Related papers: Monomial Cycle Basis on Koszul Homology Modules

For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathscr{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the…

Using reduction to positive characteristic and the method of Frobenius splitting of diagonals, due to Mehta and Ramanathan, it is shown that homogeneous coordinate rings for either proper and smooth toric varieties or Schubert varieties are…

This paper concerns the homological properties of a module $M$ over a commutative noetherian ring $R$ relative to a presentation $R\cong P/I$, where $P$ is local ring. It is proved that the Betti sequence of $M$ with respect to $P/(f)$ for…

Boij-S\"oderberg theory gives a combinatorial description of the set of Betti tables belonging to finite length modules over the polynomial ring $S = k[x_1, \ldots, x_n]$. We posit that a similar combinatorial description can be given for…

In his Ph.D. thesis, Sean Griffin introduced a family of ideals and found monomial bases for their quotient rings. These rings simultaneously generalize the Delta Conjecture coinvariant rings of Haglund-Rhoades-Shimozono and the cohomology…

Fix a poset $Q$ on $\{x_1,\ldots,x_n\}$. A $Q$-Borel monomial ideal $I \subseteq \mathbb{K}[x_1,\ldots,x_n]$ is a monomial ideal whose monomials are closed under the Borel-like moves induced by $Q$. A monomial ideal $I$ is a principal…

Consider the ring $\mathcal{S}$ of symmetric polynomials in $k$ variables over an arbitrary base ring $\mathbf{k}$. Fix $k$ scalars $a_{1},a_{2},\ldots,a_{k}\in\mathbf{k}$. Let $I$ be the ideal of $\mathcal{S}$ generated by…

In this paper, we consider homological properties of so-called graph ideals. Consider $\Gamma$ is a graph with vertices $t_1$, ..., $t_s$, without self-loops and multiple adjacencies. We can associate with such a graph an ideal…

Independent sets play a key role into the study of graphs and important problems arising in graph theory reduce to them. We define the monomial ideal of independent sets associated to a finite simple graph and describe its homological and…

This is a report on recent work of Chalupnik and Touze. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we…

Let $g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $g$ acts on a complex of vector spaces $M$ by $i_\lambda$ and $L_\lambda$, which satisfy the identities as contraction and Lie derivative do for smooth…

A monomial algebra B is defined as a quotient of a polynomial ring by a monomial ideal, which is an ideal generated by a finite set of monomials. In this paper, we determine the automorphism group of a monomial algebra B, under the…

For a commutative ring R with an ideal I, generated by a finite regular sequence, we construct differential graded algebras which provide R-free resolutions of I^s and of R/I^s for s>0 and which generalise the Koszul resolution. We derive…

When k is an algebraically closed field of characteristic 0 and H is a non-semisimple monomial Hopf algebra, we show that all Galois objects over H are determined up to H-comodule algebra isomorphism by their polynomial H-identities,…

This paper investigates the independence polynomials arising from iterated strong products of cycle graphs, examining their algebraic symmetries and combinatorial structures. Leveraging modular arithmetic and Galois theory, we establish…

We show that for cluster algebras associated with finite quivers without oriented cycles (with no coefficients), a seed is determined by its cluster, as conjectured by Fomin and Zelevinsky.We also obtain an interpretation of the monomial in…

If the inverse of a nonsingular polynomial matrix $L$ has a polynomial part then one can associate with $L$ a module over the ring of proper rational functions, which is related to the structure of $L$ at infinity. In this paper we…

The aim of this article is to investigate the cohomology (l-adic as well as Betti) of schemes, and more generally of certain algebraic stacks, that are proper and smooth over the integers and have the property that there exists a polynomial…

In this paper we investigate the question of normality for special monomial ideals in a polynomial ring over a field. We first include some expository sections that give the basics on the integral closure of a ideal, the Rees algebra on an…

We study Koszul homology over Gorenstein rings. If an ideal is strongly Cohen-Macaulay, the Koszul homology algebra satisfies Poincar\'e duality. We prove a version of this duality which holds for all ideals and allows us to give two…