Sudeshna Roy

Chennai Mathematical Institute · Chennai Mathematical Institute

This is an expository version of our paper [arXiv:1902.07384]. Our aim is to present recent Macaulay2 algorithms for computation of mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. These algorithms are based on computation of the equations of multi-Rees algebras of ideals that gene...

We study the question of finite generation of symbolic multi-Rees algebras and investigate the asymptotic behaviour of related length functions. In the setup of excellent local domains, we show that the symbolic multi-Rees algebra of a finite collection of ideals is finitely generated when the analytic spread is not maximal and the associated lengt...

Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for \(\mathbb N^{s}\)-graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for \(\mathbb N\)-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in 2-dimensional regular lo...

Let $R=K[X_1,\ldots, X_n]$ where $K$ is a field of characteristic zero, and let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. We give standard grading on $R$ and $A_n(K)$. Let $I$, $J$ be homogeneous ideals of $R$. Let $M = H^i_I(R)$ and $N = H^j_J(R)$ for some $i, j$. We show that $\Ext_{A_n(K)}^{\nu}(M,N)$ is concentrated in degree zero for all...

Let $k$ be a field of characteristic zero, and $R=k[x_1, \ldots, x_d]$ with $d \geq 3$ be a polynomial ring in $d$ variables. Let $\m=(x_1, \ldots, x_d)$ be the homogeneous maximal ideal of $R$. Let $\mathcal{K}$ be the kernel of the canonical map $\alpha: \sym(I) \rightarrow \R(I)$, where $\sym(I)$ (resp. $\R(I)$) denotes the symmetric algebra (re...

Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for ${\mathbb N}^s$-graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for $\mathbb N$-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in $2$-dimensional regular loca...

Let $A$ be a regular domain containing a field $K$ of characteristic zero, $G$ be a finite subgroup of the group of automorphisms of $A$ and $B=A^G$ be the ring of invariants of $G$. Let $S= A[X_1,\ldots, X_m]$ and $R= B[X_1, \ldots, X_m]$ be standard graded with $\ deg \ A=0$, $\ deg \ B=0$ and $\ deg \ X_i=1$ for all $i$. Extend the action of $G$...

Let A be a commutative Noetherian ring containing a field of characteristic zero. Let R = A[X1,…,Xm] be a polynomial ring and Am(A) = A〈X1,…,Xm, ∂1,…,∂m〉 be the m th Weyl algebra over A, where ∂i = ∂/∂Xi. Consider standard gradings on R and Am(A) by setting \(\deg z=0\) for all z ∈ A, \(\deg X_{i}=1\), and \(\deg \partial _{i} =-1\) for i = 1,…,m....