{\bf t},{\bf 0}}(1)=1$, $\sigma_{{\bf t},{\bf 0}}(x_{i})=x_{i}$, for
all $i\in{\mathbb{N}}$, and $\sigma_{{\bf t},{\bf 0}}(x_{j_{1}}x_{j_{2}}\cdots
x_{j_{\ell}})$ $=\prod_{k=1}^{\ell}x_{j_{k}-\sum_{s=1}^{k-1}t_{s}}$, for all
monomials $u=x_{j_{1}}x_{j_{2}}\cdots
x_{j_{\ell}}\in{\operatorname{Mon}}(T;{\bf t})$ with $2\leq\ell\leq d$.
In particular, the restriction $\sigma_{{\bf t},{\bf 0}}|_{M_{n,\ell,{\bf
t}}}$ is a injective map and its image is equal to the set
$M_{n-(t_{1}+t_{2}+\ldots+t_{\ell-1}),\ell,{\bf
0}}={\operatorname{Mon}}_{\ell}(K[x_{1},\dots,x_{n-(t_{1}+t_{2}+\ldots+t_{\ell-1})}])$.
Thus

$|M_{n,\ell,{\bf
t}}|=\binom{n+(\ell-1)-\sum_{j=1}^{\ell-1}t_{j}}{\ell},\quad\mbox{for
$0\leq\ell\leq d$}.$ (1)

For more details on this topic see [15].

If $u_{1},\dots,u_{m}$ are ${\bf t}$–spread monomials of $S$, we denote by
$B_{\bf t}(u_{1},\dots,u_{m})$ the smallest ${\bf t}$–spread strongly stable
ideal of $S$ containing $u_{1},\dots,u_{m}$ with respect to the inclusion. The
monomials $u_{1},\dots,u_{m}$ are called the ${\bf t}$–spread Borel generators
of $B_{\bf t}(u_{1},\dots,u_{m})$. If $m=1$, $B_{\bf t}(u_{1})=B_{\bf t}(u)$
is called a principal ${\bf t}$–spread Borel ideal. It is clear that for each
${\bf t}$–spread strongly stable monomial ideal $I$ of $S$, one can uniquely
determine ${\bf t}$–spread monomials $u_{1},\dots,u_{m}\in S$ such that
$I=B_{\bf t}(u_{1},\dots,u_{m})$.
Observe that for
$u=x_{n-(t_{1}+t_{2}+\ldots+t_{\ell-1})}x_{n-(t_{2}+\ldots+t_{\ell-1})}\cdots
x_{n-t_{\ell-1}}x_{n}$, the ideal $B_{\bf t}(u)$ contains all ${\bf t}$–spread
monomials of degree $\ell\leq d$. $B_{\bf t}(u)$ is called the ${\bf
t}$–spread Veronese ideal of degree $\ell$ and will be denoted by
$I_{n,\ell,{\bf t}}$. Note that $I_{n,\ell,{\bf t}}\neq(0)$ if and only if
$n\geq 1+\sum_{j=1}^{\ell-1}t_{j}$.

Now let us recall the pivotal notion of ${\bf t}$–spread support of a ${\bf
t}$–spread monomial [15, Definition 2.1].

If $n$ is a positive integer $n$, $[n]$ denotes the set $\\{1,2,\dots,n\\}$.
For $j\leq k$ positive integers, we define
$[j,k]=\\{\ell\in\mathbb{N}:j\leq\ell\leq k\\}$. We set $[j,k]=\varnothing$ if
$j>k$.
Let $u=x_{j_{1}}x_{j_{2}}\cdots x_{j_{\ell}}\in M_{n,\ell,{\bf t}}$ be a ${\bf
t}$–spread monomial of $S$. The ${\bf t}$–spread support of $u$ is the subset
of $[n]$ defined as follows

${\operatorname{supp}}_{\bf t}(u)=\textstyle\bigcup\limits_{i=1}^{\ell-1}\
\big{[}j_{i},j_{i}+(t_{i}-1)\big{]}.$

Note that ${\operatorname{supp}}_{\bf 0}(u)=\varnothing$. Furthermore, if $u$
is squarefree, ${\operatorname{supp}}_{\bf 1}(u)$
$={\operatorname{supp}}(u/x_{\max(u)})$
$=\big{\\{}j_{1},j_{2},\dots,j_{\ell-1}\big{\\}}$, where ${\bf
1}=(1,1,\dots,1)\in{\mathbb{Z}}_{\geq 0}^{d-1}$.

