Related papers: The Cartier core map for Cartier algebras

Let $R$ be a standard graded finitely generated algebra over an $F$-finite field of prime characteristic, localized at its maximal homogeneous ideal. In this note, we prove that that Frobenius complexity of $R$ is finite. Moreover, we…

We prove that the Frobenius algebra of the injective hull of a complete Stanley-Reisner ring as well as its Matlis dual notion of Cartier algebra can be only principally generated or infinitely generated. As a consequence we are able to…

We extend the notion of Frobenius Betti numbers and F-splitting ratio to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. We also prove that the strong F-regularity of a pair…

Let $R$ be a ring of prime characteristic $p$, and let $F^e_* R$ denote $R$ viewed as an $R$-module via the $e$th iterated Frobenius map. Given a surjective map $\phi : F^e_* R \to R$ (for example a Frobenius splitting), we exhibit an…

We compute a number of invariants of singularities defined via the Frobenius morphism for seminormal affine toric varieties over fields of characteristic p > 0. Our main technical tool is a combinatorial description of the potential…

Let $R$ be a commutative (Noetherian) local ring of prime characteristic $p$ that is $F$-pure. This paper is concerned with comparison of three finite sets of radical ideals of $R$, one of which is only defined in the case when $R$ is…

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. The main goal of this paper is to study in some detail when \[ \overline{W^R}:=\{\mathfrak{p}\in\operatorname{Spec} (R):\ \mathcal{F}^{E_{\mathfrak{p}}}\text{ is finitely…

We study the relative Frobenius map associated with a map of derived commutative rings over a field of positive characteristic. As part of this, we examine a relative analog of perfectness and construct a relative inverse limit perfection…

Fix any field $K$ of characteristic $p$ such that $[K:K^p]$ is finite. We discuss excellence for Noetherian domains whose fraction field is $K$, showing for example, that $R$ is excellent if and only if the Frobenius map is finite on $R$.…

We survey results produced from the interaction between methods in prime characteristic and combinatorial commutative algebra. We showcase results for edge ideals, toric varieties, Stanley-Reisner rings, and initial ideals that were proven…

This paper is concerned with ideals in a commutative Noetherian ring $R$ of prime characteristic. The main purpose is to show that the Frobenius closures of certain ideals of $R$ generated by regular sequences exhibit a desirable type of…

We give a purely combinatorial characterization of complete Stanley-Reisner rings having principally generated (equivalently, finitely generated) Cartier algebras.

We explicitly describe the Cartier dual of the $l$-th Frobenius kernel $N_l$ of the deformation group scheme, which deforms the additive group scheme to the multiplicative group scheme. Then the Cartier dual of $N_l$ is given by a certain…

It is known that the Frobenius algebra of the injective hull of the residue field of a complete Stanley--Reisner ring (i.e. a formal power series ring modulo a squarefree monomial ideal) can be only principally generated or infinitely…

Suppose $R$ is a $\mathbb{Q}$-Gorenstein $F$-finite and $F$-pure ring of prime characteristic $p>0$. We show that if $I\subseteq R$ is a compatible ideal (with all $p^{-e}$-linear maps) then there exists a module finite extension $R\to S$…

We study $F$-graded systems of ideals in $R$, which are sequences of ideals giving rise to Cartier algebras on $R$. We identify how properties of these systems (or modifications of these systems) affect the singularity properties of the…

We prove model completeness for the theory of addition and the Frobenius map for certain subrings of rational functions in positive characteristic. More precisely: Let $p$ be a prime number, $\mathbb{F}_{p}$ the prime field with $p$…

An excellent ring of prime characteristic for which the Frobenius map is pure is also Frobenius split in many commonly occurring situations in positive characteristic commutative algebra and algebraic geometry. However, using a fundamental…

We construct the Cartier duality equivalence for affine commutative group schemes $G$ whose coordinate ring is a flat Mittag-Leffler module over an arbitrary base ring $R$. The dual $G^\vee$ of $G$ turns out to be an ind-finite ind-scheme…

Let $i: A\to R$ be a ring morphism, and $\chi: R\to A$ a right $R$-linear map with $\chi(\chi(r)s)=\chi(rs)$ and $\chi(1_R)=1_A$. If $R$ is a Frobenius $A$-ring, then we can define a trace map $\tr: A\to A^R$. If there exists an element of…