7 problems

Multiplicity inequality. The following inequality holds:

Lech's conjecture. Then

Let $R$ be a local ring with maximal ideal, and let $F$ be a short complex supported on the maximal ideal. Write $\chi_{\infty}(F)$ for the Dutta multiplicity of $F$. Iyengar–Ma–Wa…

Let $R$ be a local ring, and let $M$ be a nonzero $R$-module of finite projective dimension. Iyengar–Ma–Walker length–multiplicity conjecture. … This is one of two multiplicity con…

Let $I\subset J$ be arbitrary ideals in an equidimensional and universally catenary Noetherian local ring. The multiplicity sequence is the sequence of normalized leading coefficie…

Asymptotic Lech's inequality.

Let $A$ be a regular local ring. Let $M$ and $N$ be equidimensional, finitely generated $A$-modules such that … and $\ell(M\otimes_A N)<\infty$. Write $\operatorname{gr}A$ for the…