The notion vector–spread support has been introduced in [11] in order to
classify Cohen–Macaulay $t$–spread lexsegment ideals.

## 2\. The graded Betti numbers

In [15], the graded Betti numbers of vector–spread strongly stable ideals have
been determined by means of the Koszul homology. More precisely, if $I$ is a
${\bf t}$–spread strongly stable ideal of $S$, then the graded Betti numbers
of $I$ are given by

$\beta_{i,i+j}(I)=\sum_{u\in
G(I)_{j}}\binom{\max(u)-1-\sum_{\ell=1}^{j-1}t_{\ell}}{i},\quad\mbox{for all
$i,j$}.$

The purpose of this section is to quickly obtain this formula using a
different and easier method than the one used in [15]. The vector–spread
support will be a key tool.

We need to fix some notations and recall some results from [17]. Let $I$ be a
${\bf t}$–spread strongly stable ideal of $S$. Let
$G(I)=\\{u_{1}>u_{2}>\dots>u_{m}\\}$ be the minimal generating set of $I$
ordered with respect to the pure lexicographic order [16]. For $k=1,\dots,m$,
we set $J_{k}=(u_{1},\dots,u_{k})$ and

$\textup{set}(u_{k})=\big{\\{}i\in[n]:x_{i}\in(u_{1},\dots,u_{k-1}):u_{k}\big{\\}}.$

Note that $\textup{set}(u_{1})=\varnothing$. Our aim is to prove that $I$ has
linear quotients, _i.e._ ,

$J_{k-1}:u_{k}=(u_{1},u_{2},\dots,u_{k-1}):u_{k},$

is generated by variables, for all $k=2,\dots,m$. In such a case, by [17,
Lemma 1.5], one has

$\beta_{i,i+j}(I)=\sum_{u\in
G(I)_{j}}\binom{|\textup{set}(u)|}{i},\quad\mbox{for all $i,j\geq 0$}.$ (2)

We quote the next crucial lemma from [15] (see also [13, Lemma 1.3]). It
provides the existence of a standard decomposition for a ${\bf t}$–spread
monomial belonging to a ${\bf t}$–spread strongly stable ideal.

###### Lemma 2.1

Let $I$ be a ${\bf t}$–spread strongly stable ideal of $S$, and $w\in I$ a
${\bf t}$–spread monomial. Then, there exist $u\in G(I)$ and
$v\in{\operatorname{Mon}}(S)$ such that $w=uv$ and $\max(u)\leq\min(v)$.

Now we are in position to prove the main result of this section.

###### Theorem 2.2

Let $I$ be a ${\bf t}$–spread strongly stable ideal of $S$. Then $I$ has
linear quotients, in particular it is componentwise linear.

Proof. Let $G(I)=\\{u_{1}>u_{2}>\dots>u_{m}\\}$ ordered with respect to the
pure lexicographic order. Fix $k\in\\{2,\dots,m\\}$. A set of generators for
$J_{k-1}:u_{k}=(u_{1},\dots,u_{k-1}):u_{k}$ is
$\\{u_{s}/\textup{gcd}(u_{k},u_{s}):s=1,\ldots,k-1\\}$ [16, Proposition
1.2.2]. Hence, it suffices to show that for each $s$, there exists a variable
$x_{i}\in J_{k-1}:u_{k}$ with $x_{i}$ dividing
$u_{s}/\textup{gcd}(u_{k},u_{s})$. Let $u_{s}=x_{i_{1}}x_{i_{2}}\cdots
x_{i_{p}},\ u_{k}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{q}},$ with
$u_{s}>_{{\operatorname{lex}}}u_{k}$. Hence, there exists an integer
$\ell\in[q]$ such that

$i_{1}=j_{1},\ \ i_{2}=j_{2},\ \ \ldots,\ \ i_{\ell-1}=j_{\ell-1},\ \
i_{\ell}<j_{\ell}.$ (3)

