Cohomological Comparison Theorem

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