Related papers: Gr\"obner bases and Betti numbers of monoidal comp…

Let $I_1\subset I_2\subset\dots$ be an increasing sequence of ideals of the ring $\Bbb Z[X]$, $X=(x_1,\dots,x_n)$ and let $I$ be their union. We propose an algorithm to compute the Gr\"obner base of $I$ under the assumption that the…

This paper considers graded near-rings over a monoid G as a generalizations of the graded rings over groups, introduce certain innovative graded weakly prime ideals and graded almost prime ideals as a generalizations of graded prime ideals…

If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,\k)$ is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, $\phi_k$, of the lower central series quotients of $\pi_1(M)$ can be computed from the…

It has been shown previously that a large class of monomial maps equivariant under the action of an infinite symmetric group have finitely generated kernels up to the symmetric action. We prove that these symmetric toric ideals also have…

To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By…

This paper provides a formula for the Mather-Jacobian multiplier ideals of torus invariant ideals on (not necessarily normal) toric varieties that generalizes Howald's formula for the multiplier ideal of monomial ideals in a polynomial…

Monomial ideals and toric rings are closely related. By consider a Grobner basis we can always associated to any ideal $I$ in a polynomial ring a monomial ideal ${\rm in}_\prec I$, in some special situations the monomial ideal ${\rm…

The asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field has recently been reviewed. We extend quasi polynomial behavior of graded Betti numbers of powers of homogenous ideals to…

We study the structure of a family of algebras which encodes a generalization of the Pieri Rule for the complex orthogonal group. In particular, we show that each of these algebras has a standard monomial basis and has a flat deformation to…

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…

We study ideals generated by $n+1$ powers of general linear forms in $R= k[x_1,\dots,x_n]$. By generalizing the ideas in a recent paper of Diethorn et al., we determine the Betti numbers of such ideals when at least one generator is a…

From a system consisting of a right non-degenerate ring $R$, a pair of $R$-bimodules $Q$ and $P$ and an $R$-bimodule homomorphism $\psi:P\otimes Q\longrightarrow R$ we construct a $\Z$-graded ring $\mathcal{T}_{(P,Q,\psi)}$ called the…

We consider the relationship between the Stanley-Reisner ring (a.k.a. face ring) of a simplicial or boolean complex $\Delta$ and that of its barycentric subdivision. These rings share a distinguished parameter subring. S. Murai asked if…

Relying on the combinatorial classification of toric ideals using their bouquet structure, we focus on toric ideals of hypergraphs and study how they relate to general toric ideals. We show that hypergraphs exhibit a surprisingly general…

We express the multigraded Betti numbers of monomial ideals in 4 variables in terms of the multigraded Betti numbers of 66 squarefree monomial ideals, also in 4 variables. We use this class of 66 ideals to prove that monomial resolutions in…

We develop the theory of integrable representations for an arbitrary maximal parabolic subalgebra of an affine Lie algebra. We see that such subalgebras can be thought of as arising in a natural way from a Borel--de Siebenthal pair of…

We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and…

Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the…

Theory of representations of F-algebra is a natural development of the theory of F-algebra. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. In the book I considered the…

We give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of any homogeneous ideal in a polynomial ring over a field.