On additivity of local entropy under flat extensions

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

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