Narrow Results

In this paper, we study the Dao numbers ${𝔡}_1(I),{𝔡}_2(I)$ and ${𝔡}_3(I)$ of an ideal $I$ of a Noetherian local ring $(R,𝔪,K)$ or a standard graded Noetherian $K$-algebra. They are defined as the smallest $\ell \ge 0$ such that $I{𝔪}^k$ is $𝔪$-full, full, weakly $𝔪$-full, respectively, for all $k\ge \ell$. We provide general bounds for the Dao numbers in terms of the Castelnuovo–Mumford regularity of certain modules over the Rees algebra ${ℛ}(𝔪)$. If $R$ is a Koszul algebra, we prove that the Dao numbers are less or equal to ${\mathrm{\text{reg}}}_{{\mathrm{\text{gr}}}_{𝔪}(R)}{\mathrm{\text{gr}}}_{𝔪}(I)$, where ${\mathrm{\text{gr}}}_{𝔪}(I)$ is the associated graded module of $I$. Finally, for monomial ideals, we combinatorially bound the Dao numbers in terms of asymptotic linear quotients and bounding multidegrees.

I construct a Koszul algebra A and a finitely generated graded A-module M that together form a counterexample to a recently published claim. M is generated in degree 0 and has a pure resolution, and the graded Jacobson radical of the Yoneda algebra of A does not annihilate the Ext module of M, but nonetheless M is not a Koszul module.

In this paper, we describe the algebras $A({\mathrm{\Gamma }}_{K_n})$ associated with the Hasse graph of the poset of the faces of the complete graph at n vertices $K_n$. Present the relations that define this algebra, calculate the Hilbert series of quadratic grade algebra $A({\mathrm{\Gamma }}_{K_n})$ and the graded trace generating functions of $D_n$ acting on $A({\mathrm{\Gamma }}_{K_n})$ and show that $A({\mathrm{\Gamma }}_{K_n})$ is a Koszul algebra. The methodology adopted in this paper consists of building the algebra $A({\mathrm{\Gamma }}_{K_n})$ using [I. Gelfand, V. Retakh, S. Serconek and R. L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. 11(2) (2005) 281], giving a presentation based on what was established by [V. Retakh, S. Serconek and R. L. Wilson, On a class of koszul algebras associated to directed graphs, J. Algebra 304(2) (2006) 1114–1129] in terms of the vertices of the graph ${\mathrm{\Gamma }}_{K_n}$. After that, we use [V. Retakh, S. Serconek and R. L. Wilson, Hilbert series of algebras associated to directed graphs, J. Algebra 312(1) (2007) 142–151] to determine the Hilbert series of $A({\mathrm{\Gamma }}_{K_n})$ and use [9] again to show that this algebra is a Koszul algebra. We use [J. Caldeira, A. De Souza Lima and J. Eder Salvador De Vasconcelos, Representations of automorphism groups of algebras associated to star polygons, J. Algebra Appl. 18(10) (2019) 1950197; C. Duffy, Representations of Aut(A($\mathrm{\Gamma }$)) acting on homogeneous components of $A(\mathrm{\Gamma })$ and $A(\mathrm{\Gamma })!$, Adv. Appl. Math. 42(1) (2009) 94–122] to determine the generating functions of the graduated trace of $D_n$ acting on $A({\mathrm{\Gamma }}_{K_n})$.

In this note we describe a representation theoretic approach to functorial functor valued knot invariants with the focus on (categorified) Schur-Weyl dualities. Applications include categorified Reshetikhin-Turaev invariants, an extension of Khovanov homology and a diagrammatical description of the category of finite dimensional GL(m|n)-modules.