10 problems

Let $A$ and $B$ be the graded rings and $R=A\times_T B$ the graded fiber product considered above, with canonical modules $\omega_R$ and $\omega_{A,B}:=\omega_A\oplus\omega_B$. Ass…

Let $S$ be either a regular local ring with residue class field $K$ or a regular positively graded $K$-algebra. Let $I$ be a perfect ideal of grade two, and let $A$ be an Hilbert–B…

Let $(R,\mathfrak{m})$ be a Cohen–Macaulay local ring with a canonical ideal $\mathcal{C}$. Let $x$ be an element of $R$ that is regular modulo $\mathcal{C}$, and suppose that…

Let $(R,\mathfrak{m})$ be a Cohen–Macaulay local ring that has a canonical ideal $\mathcal{C}$. The canonical degree and bi-canonical degree are denoted by $\operatorname{cdeg}(R)$…

Comparison conjecture. In general,

Hyperplane-section conjecture. Under these conditions,

Comparison conjecture. The inequality

Canonical-module self-duality conjecture. If $\omega_R$ has analytic spread $1$, then some power of $\omega_R$, regarded as a fractional ideal, is $\omega_R$-self-dual up to a shif…

Let $R$ be a Cohen–Macaulay local or $^*\!$local ring, and let $I$ be a strongly Cohen–Macaulay ideal, or more generally an ideal satisfying sliding depth. Let…

Let $(R, \mathfrak{m}, k)$ be a reduced Cohen–Macaulay $F$-finite local ring of characteristic $p>0$, and let $\omega_R$ be a canonical module. The dual F-signature conjecture. $R$…