The Dutta-multiplicity bound for short complexes

Let $(R,\mathfrak m)$ be a local ring, let $e(R)$ denote its Hilbert–Samuel multiplicity, and let $F$ be a short complex supported on the maximal ideal, meaning a non-exact complex of finite free $R$-modules

$$0\longrightarrow F_{\dim R}\longrightarrow\cdots\longrightarrow F_1\longrightarrow F_0\longrightarrow0$$

whose homology modules have finite length. In positive characteristic, let $\chi_\infty(F)$ denote its Dutta multiplicity. Dutta-multiplicity bound. Every short complex $F$ supported on the maximal ideal satisfies

$$\chi_\infty(F)\geq e(R).$$

The paper proves this bound when $R$ is the localization at its homogeneous maximal ideal of a standard graded algebra over a field of positive characteristic. The proposed extension to arbitrary local rings remains open.