Coefficient and elimination algebras in Resolution of Singularities

TL;DR

This paper compares various algebras used in resolving singularities, introduces a new algebraic structure encoding similar information, and discusses their application in characteristic zero cases.

Contribution

It constructs an algebra equivalent to existing structures, providing a unified approach to resolution of singularities in positive and zero characteristic.

Findings

Constructed an algebra encoding previous structures' information

Established equivalence up to integral closure of the algebras

Applied structures to obtain resolution in characteristic zero

Abstract

We compare some algebras appeared in the recent attempts to prove resolution of singularities in positive characteristic. We also construct an algebra which encodes the same information and it is equivalent, up to integral closure, to the previous structures. In the case of characteristic zero one may use these structures to obtain a resolution of singularities.