CyberSec Research

Let $I$ be a polymatroidal ideal. In this paper, we study the asymptotic behavior of the homological shift ideals of powers of polymatroidal ideals. We prove that the first homological shift algebra $\text{HS}_1(\mathcal{R}(I))$ of $I$ is generated in degree one as a module over the Rees algebra $\mathcal{R}(I)$ of $I$. We conjecture that the $i$th homological shift algebra $\text{HS}_i(\mathcal{R}(I))$ of $I$ is generated in degrees $\le i$, and we confirm it in many significant cases. We show that $I$ has the $1$st homological strong persistence property, and we conjecture that the sequence $\{\text{Ass}\,\text{HS}_i(I^k)\}_{k>0}$ of associated primes of $\text{HS}_i(I^k)$ becomes an increasing chain for $k\ge i$. This conjecture is established when $i=1$ and for many families of polymatroidal ideals. Finally, we explore componentwise polymatroidal ideals, and we prove that $\text{HS}_1(I)$ is again componentwise polymatroidal, if $I$ is componentwise polymatroidal.

We give results on reduced complex-analytic curve germs which relate their indecomposable maximal Cohen-Macaulay (MCM) modules to their lattice homology groups and related invariants, thereby providing a connection between the algebraic theory of MCM modules and techniques arising from low-dimensional topology. In particular, we characterize the germs $(C, o)$ of finite Cohen-Macaulay type in terms of the lattice homology $\mathbb{H}_*(C, o)$, and those of tame type in terms of the lattice homologies and associated spectral sequences of $(C, o)$ and its subcurves, including the distinction between germs of finite and infinite growth. As a consequence of these results, we obtain corresponding characterizations of a germ's Cohen-Macaulay type in terms of the motivic Poincar\'e series.

We introduce a new class of rings, pseudo-krullian orders, consider the Serre quotients of their module categories with respect to pseudo-isomorphisms and describe injective objects in such quotient categories and its global homological dimension. These results generalize the results of I. Beck for the case of Krull rings. In particular, we establish the global homological dimension of the category of maximal Cohen-Macaulay modules over an order over a noetherian ring of Krull dimension 2.

Hanlon, Hicks and Lazarev constructed resolutions of structure sheaves of toric substacks by certain line bundles on the ambient toric stacks. In this paper, we give a new and substantially simpler proof of their result.

For a henselian valued field $(K,v)$ and a separable-algebraic element $a\in\overline{K}\setminus K$, we consider the set $S_K(a):= \{ v(a-a^\prime) \mid a^\prime\neq a \text{ is a $K$-conjugate of $a$} \}$. The central aim of this paper is to provide a bound for the cardinality of the set $S_K(a)$, and to characterize the elements $a$ for which this set is a singleton. Connections of this set with the notion of \textit{depth} of $a$ has also been explored. We show that $S_K(a)$ is a singleton whenever $K(a)|K$ is a minimal extension. A stronger version of this result is obtained when $a$ has depth one over $K$. We also provide a host of examples illustrating that the bounds obtained are strict. Apart from being of independent interest, another primary motivation for considering this problem comes from the study of ramification ideals. In the depth one case, when $K(a)|K$ is a Galois extension, we obtain intimate connections between the cardinalities of $S_K(a)$ and the number of ramification ideals of the extension $(K(a)|K,v)$. In particular, we show that these cardinalities are same whenever the extension is defectless and non-tame, or whenever $(K,v)$ has rank one. In order to obtain these results, we provide comprehensive descriptions of the ramification ideals of $(K(a)|K,v)$ which extend the known results in this direction.

We establish a bulk--boundary correspondence for translation-invariant stabilizer states in arbitrary spatial dimension, formulated in the framework of modules over Laurent polynomial rings. To each stabilizer state restricted to half-space geometry we associate a boundary operator module. Boundary operator modules provide examples of quasi-symplectic modules, which are objects of independent mathematical interest. In their study, we use ideas from algebraic L-theory in a setting involving non-projective modules and non-unimodular forms. Our results about quasi-symplectic modules in one spatial dimension allow us to resolve the conjecture that every stabilizer state in two dimensions is characterized by a corresponding abelian anyon model with gappable boundary. Our techniques are also applicable beyond two dimensions, such as in the study of fractons.

Let $X$ be a matrix of indeterminates, $t$ an integer, and $P_t(X)$ define the ideal generated by the permanents of all $t\times t$ submatrix of $X$. $P_t(X)$ is called a permanental ideal. In this article, we study the algebras $\Bbbk[X]/P_t(X)$ where $X$ is a generic, symmetric, or a Hankel matrix of indeterminates. When $\operatorname{char}\Bbbk = 2$, $P_t(X)$ is also known as a determinantal ideal, a popular class in commutative algebra and algebraic geometry, and thus many properties of $P_t(X)$ are known in this case. We prove that, if $X$ is an $n\times n$ matrix and $\operatorname{char} \Bbbk>2$, the algebra $\Bbbk[X]/P_n(X)$ is $F$-regular, just like when $\operatorname{char} \Bbbk = 2$. On the other hand, we obtain a full characterization of when $\Bbbk[X]/P_2(X)$ is $F$-pure or $F$-regular, when $\operatorname{char} \Bbbk >2$, and the answer is different than that in even characteristic.

