# The Hochschild cohomology ring of the numerical semigroup algebras of embedding dimension two

## TL;DR

This paper explicitly determines the Hochschild cohomology ring structure for numerical semigroup algebras generated by two coprime integers, including generators, relations, and Hilbert series, enriching understanding of their algebraic properties.

## Contribution

It provides a complete description of the Hochschild cohomology ring for two-generated numerical semigroup algebras, including generators, relations, and Hilbert series.

## Key findings

- Ring structure explicitly determined
- Generators and relations identified
- Hilbert series computed

## Abstract

Let $a$ and $b$ be two coprime positive integers and $k$ an arbitrary field. We determine the ring structure of the Hochschild cohomology of the numerical semigroup algebras $k[s^{a},s^{b}]$ of embedding dimension two (thus also complete intersections) in terms of generators and relations. In addition, we compute the Hilbert series for this cohomology ring.