Algebra retracts and Stanley-Reisner rings

TL;DR

This paper proves that graded algebra retracts of Stanley-Reisner rings are again Stanley-Reisner rings, providing evidence for a broader conjecture about monomial quotients of polynomial rings and their behavior under algebra retracts.

Contribution

It establishes that Stanley-Reisner rings are closed under graded algebra retracts, extending previous conjectures and providing partial evidence for a generalization.

Findings

Graded algebra retracts of Stanley-Reisner rings are Stanley-Reisner rings.

Supports conjecture on monomial quotients descending along algebra retracts.

Provides partial evidence for a broader conjecture in algebraic geometry.

Abstract

In a paper from 2002, Bruns and Gubeladze conjectured that graded algebra retracts of polytopal algebras over a field $k$ are again polytopal algebras. Motivated by this conjecture, we prove that graded algebra retracts of Stanley-Reisner rings over a field $k$ are again Stanley-Reisner rings. Extending this result further, we give partial evidence for a conjecture saying that monomial quotients of standard graded polynomial rings over $k$ descend along graded algebra retracts.