11 problems

Let $R$ be the complete intersection ring and let $\mathcal C_{\bar{\psi}}(r)$ be the generalized Koszul complex appearing above, with cohomology modules…

Let $S_n$ be the symmetric group, let $R$ be the polynomial ring used to define the complexes $K_r$, and let $\mathcal{A}_w$ be the dg algebra whose modules are being considered. F…

Let $A=k\langle x^{\pm1},y^{\pm1}\rangle$, with both generators invertible, and let $\vartheta_{ij}$ denote the Koszul variables for the commutator relation. Write $X=(x_{ij})$ and…

Let $A=k\langle x,y\rangle$ and let $\mathcal{K}_n(A)$ be the Koszul complex for the commuting-matrix relation $[x,y]=0$. Write $x_{uv},y_{uv}$ for the matrix-entry variables,…

Let $R$ be a Cohen–Macaulay ring, let $\mathfrak a\subseteq I$ be ideals, and write … Assume that $J$ is an algebraic $s$-residual intersection. The Kitt ideal…

Let $R$ be a Cohen–Macaulay local ring of dimension $d$, let $I$ be an ideal satisfying sliding depth, and let $s$ be the residual-intersection parameter. An algebraic $s$-residual…

Let $R$ be a commutative ring, let $I$ be an ideal with sliding depth, and let $J=\mathfrak a:_R I$ be an algebraic $s$-residual intersection. A disguised residual intersection is…

Sancho's acyclicity conjecture. The complex $\operatorname{Kos}(M)$ is always acyclic.

The generation conjecture. The homology of $d_N$ is generated as an algebra by $x_k$ and $mu_k$. This gives a proposed algebraic description of stable $\operatorname{sl}_N$-homolog…

Let $f$ be a homogeneous polynomial in $R={\mathbf C}[x_1,\dots,x_n]$, with $\deg x_i=1$, $n\geqslant 2$, and $d=\deg f$. Assume … For the shifted Koszul complex…

Let $P=\Bbbk[x_1,\ldots,x_d]$, let $K_\bullet$ be the Koszul complex on $x_1,\ldots,x_d$, let $C_i=\operatorname{Im}(\delta_i)$ be its $i$th cycle module, and let $S(C_i)$ and…