Density function associated to a module and integral dependence

Time:

Venue/Location: Phòng C101, VIASM, 157 phố Chùa Láng, Hà Nội

Báo cáo viên: Sudeshna Roy, TIFR, India

Tóm tắt: A density function for an algebraic invariant is a measurable function on the set of real numbers which measures the invariant on a real scale. In this talk, we will discuss density functions for Noetherian filtrations of homogeneous ideals in a standard graded Noetherian domain over a field $k$. This was inspired by the Hilbert-Kunz density functions developed by V. Trivedi. We will also demonstrate a density function for a finitely generated bigraded module over a bigraded Noetherian $k$-algebra, which is generated in bidegrees $(1,0), (d_1, 1), \ldots, (d_r,1)$ for some $d_i \geq 0$. Our main ingredients are the method of Gr\"obner bases and Sturmfels' structure theorem for vector partition functions. As an application, we will provide a new numerical criterion for integral dependence of arbitrary homogeneous ideals in terms of computable and well-studied invariants, such as, mixed multiplicities of ideals and Hilbert-Samuel multiplicities of certain standard graded algebras. A novelty of our approach is that it does not involve localizations. This talk is based on joint works with Suprajo Das and Vijaylaxmi Trivedi.