Related papers: Combinatorial presentation of multidimensional per…

Let $ R=k[x_1...x_r]$ and $M$ a multigraded $R-$module. In this work we interpret $M$ as a multipersistent homology module and give a multigraded resolution of it. The construction involves cellular resolutions of monomial ideals and…

Let $R$ be an associative ring with identity. This paper investigates the structure of the monomorphism category of large $R$-modules and establishes connections with the category of contravariant functors defined on finitely presented…

Persistent homology encodes the evolution of homological features of a multifiltered cell complex in the form of a multigraded module over a polynomial ring, called a multiparameter persistence module, and quantifies it through invariants…

A finitely generated module C over a commutative noetherian ring R is semidualizing if Hom_R(C,C) \cong R and Ext^i_R(C,C) = 0 for all i \geq 1. For certain local Cohen-Macaulay rings (R,m), we verify the equality of Hilbert-Samuel…

This paper deals with properties of filtrations on vector spaces indexed by partially ordered finitely generated abelian groups, which we call multifiltrations. We discuss the usual properties of filtrations, like exhaustivity and…

In this work, we develop a new theory of multivariate V-filtration on D-modules along a simple normal crossing divisor and relate it with Sabbah's multi-filtration. We establish several new structural results and relate them with the Hodge…

We introduce an exact functor defined on multigraded modules which we call the expansion functor and study its homological properties. The expansion functor applied to a monomial ideal amounts to substitute the variables by monomial prime…

Let $R$ be a ring essentially of finite type over an $F$-finite field. Given an ideal $\mathfrak{a}$ and a principal Cartier module $M$ we introduce the notion of a $V$-filtration of $M$ along $\mathfrak{a}$. If $M$ is $F$-regular then this…

For $i : Z \to X$ a closed immersion of smooth varieties, we study how the $V$-filtration along $Z$ and the Hodge filtration on a mixed Hodge module $M$ on $X$ interact with each other. We also give a formula for the functors $i^*$, $i^!$…

A differential modality is a comonad on an additive symmetric monoidal category $(\mathsf{C},\otimes,I)$, whose underlying functor we denote $!\colon\mathsf{C} \rightarrow \mathsf{C}$, together with some additional structure including a…

Multisorted modules, equivalently representations of quivers, equivalently additive functors on preadditive categories, encompass a wide variety of additive structures. In addition, every module has a natural and useful multisorted…

A multiparameter filtration, or a multifiltration, may in many cases be seen as the collection of sublevel sets of a vector function, which we call a multifiltering function. The main objective of this paper is to obtain a better…

A filtration of a representation whose successive quotients are isomorphic to Demazure modules is called an excellent filtration. In this paper we study graded multiplicities in excellent filtrations of fusion products for the current…

Let $R\to A$ be a homomorphism of associative rings, and let $(\mathcal F,\mathcal C)$ be a hereditary complete cotorsion pair in $R\mathsf{-Mod}$. Let $(\mathcal F_A,\mathcal C_A)$ be the cotorsion pair in $A\mathsf{-Mod}$ in which…

The coprimary filtration is a basic construction in commutative algebra. In this article, we prove the existence and uniqueness of coprimary filtration of modules (not necessarily finitely generated) over a Noetherian ring. Moreover, we…

In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, generalizing the classical theory for $m_R$-primary ideals. We…

The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth…

Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this…

Given a finite set $A\subset\mathbb{R}^d$, let Cov$_{r,k}$ denote the set of all points within distance $r$ to at least $k$ points of $A$. Allowing $r$ and $k$ to vary, we obtain a 2-parameter family of spaces that grow larger when $r$…

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ be an ideal of $R$. Suppose $M$ is a finitely generated $R$-module and $N$ is an Artinian $R$-module. We define the concept of filter coregular sequence to determine the infimum of…