12 problems

Let $A=S/I$ be a Cohen–Macaulay quotient ring of codimension $c$. In each homological term of its minimal free resolution, let $a_i$ be the maximal twist and $b_i$ the minimal twis…

Lech's conjecture. Then

Let $R$ be a local ring, and let $M$ be a nonzero $R$-module of finite length and finite projective dimension. Let $\ell_R(M)$ denote its length and let $e(R)$ denote the Hilbert–S…

Asymptotic Lech inequality. In this case, the asymptotic supremum of the normalized multiplicities $e(J)/d!$ over $\mu$-primary ideals of length tending to infinity should satisfy

Multiplicity inequality. The following inequality holds:

Multiplicative-formula conjecture. There exists a multiplicative formula in $X_{d,n}$ similar to the known formula for a face that is a single suitable block, for a face that is a…

Let $S$ be a polynomial ring and $I\subset S$ an ideal. Let $e(S/I)$ denote the multiplicity, and let the lower and upper bounds be the quantities in the Herzog–Huneke–Srinivasan m…

Teissier's conjecture. The Minkowski inequality

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

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…

Let $R=K[x_0,\ldots,x_n]$, let ${\mathfrak p}\in\operatorname{mProj}(R)$, let ${\mathfrak a}\subseteq R$ be a graded ideal with ${\mathfrak a}\not\subseteq{\mathfrak p}$, and let…