3 problems

Let $R$ be a complete local ring, and let $F$ be a short complex in the perfect-complex category, meaning a short complex in $\operatorname{perf} R$ supported on the maximal ideal.…

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…

Let $(R,\mathfrak m)$ be a local ring, let $e(R)$ denote its Hilbert–Samuel multiplicity, and let $\ell_R M$ denote the length of an $R$-module $M$. Minimal length conjecture. For…