7 problems

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring with its standard grading, let $I\subset S$ be a graded ideal, and let $R=S/I$ be a standard graded $K$-algebra. Write…

Let $V$ be a finite set, let $R=\Bbbk[V]$ be a standard graded polynomial ring, and let $I$ be a homogeneous ideal of height $c$. Let $e(R/I)$ be the Hilbert-Samuel multiplicity, a…

Huneke–Srinivasan–Herzog–Srinivasan conjecture. If $R/I$ is Cohen–Macaulay, then

Let $I$ be the ideal generated by the entries of $XY-YX$ for generic $n$ by $n$ matrices, and let $J$ be generated by the off-diagonal entries. Let $d_{\min}$ be the quantity from…

Let $p$ be a prime. For each $r\geq 1$, define … For an integer $i\geq 0$, let $A_i^R(M)$ be the multiset in which $j$ occurs with multiplicity $\beta_{i,j}^R(M)$, for a graded mod…

Polynomial Betti-bound conjecture. All Betti numbers of $S/I$ are bounded, as functions of $d$, by polynomials of degree at most $p-1$.

Herzog-Srinivasan's Taylor multiplicity bound conjecture.