Related papers: The first Cotangent Cohomology Module for Matroids

We describe the action of the mod $2$ Steenrod algebra on the cohomology of various polyhedral products and related spaces. We carry this out for Davis-Januszkiewicz spaces and their generalizations, for moment-angle complexes as well as…

It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping…

In the present paper, we provide a cohomology group as a categorification of the characteristic polynomial of matroids. The construction depends on the ``quasi-representation'' of a matroid. For a certain choice of the quasi-representation,…

We introduce the concept of quotient-convergence for sequences of submodular set functions, providing, among others, a new framework for the study of convergence of matroids through their rank functions. Extending the limit theory of…

Let G be a finite group of Lie type, defined over a field k of characteristic p > 0. We find explicit bounds for the dimension of the first cohomology group for G with coefficients in a simple kG-module. We proceed by bounding the number of…

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to…

We initiate a systematic study on the cohomology rings of the moduli stack $\mathfrak{M}_{d,\chi}$ of semistable one-dimensional sheaves on the projective plane. We introduce a set of tautological relations of geometric origin, including…

We compute the Hochschild cohomology ring of the algebras $A= k\langle X, Y\rangle/ (X^a, XY-qYX, Y^a)$ over a field $k$ where $a\geq 2$ and where $q\in k$ is a primitive $a$-th root of unity. We find the the dimension of $\mathrm{HH}^n(A)$…

Quasiperiodic patterns described by polyhedral "atomic surfaces" and admitting matching rules are considered. It is shown that the cohomology ring of the continuous hull of such patterns is isomorphic to that of the complement of a torus…

Unstable modules over the Steenrod algebra with only the top $k$ operations are introduced in the language of ringoids. We prove the category of such modules has homological dimension at most $k$. A pratical method, which generalizes the…

In this paper, the principal tool to describe transversal polymatroids with Gorenstein base ring is polyhedral geometry, especially the $Danilov-Stanley$ theorem for the characterization of canonical module. Also, we compute the…

When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d>1, one obtains a finite group G^d which is often the automorphism group of an abstract regular polytope. Building…

In this PhD thesis we will discuss some aspects in Commutative Algebra which have interactions with Algebraic Geometry, Representation Theory and Combinatorics. In particular, in the first chapter we will focus on understanding when certain…

We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M, in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always…

We give a criterion for the complete reducibility of modules satisfying a composability condition for a meromorphic open-string vertex algebra $V$ using the first cohomology of the algebra. For a $V$-bimodule $M$, let…

One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this…

In this paper we completely classify which graded polynomial R-algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring…

Following the work of Gao and Xie in [2], we state some properties of the inverse Kazhdan-Lusztig polynomial of a matroid. We also give partial answers to a conjecture that states that regular connected matroids are non-degenerate. We link…

The Orlik-Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first we study homological properties of E-modules as e.g. complexity, depth and regularity. In particular, we consider modules with…

We compute the invariant subspace of the rational group ring of a surface, truncated by powers of the augmentation ideal, under the action of the mapping class group. The surface is compact, oriented with one boundary component. This…