Set $v=x_{i_{\ell}}(u_{k}/x_{j_{\ell}})$. Then $v=x_{j_{1}}x_{j_{2}}\cdots
x_{j_{\ell-1}}x_{i_{\ell}}x_{j_{\ell+1}}\cdots x_{j_{q}}$. We need to show
that $v$ is ${\bf t}$–spread. This is the case, as
$i_{\ell}-j_{\ell-1}=i_{\ell}-i_{\ell-1}\geq t_{\ell-1}$ and
$j_{\ell+1}-i_{\ell}>j_{\ell+1}-j_{\ell}\geq t_{\ell}$. On the other hand, $I$
is a ${\bf t}$–spread strongly stable ideal. Therefore $v\in I$ and,
furthermore, $v>_{{\operatorname{lex}}}u_{k}$. Now we show that $v\in
J_{k-1}$. By Lemma 2.1, $v=u_{r}w$ with $u_{r}\in G(I)$ and
$w\in{\operatorname{Mon}}(S)$ such that $\max(u_{r})\leq\min(w)$. If $v\notin
J_{k-1}=(u_{1},\dots,u_{k-1})$, then $u_{r}\in
G(I)\setminus\\{u_{1},\dots,u_{k-1}\\}$, and consequently
$u_{r}\leq_{{\operatorname{lex}}}u_{k}$. As $v=u_{r}w$, then
$v\leq_{{\operatorname{lex}}}u_{k}$, a contradiction. Therefore, $v\in
J_{k-1}$ and $x_{i_{\ell}}u_{k}=vx_{j_{\ell}}\in J_{k-1}$. Thus
$x_{i_{\ell}}\in J_{k-1}:u_{k}$ and, by (3), $x_{i_{\ell}}$ divides
$u_{s}/\textup{gcd}(u_{k},u_{s})$. It follows that $I$ has linear quotients.
Finally, it is well known that ideals with linear quotients are componentwise
linear [16, Theorem 8.2.15]. $\Box$

The following corollary highlights the role of the vector–spread supports.

###### Corollary 2.3

In the previous setting, for all $k=2,\dots,m$, we have

$\textup{set}(u_{k})=[\max(u_{k})-1]\setminus{\operatorname{supp}}_{\bf
t}(u_{k}).$

Proof. Let $u_{k}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{d}}$. We have
${\operatorname{supp}}_{\bf
t}(u_{k})=\bigcup_{i=1}^{d-1}\big{(}\bigcup_{q=0}^{t_{i}-1}\\{j_{i}+q\\}\big{)}$.
Let $\ell\in[n]$. To determine if $\ell\in\textup{set}(u_{k})$, we consider
two cases.
Case 1. Let $\max(u_{k})\leq\ell\leq n$, then
$x_{\ell}u_{k}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{d}}x_{\ell}$. First, observe
that the ideal $J_{k-1}=(u_{1},u_{2},\dots,u_{k-1})$ is ${\bf t}$–spread
strongly stable. If for absurd $\ell\in\textup{set}(u_{k})$, then
$x_{\ell}u_{k}\in J_{k-1}$. Hence $u_{r}$ divides $x_{\ell}u_{k}$, for some
$r\in\\{1,\dots,k-1\\}$. Since $\deg(u_{r})\leq\deg(u_{k})$, then $u_{r}$
divides $u_{k}$. An absurd, since $u_{k}$ is a minimal generator of $I$.
Case 2. Let $\ell\in{\operatorname{supp}}_{\bf t}(u_{k})$, then
$\ell=j_{r}+s$, for some $r\in\\{1,\dots,d-1\\}$, $0\leq s\leq t_{r}-1$,
$t_{r}\geq 1$. If $x_{\ell}u_{k}\in J_{k}=(u_{1},u_{2},\dots,u_{k})$, then
$u_{p}$ divides $x_{\ell}u_{k}$, for some $p\in\\{1,\dots,k-1\\}$. As
$j_{r}+s-j_{r}=s<t_{r}$, and $u_{p}$ is ${\bf t}$–spread, necessarily $u_{p}$
divides $u_{k}$. An absurd, since $u_{k}$ is a minimal generator of $I$.

The above cases imply that if $\ell\in\textup{set}(u_{k})$, then
$\ell\notin[\max(u_{k}),n]$ and $\ell\notin{\operatorname{supp}}_{\bf
t}(u_{k})$.

