Cohomological Comparison Theorem

TL;DR

This paper establishes conditions under which the cohomology rings of a ring and a related subring are eventually isomorphic, enabling comparison of their algebraic properties such as finite generation, GK dimension, and global dimension.

Contribution

It provides new sufficient conditions for cohomology ring isomorphism and compares algebraic invariants between a ring and its idempotent subring.

Findings

Cohomology rings are eventually isomorphic under certain conditions.

Finite generation and GK dimension are comparable between the rings.

Global dimensions of the rings can be compared.

Abstract

If $f$ is an idempotent in a ring $\Lambda$, then we find sufficient \linebreak conditions which imply that the cohomology rings $\oplus_{n\ge 0}Ext^n_{\Lambda}(\Lambda/{\br},\Lambda/{\br})$ and \linebreak $\oplus_{n\ge 0}Ext^n_{f\Lambda f}(f\Lambda f/f{\br} f,f\Lambda f/f{\br} f)$ are eventually isomorphic. This result allows us to compare finite generation and GK dimension of the cohomology rings $\Lambda$ and $f\Lambda f$. We are also able to compare the global dimensions of $\Lambda$ and $f\Lambda f$.