6 problems

Let $P_0,\ldots,P_n$ be homogeneous polynomials in $cmathbb{C}[x_0,\ldots,x_n]$ with no common zeroes in $cmathbb{P}^n$, and set … For a general linear form $cell$ in $A_1$, consid…

Let $K$ be an algebraically closed field of characteristic zero, and let $A$ and $B$ be $K$-algebras essentially of finite type over $K$. Assume that $B$ is a finite-dimensional pr…

Pure-semisimplicity conjecture. Right pure-semisimplicity is left-right symmetric: every right pure-semisimple ring is left pure-semisimple. Equivalently, all right pure-semisimple…

Let $(R,\mathfrak{m})$ be an artinian Gorenstein ring whose embedding dimension is $3$. Ikeda's conjecture. … The conjecture concerns equality between the Dilworth number and the s…

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…

Finitistic dimension conjecture. The number $\operatorname{pfd}(A)$ is finite for every left artinian ring $A$.