Now assume $\ell\in[\max(u_{k})-1]\setminus{\operatorname{supp}}_{\bf
t}(u_{k})$. Then $\ell=j_{r}+s$, for some $r\in\\{1,\dots,d-1\\}$ and $s\geq
t_{r}$. Let
$j_{q}=\min\big{\\{}j\in{\operatorname{supp}}(u_{k}):j>\ell\big{\\}}$. Then
$x_{\ell}(u_{k}/x_{j_{q}})$ is a ${\bf t}$–spread monomial that belongs to the
ideal $(J_{k-1},u_{k})=(u_{1},u_{2},\dots,u_{k-1},u_{k})=J_{k}$, as
$\ell<j_{q}$ and $(J_{k-1},u_{k})$ is a ${\bf t}$–spread strongly stable
ideal. By Lemma 2.1, $x_{\ell}(u_{k}/x_{j_{q}})=u_{p}w$ with
$p\in\\{1,\dots,k\\}$ and $w\in{\operatorname{Mon}}(S)$ such that
$\max(u_{p})\leq\min(w)$. Clearly, $p\neq k$ and so
$x_{\ell}(u_{k}/x_{j_{q}})\in J_{k-1}=(u_{1},u_{2},\dots,u_{k-1})$. Finally
$x_{\ell}u_{k}=x_{\ell}(u_{k}/x_{j_{q}})x_{j_{q}}\in J_{k-1}$ and
$\ell\in\textup{set}(u_{k})$. The assertion follows. $\Box$

As a consequence of the above result, one has

$J_{k-1}:u_{k}=\big{(}x_{i}:i\in[\max(u_{k})-1]\setminus{\operatorname{supp}}_{\bf
t}(u_{k})\big{)},\quad\mbox{for $k=2,\dots,m$.}$

On the other hand, for each $k$

$\big{|}[\max(u_{k})-1]\setminus{\operatorname{supp}}_{\bf
t}(u_{k})\big{|}=\max(u_{k})-1-\sum_{\ell=1}^{\deg(u_{k})-1}t_{\ell}.$

Thus, formula (2) yields the result we are looking for.

###### Corollary 2.4

Let $I$ be a ${\bf t}$–spread strongly stable ideal of $S$. Then,

$\beta_{i,i+j}(I)=\sum_{u\in
G(I)_{j}}\binom{\max(u)-1-\sum_{\ell=1}^{j-1}t_{\ell}}{i},\quad\mbox{for all
$i,j\geq 0$}.$

As a consequence, we obtain:

###### Corollary 2.5

Let $I$ be a ${\bf t}$–spread strongly stable ideal of $S$. Then

$P_{S/I}(y,z)=1+\sum_{u\in
G(I)}(1+y)^{\max(u)-1-\sum_{s=1}^{\deg(u)-1}t_{s}}yz^{\deg(u)}.$

Proof. Let $P_{S/I}(y,z)=\sum_{i,j}\beta_{i,j}(I)y^{i}z^{j}$ be the bigraded
Poincaré series of $S/I$. From [17, Corollary 1.6], for an ideal $I$ with
linear quotients, one has

$P_{S/I}(y,z)=1+\sum_{u\in G(I)}(1+y)^{|\textup{set}(u)|}yz^{\deg(u)}.$

The assertion follows from Corollary 2.4. $\Box$

## 3\. Extremal Betti numbers

In this section we characterize the extremal Betti numbers of the class of
${\bf t}$–spread Borel ideals. Our results generalizes some statements in [1]
(see also [2, 3, 9, 10] and the reference therein).

First, one can note that Corollary 2.4 implies

$\displaystyle{\operatorname{reg}}(I)\ $ $\displaystyle=\
\max\big{\\{}\deg(u):u\in G(I)\big{\\}},$ (4)
$\displaystyle{\operatorname{pd}}(I)\ $ $\displaystyle=\
\max\big{\\{}\max(u)-1-\textstyle\sum_{j=1}^{\deg(u)-1}t_{j}:u\in
G(I)\big{\\}}.$ (5)

For our aim, the next lemma will play a crucial role.

###### Lemma 3.1

Let $I$ be a ${\bf t}$–spread strongly stable ideal of $S$. If
$\beta_{i,i+j}(I)\neq 0$, then $\beta_{k,k+j}(I)\neq 0$ for all $k=0,\dots,i$.

Proof. Let $\beta_{i,i+j}(I)\neq 0$. From Corollary 2.4, there exists a
monomial $u\in G(I)_{j}$ such that $\max(u)-1-\sum_{\ell=1}^{j-1}t_{\ell}\geq
i$. Hence, $\max(u)-1-\sum_{\ell=1}^{j-1}t_{\ell}\geq k$, for all
$k=0,\dots,i$. This implies that $\beta_{k,k+j}(I)\neq 0$, for all
$k=0,\dots,i$. $\Box$

The previous lemma and the definition of extremal Betti number imply the next
result.

