21 problems

Let $(R,\mathfrak{m},k)$ be an $F$-finite local ring of prime characteristic $p>0$. An ideal of $R$ is tightly closed if it equals its tight closure, and $R$ is strongly $F$-regula…

Throughout, let $(R,m,K)$ be a local ring of characteristic $p>0$, let $M$ be an $R$-module, and let $N\subset M$ be a submodule. Write $N^\ast_M$ for the tight closure of $N$ in…

Injective-hull tight-closure conjecture. One has

Test ideal conjecture. The test ideal defines precisely the non-F-regular locus in $\operatorname{Spec} R$.

Let $R$ be a local $\mathbb C$-affine domain, let $I$ be an ideal of $R$, and let $\mathcal B(R)$ denote the canonical big Cohen–Macaulay algebra associated to $R$. The canonical b…

Let $(R,\frak{m})$ be an excellent Noetherian local ring of characteristic $p>0$ such that its completion $\widehat{R}$ is reduced and satisfies $(S_2)$. For a parameter ideal…

A ring $R$ has prime characteristic if its characteristic is a prime number, and it is excellent and reduced in the usual commutative-algebraic senses. Test-element existence conje…

Equality of tight closures in injective hulls. One should have

Let $R$ be a Noetherian ring of prime characteristic $p>0$. Write $I^*$ for the tight closure of an ideal $I$, call $R$ weakly $F$-regular if every ideal satisfies $I=I^*$, call…

Test-ideal equality conjecture. One has

Infinitude conjecture. The set $\mathcal{P}$ has infinite cardinality.

Let $R$ be an excellent Noetherian domain of characteristic $p>0$. Denote by $\tau_{\rm fg}(R)$ the finitistic test ideal and by $\tau_{\rm b}(R)$ the big test ideal of $R$. Test i…

Let $(R,\operatorname{\mathfrak{m}})$ be a $d$-dimensional complete local domain of mixed characteristic $p$. Assume that $p,x_2,\dots,x_d$ is a system of parameters. Set … and, fo…

Let $R$ be an $F$-finite weakly $F$-regular ring. Strong $F$-regularity conjecture. The ring $R$ is strongly $F$-regular. Strong $F$-regularity is a stronger Frobenius-splitting co…

Equality of test ideals conjecture. All versions of test ideals in tight closure theory should be equal.

Let $R$ be a Noetherian ring, let … be an injective cogenerator of $R$, and let $0^{*{\rm f}{\rm g}}_E$ denote the finitistic tight closure of zero in $E$, namely the union of…

Let $R$ be a ring in the setting above, and write $\tld\tau$ for the big test ideal and $\tau$ for the finitistic test ideal. Big–finitistic test ideal conjecture. It is conjecture…

Finitistic test ideal conjecture. The two test ideals coincide:

Strong-versus-weak F-regularity conjecture. $R$ is strongly $F$-regular if and only if

Let $(A,\mathfrak m)$ be a local ring, let $E$ be the injective hull of its residue field, and let the finitistic tight closure and regular tight closure of $0$ in $E$ be denoted b…

Let $(R,\mathfrak{m})$ be a regular local ring essentially of finite type over a perfect field of prime characteristic $p$, and let $I \subseteq R$ be a nonzero unmixed ideal. Let…