Normality and Related Properties of Forcing Algebras

TL;DR

This paper establishes criteria for normality and irreducibility of forcing algebras, providing geometric insights and explicit computations, thereby advancing the understanding of their algebraic and geometric properties.

Contribution

It introduces new criteria for normality and irreducibility of forcing algebras, including explicit normalization computations and a detailed example.

Findings

Provided a sufficient condition for irreducibility.

Proved a normality criterion over polynomial base rings.

Explicitly computed the normalization of a specific forcing algebra.

Abstract

We present a sufficient condition for irreducibility of forcing algebras and study the (non)-reducedness phenomenon. Furthermore, we prove a criterion for normality for forcing algebras over a polynomial base ring with coefficients in a perfect field. This gives a geometrical normality criterion for algebraic (forcing) varieties over algebraically closed fields. Besides, we examine in detail an specific (enlightening) example with several forcing equations. Finally, we compute explicitly the normalization of a particular forcing algebra by means of finding explicitly the generators of the ideal defining it as an affine ring.