###### Corollary 3.2

Let $I$ be a ${\bf t}$–spread strongly stable ideal of $S$. The following
conditions are equivalent:

  1. (i)

$\beta_{k,k+\ell}(I)$ is extremal;

  2. (ii)

$\beta_{k,k+\ell}(I)\neq 0$, and $\beta_{i,i+\ell}(I),\beta_{k,k+j}(I)=0$, for
all $i>k$ and all $j>\ell$.

The following characterization generalizes a result in [1, Theorem 1].

###### Proposition 3.3

Let $I$ be a ${\bf t}$–spread strongly stable ideal of $S$. The following
conditions are equivalent:

  1. (i)

$\beta_{k,k+\ell}(I)$ is extremal;

  2. (ii)

$\max\big{\\{}\max(u):u\in G(I)_{\ell}\big{\\}}=k+\sum_{j=1}^{\ell-1}t_{j}+1$,
and $\max(u)<k+\sum_{r=1}^{j-1}t_{r}+1$, for all $u\in G(I)_{j}$ and all
$j>\ell$.

Proof. (i)$\implies$(ii). Since $I$ is ${\bf t}$–spread strongly stable,
Corollary 2.4 implies that $\beta_{k,k+\ell}(I)$ is non zero if and only if
there exists a monomial $u_{0}\in G(I)_{\ell}$ such that $\max(u_{0})\geq
k+\sum_{j=1}^{\ell-1}t_{j}+1$. Therefore,

$\max\big{\\{}\max(u):u\in G(I)_{\ell}\big{\\}}\ \geq\ \max(u_{0})\ \geq\
k+\textstyle\sum_{j=1}^{\ell-1}t_{j}+1.$ (6)

Suppose that for a monomial $u_{1}\in G(I)_{\ell}$,
$\max(u_{1})=j+\sum_{j=1}^{\ell-1}t_{j}+1>k+\sum_{j=1}^{\ell-1}t_{j}+1$, for
some $j>k$. Then $\beta_{j,j+\ell}(I)\neq 0$, which contradicts the fact that
$\beta_{k,k+\ell}(I)$ is extremal (Corollary 3.2). Hence,
$k+\sum_{j=1}^{\ell-1}t_{j}+1=\max\big{\\{}\max(u):u\in G(I)_{\ell}\big{\\}}$.
Assume there exists a monomial $v\in G(I)_{j}$ such that $\max(v)\geq
k+\sum_{r=1}^{j-1}t_{r}+1$, for some $j>\ell$. Then $\beta_{k,k+j}(I)\neq 0$.
A contradiction, since $\beta_{k,k+\ell}(I)$ is extremal (Corollary 3.2).
Hence, condition (ii) holds.
(ii)$\implies$(i). Since $\max\big{\\{}\max(u):u\in
G(I)_{\ell}\big{\\}}=k+\sum_{j=1}^{\ell-1}t_{j}+1$, then
$\beta_{k,k+\ell}(I)\neq 0$ and $\beta_{j,j+\ell}(I)=0$, for all $j>k$.
Moreover, $\max(u)<k+\sum_{r=1}^{j-1}t_{r}+1$, for all $u\in G(I)_{j}$ and all
$j>\ell$, implies that $\beta_{k,k+j}(I)=0$, for all $j>\ell$. Finally, by
Corollary 3.2, $\beta_{k,k+\ell}(I)$ is extremal. $\Box$

The previous result yields the following useful corollary.

###### Corollary 3.4

Let $I$ be a ${\bf t}$–spread strongly stable ideal of $S$, and let
$\beta_{k,k+\ell}(I)$ be an extremal Betti number of $I$. Then

$\beta_{k,k+\ell}(I)=\big{|}\big{\\{}u\in
G(I)_{\ell}:\max(u)=k+\textstyle\sum_{j=1}^{\ell-1}t_{j}+1\big{\\}}\big{|}\leq\
\tbinom{k+\ell-1}{\ell-1}.$

Proof. The equality follows immediately from Proposition 3.3. For the
inequality, it sufficies to observe that

$\displaystyle\big{|}\big{\\{}u\in M_{n,\ell,{\bf
t}}:\max(u)=k+\sum_{j=1}^{\ell-1}t_{j}+1\big{\\}}\big{|}=$
$\displaystyle=\binom{k+\sum_{j=1}^{\ell-1}t_{j}+1+(\ell-1)-\sum_{j=1}^{\ell-1}t_{j}-1}{\ell-1}=\binom{k+\ell-1}{\ell-1}.$

