# Some Results on Betti Series of Universal Modules of Differential Operators

## TL;DR

This paper investigates the rationality of Betti series for universal modules of differential operators, establishing conditions under which the series is rational for certain algebraic structures.

## Contribution

It proves the rationality of Betti series for universal modules of derivations over coordinate rings of affine irreducible curves with at most one singularity.

## Key findings

- Betti series is rational for coordinate rings of affine irreducible curves with at most one singularity
- Established conditions for the rationality of Betti series in this context
- Provides new insights into the structure of universal modules of differential operators

## Abstract

In this article, we discuss the rationality of the Betti Series of the universal module of nth order derivations of R_{m} where m is a maximal ideal of R. We proved that if R is a coordinate ring of an affine irreducible curve and if it has at most one singularity point, then the Betti series is rational.