On additivity of local entropy under flat extensions

Abstract

Let $f:(R,\mathfrak{m})\to S$ be a local homomorphism of Noetherian local rings. Consider two endomorphisms \textit{of finite length} (i.e., with zero-dimensional closed fibers) $\phi :R\to R$ and $\psi :S\to S$, satisfying $\psi \circ f=f\circ \phi$. Then $\psi$ induces a finite length endomorphism $\overline{\psi }:S/f(\mathfrak{m})S\to S/f(\mathfrak{m})S$. When $f$ is flat, under the assumption that $S$ is Cohen-Macaulay we prove an additivity formula: $h_{\mathrm{loc}}(\psi )=h_{\mathrm{loc}}(\phi )+h_{\mathrm{loc}}(\overline{\psi })$ for \textit{local entropy}.

5 pages, comments are welcome. In the second version the proof of the main result is re-written to provide more details. An example has been added to show a possible application of the main result in calculating local entropy

Cited by in corpus (1)

• Entropy in the category of perfect complexes with cohomology of finite length