$\Box$

###### Example 3.5

Let $I=(x_{1}x_{2},x_{1}x_{3},x_{1}x_{4}^{2})$ be a $(1,0)$–spread strongly
stable ideal of $S=K[x_{1},x_{2},x_{3},x_{4}]$

The Betti table of $I$ is [14]:

$\begin{matrix}&0&1&2\\\ \text{total:}&3&3&1\\\ \text{2:}&2&1&\text{-}\\\
\text{3:}&1&2&1\end{matrix}$

Hence, ${\operatorname{pd}}(I)=2$, ${\operatorname{reg}}(I)=3$. Moreover,
$\beta_{2,2+3}(I)$ is the unique extremal Betti number of $I$ and by Corollary
3.4,

$\beta_{2,2+3}(I)=\big{|}\big{\\{}u\in
G(I)_{3}:\max(u)=2+\textstyle\sum_{j=1}^{3-1}t_{j}+1=4\big{\\}}\big{|}=\big{|}\big{\\{}x_{1}x_{4}^{2}\big{\\}}\big{|}=1.$

## 4\. Cohen–Macaulay vector–spread Borel ideals

In this section we investigate the Cohen–Macaulayness of vector–spread Borel
ideals by tools from [11, 13]. To obtain a characterization, we only need to
investigate the height of a vector–spread Borel ideal. Indeed, from Corollary
2.4 and the Auslander–Buchsbaum formula, one has that

$0pt(I)=n-\max\big{\\{}\max(u)-1-\textstyle\sum_{j=1}^{\deg(u)-1}t_{j}:u\in
G(I)\big{\\}}.$

###### Proposition 4.1

Let $I_{n,d,{\bf t}}$ be the ${\bf t}$–spread Veronese ideal of degree $d$ of
$S$. Then

$\textup{height}(I_{n,d,{\bf t}})=n-\textstyle\sum_{j=1}^{d-1}t_{j}.$

Proof. First, observe that $I_{n,d,{\bf t}}\neq(0)$ if and only if $n\geq
1+\sum_{j=1}^{d-1}t_{j}$. The monomial prime ideal
$\mathfrak{p}=(x_{i}:i=1,2,\dots,n-\sum_{j=1}^{d-1}t_{j})$ is a minimal prime
of $I_{n,d,{\bf t}}$, as each monomial generator $u\in G(I_{n,d,{\bf
t}})=M_{n,d,{\bf t}}$ has minimum belonging to the set
$\big{\\{}1,2,\dots,n-\sum_{j=1}^{d-1}t_{j}\big{\\}}$. Therefore

$\textup{height}(I_{n,d,{\bf t}})\leq n-\textstyle\sum_{j=1}^{d-1}t_{j}.$

Suppose $\textup{height}(I_{n,d,{\bf t}})=k<n-\sum_{j=1}^{d-1}t_{j}$. Then,
there would be a minimal prime $\mathfrak{q}=(x_{i_{j}}:j=1,\dots,k)$ of
$I_{n,d,{\bf t}}$ of height $k\leq n-\sum_{j=1}^{d-1}t_{j}-1$, with all
$i_{j}$ distinct. The set $A=[n]\setminus\\{i_{j}:j=1,\dots,k\\}$ has
cardinality

$|A|=n-k\geq
n-\big{(}n-\textstyle\sum_{j=1}^{d-1}t_{j}-1\big{)}=1+\textstyle\sum_{j=1}^{d-1}t_{j}.$

We set $\ell_{1}=\min(A)$ and $\ell_{j}=\min\\{\ell\in
A:\ell\geq\ell_{j-1}+t_{j}\\}$, for all $j\geq 2$. The sequence
$\ell_{1}\leq\ell_{2}\leq\dots\leq\ell_{k}$ has at least $d$ terms, otherwise
$|A|<1+\textstyle\sum_{j=1}^{d-1}t_{j}$. So, the monomial
$w=x_{\ell_{1}}x_{\ell_{2}}\cdots x_{\ell_{d}}$ is ${\bf t}$–spread of degree
$d$, but $w\in G(I_{n,d,{\bf t}})\setminus\mathfrak{q}$, an absurd. Finally,
we have $\textup{height}(I_{n,d,{\bf t}})=n-\sum_{j=1}^{d-1}t_{j}$, as
desired. $\Box$

