3 problems

Vasconcelos' SCM conjecture. If $I$ has deviation at least $3$, is generically a complete intersection, is not Gorenstein, and the resolution of $S/I$ is pure, then $I$ is not SCM.

Let $(A,m)$ be an Artinian local ring, and let $x_1,\ldots,x_n\in m$. Write $H_1(x_1,\ldots,x_n;A)$ for the first Koszul homology module of the sequence on $A$. Artinian Koszul hom…

Let $(R,m)$ be a regular local ring with Krull dimension $n$. Suppose $I+J$ is an $m$-primary ideal in $R$ and $R/I$ and $R/J$ are Cohen–Macaulay. The Tor generator conjecture. The…