Semigroup rings

Abstract: Let $A = \{a_{ij}\}$ be an $n \times m$ matrix of natural numbers $\mathbb N.$ The $S(A)$ denotes the sub-semigroup of $\mathbb N^n$ generated by the columns of $A$. The semigroup ring of $A$ over a field $k$, denoted by $k[A]$ is the homomorphic image of $\phi: k[x_1, \ldots, x_m] \to k[t_1, \ldots, t_n]$ defined by $\phi (x_j) = \prod_{i=1}^nt_i^{a_{ij}}$ and hence $k[A]$ is isomorphic to $k[x_1, \ldots, x_m]/I_A$. In this talk, we will discuss various invariants of $k[A]$, such as depth, dimension, Frobenius numbers and homological properties, such as resolutions, Betti Numbers, regularity and Hilbert Series. Recent work on gluing and its relation to these invariants will be outlined. We will compare the situation in numerical semigroups (subgroups of $\mathbb N$) to semigroups of higher dimension and which of the many formulas and structures generalize to higher dimensions.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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