###### Theorem 4.2

Let $I$ be a ${\bf t}$–spread strongly stable ideal of $S$. Then

$\textup{height}(I)=\max\big{\\{}\min(u):u\in G(I)\big{\\}}.$

Proof. First we prove that $\textup{height}(I)\leq\max\\{\min(u):u\in
G(I)\\}$. Indeed, let $u_{0}\in G(I)$ such that
$\min(u_{0})=\max\\{\min(u):u\in G(I)\\}$. In such a case the monomial prime
ideal $\mathfrak{p}_{[\min(u_{0})]}=(x_{1},x_{2},\dots,x_{\min(u_{0})})$ is a
minimal prime of $I$, thus proving the inequality. For the other inequality,
write $u_{0}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{d}}$ and consider the monomial
$v_{0}=x_{j_{1}}x_{j_{1}+t_{1}}\cdots x_{j_{1}+(t_{1}+t_{2}+\ldots+t_{d-1})}$.
Then $v_{0}\in B_{\bf t}(u_{0})\subseteq I$. Moreover, $I^{\prime}=B_{\bf
t}(v_{0})=I_{j_{1}+t_{1}+t_{2}+\ldots+t_{d-1},d,{\bf t}}$. Hence, by
Proposition 4.1

$\textup{height}(I)\geq\textup{height}(I^{\prime})=j_{1}+\sum_{\ell=1}^{d-1}t_{\ell}-\sum_{\ell=1}^{d-1}t_{\ell}=j_{1}=\min(v_{0})=\min(u_{0}).$

$\Box$

Using what we have shown thus far, we are able to classify all Cohen–Macaulay
vector–spread strongly stable ideals of $S$.

###### Theorem 4.3

Let $I\subseteq S$ be a ${\bf t}$–spread strongly stable ideal of $S$, ${\bf
t}=(t_{1},t_{2},\dots,t_{d-1})$. Then $S/I$ is a Cohen–Macaulay ring if and
only if there exists $u\in G(I)$ of degree $\ell\leq d$ such that

$u=x_{n-(t_{1}+t_{2}+\ldots+t_{\ell-1})}x_{n-(t_{2}+t_{3}+\ldots+t_{\ell-1})}\cdots
x_{n-t_{\ell-1}}x_{n}\in G(I).$

In particular, if $I$ is equigenerated in degree $\ell$ then $S/I$ is
Cohen–Macaulay if and only if $I=I_{n,\ell,{\bf t}}$ is a vector–spread
Veronese ideal.

Proof. Firstly, we may suppose $n\in\bigcup_{u\in
G(I)}{\operatorname{supp}}(u)$. On the contrary, setting $\widetilde{n}=$
$\max\bigcup_{u\in G(I)}{\operatorname{supp}}(u)$, we may replace $S$ with
$\widetilde{S}=K[x_{1},\dots,x_{\widetilde{n}}]$ and consider
$I\cap\widetilde{S}$, instead. Since
${\operatorname{pd}}(S/I)={\operatorname{pd}}(I)+1$, by (5), we have

$\displaystyle{\operatorname{pd}}(S/I)\ $ $\displaystyle=\
{\textstyle\max\big{\\{}\max(u)-\sum_{j=1}^{\deg(u)-1}t_{j}:u\in
G(I)\big{\\}}}.$ On the other hand, by Theorem 4.2, $\displaystyle\dim(S/I)\ $
$\displaystyle=\ n-\max\big{\\{}\min(u):u\in G(I)\big{\\}}.$

By the Auslander–Buchsbaum formula, $n-{\operatorname{pd}}(S/I)=0pt(S/I)$.
Since $S/I$ is Cohen–Macaulay if and only if $0pt(S/I)=\dim(S/I)$, by the
previous formulas, $S/I$ is Cohen–Macaulay if and only if

$\max\Big{\\{}\max(u)-\sum_{j=1}^{\deg(u)-1}t_{j}:u\in
G(I)\Big{\\}}=\max\big{\\{}\min(u):u\in G(I)\big{\\}}.$ (7)

Let $u_{0}\in G(I)$ such that $\min(u_{0})=\max\big{\\{}\min(u):u\in
G(I)\big{\\}}$. Clearly, for all $u\in G(I)$,
$\min(u)\leq\max(u)-\sum_{j=1}^{\deg(u)-1}t_{j}$. Hence, equation (7) holds if
and only if

$\min(u_{0})=\max(u_{0})-\textstyle\sum_{j=1}^{\deg(u)-1}t_{j}.$

Thus, $S/I$ is Cohen–Macaulay if and only if there exist a monomial $u_{0}\in
G(I)$ and an integer $\ell\leq d$ such that