This paper systematically develops a notion of regular sequences in the context of $R$-linear triangulated categories for a graded-commutative ring $R$. The notion has equivalent characterizations involving Koszul objects and local cohomology. The main examples are in the context of the Hochschild cohomology ring or the group cohomology ring acting on derived or stable categories. As applications, lengths of regular sequences provide lower bounds for level and Rouquier dimension.

In this note, we provide three new, very short proofs of two interesting congruences for Merca's partition function $a(n)$, which enumerates integer partitions where the odd parts have multiplicity at most 2. These modulo 2 congruences were first shown elementarily by Sellers. We then frame $a(n)$ into the much broader context of eta-quotients, and suggest how to comprehensively describe its parity behavior. In particular, extensive computations suggest that $a(n)$ is odd precisely 25\% of the time.

Boolean matrix factorization (BMF) has many applications in data mining, bioinformatics, and network analysis. The goal of BMF is to decompose a given binary matrix as the Boolean product of two smaller binary matrices, revealing underlying structure in the data. When interpreting a binary matrix as the adjacency matrix of a bipartite graph, BMF is equivalent to the NP-hard biclique cover problem. By approaching this problem through the lens of commutative algebra, we utilize algebraic structures and techniques--particularly the Castelnuovo-Mumford regularity of combinatorially defined ideals--to establish new lower bounds for Boolean matrix rank.

A foundational result by C. Huneke and V. Trivedi provides a formula for the depth of an ideal in terms of height, computed over a finite set of prime ideals, for rings that are homomorphic images of regular rings. Building on a result by the first author for local quotients of Cohen-Macaulay rings, this paper first gives a new proof and derives a similar formula for the finiteness dimension. Our main result then establishes the depth formula for non-local rings that are homomorphic images of a finite-dimensional Gorenstein ring.

Parke-Taylor functions are certain rational functions on the Grassmannian of lines encoding MHV amplitudes in particle physics. For $n$ particles there are $n!$ Parke-Taylor functions, corresponding to all orderings of the particles. Linear relations between these functions have been extensively studied in the last years. We here describe all non-linear polynomial relations between these functions in a simple combinatorial way and study the variety parametrized by them, called the Parke-Taylor variety. We show that the Parke-Taylor variety is linearly isomorphic to the log canonical embedding of the moduli space $\overline{\mathcal{M}}_{0,n}$ due to Keel and Tevelev, and that the intersection with the algebraic torus recovers the open part, $\mathcal{M}_{0,n}$. We give an explicit description of this isomorphism. Unlike the log canonical embedding, this Parke-Taylor embedding respects the symmetry of the $n$ marked points and is constructed in a single-step procedure, avoiding the intermediate embedding into a product of projective spaces.

Let $D$ be an integrally closed domain with quotient field $K$ and $A$ a torsion-free $D$-algebra that is finitely generated as a $D$-module and such that $A\cap K=D$. We give a complete classification of those $D$ and $A$ for which the ring $\text{Int}_K(A)=\{f\in K[X] \mid f(A)\subseteq A\}$ is a Pr\"ufer domain. If $D$ is a semiprimitive domain, then we prove that $\text{Int}_K(A)$ is Pr\"ufer if and only if $A$ is commutative and isomorphic to a finite direct product of almost Dedekind domains with finite residue fields, each of them satisfying a double-boundedness condition on its ramification indices and residue field degrees.

By using logarithmic $\mathcal D$-modules and Gr\"obner bases, we prove that Bernstein-Sato ideals satisfy some symmetric intersection property, answering a question posed by Budur. As an application, we obtain a formula for the Bernstein-Sato polynomials of $f^n$, the integer powers of a multi-variable polynomial $f$.

We identify a close relationship between stable sheaf cohomology for polynomial functors applied to the cotangent bundle on projective space, and Koszul--Ringel duality on the category of strict polynomial functors as described in the work of Cha\l upnik, Krause, and Touz\'e. Combining this with recent results of Maliakas--Stergiopoulou we confirm a conjectured periodicity statement for stable cohomology. In a different direction, we find a remarkable invariance property for $\Ext$ groups between Schur functors associated to hook partitions, and compute all such extension groups over a field of arbitrary characteristic. We show that this is further equivalent to the calculation of $\Ext$ groups for partitions with $2$ rows (or $2$ columns), and as such it relates to Parker's recursive description of $\Ext$ groups for $\SL_2$-representations. Finally, we give a general sharp bound for the interval of degrees where stable cohomology of a Schur functor can be non-zero.

For an increasing weighted tree $G_\omega$, we obtain an asymptotic value and a sharp bound on the index stability of the depth function of its edge ideal $I(G_\omega)$. Moreover, if $G_\omega$ is a strictly increasing weighted tree, we provide the minimal free resolution of $I(G_\omega)$ and an exact formula for the regularity of all powers of $I(G_\omega)$.

For the edge ideal $I(\D)$ of a weighted oriented graph $\D$, we prove that its symbolic powers $I(\D)^{(t)}$ are Cohen-Macaulay for all $t\geqslant 1$ if and only if the underlying graph $G$ is composed of a disjoint union of some complete graphs. We also completely characterize the Cohen-Macaulayness of the ordinary powers $I(\D)^t$ for all $t\geqslant 2$. Furthermore, we provide a criterion for determining whether $I(\D)^t=I(\D)^{(t)}$.

We consider the finest grading of the algebra of regular functions of a trinomial variety. An explicit description of locally nilpotent derivations that are homogeneous with respect to this grading is obtained.