8 problems

Let $(R,\mathfrak m,k)$ be a commutative noetherian local ring with a dualizing complex $\operatorname{D}(R)$, shifted so that $\operatorname{H}_i(\operatorname{D}(R))=0$ for $i<0$…

Balanced dualizing dg-module conjecture. The graded commutativity hypothesis can be removed from the theorem asserting that every Gorenstein dg-algebra as in the setup admits a bal…

Let $f:X\to Y$ be a morphism between quasi-smooth, locally almost of finite presentation derived Artin stacks. Let $L_{X/Y}$ be the relative cotangent complex and let…

Existence conjecture. Any sequence-regular DG-ring has a dualizing DG-module.

Functorial rigidity conjecture. In the pseudo-finite case, $R_B$ has an induced rigidifying isomorphism $\rho_B$ relative to $\mathbb{K}$; in the essentially smooth case, $R_B$ lik…

Let $A$ be a tractable commutative DG ring with traction $\mathbb{K}\to A$, and let $(M,\rho)$ be a rigid DG module over $A$ relative to $\mathbb{K}$. Write…

Let $A$ be a tractable commutative DG ring, with traction $\mathbb{K} \to A$. A rigid dualizing DG module over $A$ relative to $\mathbb{K}$ is a pair $(R,\rho)$ consisting of a dua…

Let $R$) be a local ring admitting a dualizing complex $D$ with $\inf(D)=0$. If … for all $n\geqslant 1$, then $R$ is Gorenstein. Conjecture. This is a strengthening of the precedi…