$\min(u_{0})=\max\big{\\{}\min(u):u\in G(I)\big{\\}},\
u_{0}=x_{j_{1}}x_{j_{1}+t_{1}}\cdots
x_{j_{1}+(t_{1}+t_{2}+\ldots+t_{\ell-1})}.$

It remains to show that $j_{1}=n-\sum_{r=1}^{\ell-1}t_{r}$. Since
$n\in\bigcup_{u\in G(I)}{\operatorname{supp}}(u)$, it follows the existence of
a monomial $u\in G(I)$ with $\max(u)=n$. Moreover, $\min(u)\leq\min(u_{0})$,
as $\min(u_{0})=\max\\{\min(u):u\in G(I)\\}$. Note that

$\max(u)-\textstyle\sum_{r=1}^{\deg(u)-1}t_{r}\leq j_{1}=\min(u_{0}).$

Hence, as $\max(u)=n$, we have $j_{1}\geq n-\sum_{r=1}^{\deg(u)-1}t_{r}$. If
$\deg(u)\leq\deg(u_{0})$, then $j_{1}\geq n-\sum_{r=1}^{\deg(u)-1}t_{r}\geq
n-\sum_{r=1}^{\deg(u_{0})-1}t_{r}$. On the other hand, $j_{1}\leq
n-\sum_{r=1}^{\deg(u_{0})-1}t_{r}$. So, in such a case,
$j_{1}=n-\sum_{r=1}^{\deg(u_{0})-1}t_{r}$, as desired.
Suppose now $\deg(u)>\deg(u_{0})$. If $u=x_{k_{1}}x_{k_{2}}\cdots
x_{k_{\ell}}x_{k_{\ell+1}}\cdots x_{k_{\deg(u)}}$, where $\ell=\deg(u_{0})$,
we set $u_{1}=x_{k_{1}}x_{k_{2}}\cdots x_{k_{\ell}}$. If $k_{\ell}\leq
j_{1}+\sum_{r=1}^{\ell-1}t_{r}$, then $u_{1}$ is ${\bf t}$–spread and
$k_{1}\leq j_{1},\ k_{2}\leq j_{2},\ \dots,\ k_{\ell}\leq j_{\ell}$, where
$j_{i}=j_{1}+\sum_{j=1}^{i-1}t_{j}$ ($i=1,\ldots,\ell$). Since $I$ is ${\bf
t}$–spread strongly stable, then $u_{1}\in I$. A contradiction, as $u_{1}$
divides $u$ and $u$ is a minimal generator. Therefore, we must have
$k_{\ell}>j_{1}+\sum_{r=1}^{\ell-1}t_{r}$ and consequently

$\max(u)-\sum_{r=1}^{\deg(u)-1}t_{r}\geq\max(u_{1})-\sum_{r=1}^{\deg(u_{1})-1}t_{r}>j_{1},$

which contradicts (7). $\Box$

###### Remark 4.4

From Theorem 4.3, the vector–spread Veronese ideal $I_{n,d,{\bf t}}$ is
Cohen–Macaulay. Moreover, ${\operatorname{pd}}(S/I_{n,d,{\bf
t}})=n-\sum_{j=1}^{d-1}t_{j}$. So $I_{n,d,{\bf t}}$ is a Cohen–Macaulay ideal
with pure resolution of type $(d_{1},\dots,d_{p})$, with $d_{j}=d+j-1$, for
$j=1,\dots,p$. Therefore, by [8, Theorem 4.1.15], we have for all $i\geq 1$,

$\displaystyle\beta_{i}(S/I_{n,d,{\bf t}})\ $ $\displaystyle=\
(-1)^{i+1}\prod_{j\neq
i}\frac{d_{j}}{d_{j}-d_{i}}=(-1)^{i+1}\prod_{j=1}^{i-1}\frac{d+j-1}{j-i}\prod_{j=i+1}^{p}\frac{d+j-1}{j-i}$
$\displaystyle=\
\binom{d+i-2}{d-1}\binom{n-\sum_{j=1}^{d-1}t_{j}+d-1}{d+i-1}.$

Note that for $i=1$, we obtain $\mu(I_{n,d,{\bf t}})=\beta_{1}(S/I_{n,d,{\bf
t}})=\binom{n-\sum_{j=1}^{d-1}t_{j}+d-1}{d}=|M_{n,d,{\bf t}}|$ and we get
again formula (1).

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