Lim Cohen–Macaulay sequences of modules

Journal für die reine und angewandte Mathematik (Crelles Journal) · 2026-05-13

Abstract We introduce the notion of a lim Cohen–Macaulay sequence of modules. We prove the existence of such sequences in positive characteristic, and show that their existence in mixed characteristic implies the long open conjecture about positivity of Serre intersection multiplicities for all regular local rings, as well as a new proof of the existence of big Cohen–Macaulay modules. We describe how such a sequence leads to a notion of closure for submodules of finitely generated modules: this family of closure operations includes the usual notion of tight closure in characteristic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> p&gt;0 , and all of them have the property of capturing colon ideals. In fact, they satisfy axioms formulated by Dietz from which it follows that if a local ring 𝑅 has a lim Cohen–Macaulay sequence then it has a big Cohen–Macaulay module. We also prove the existence of lim Cohen–Macaulay sequences for certain rings of mixed characteristic.

$F$-finite schemes have a dualizing complex

arXiv (Cornell University) · 2026-04-21

In this paper we show that any Noetherian $F$-finite scheme has a dualizing complex $ω^{\bullet}_{X}$ with the property that for all finite type maps $f \colon X \to Y$ between $F$-finite Noetherian schemes there is a canonical isomorphism $ω^{\bullet}_{X} \xrightarrow{\cong} f^!ω^{\bullet}_{Y}$ in $D^b_{coh}(X)$. This, in particular, applies to the Frobenius morphism $F \colon X \to X$ so that we obtain a canonical isomorphism $ω^{\bullet}_{X} \xrightarrow{\cong} F^!ω^{\bullet}_{X}$. To prove this, we rely on a result of Gabber that every Noetherian $F$-finite ring is a quotient of a regular ring, from which it follows that every $F$-finite Noetherian scheme has a (potentially non-canonical) dualizing complex. To make this canonical, we identify the dualizing complex of any $F$-finite Noetherian scheme as a unit of an alternate symmetric monoidal structure on $D^b_{coh}(X)$ we call the $!$-tensor product. We also sketch an alternate approach to finding this canonical dualizing complex following the more classical approach to Grothendieck duality.

$F$-finite schemes have a dualizing complex

ArXiv.org · 2026-04-21

In this paper we show that any Noetherian $F$-finite scheme has a dualizing complex $ω^{\bullet}_{X}$ with the property that for all finite type maps $f \colon X \to Y$ between $F$-finite Noetherian schemes there is a canonical isomorphism $ω^{\bullet}_{X} \xrightarrow{\cong} f^!ω^{\bullet}_{Y}$ in $D^b_{coh}(X)$. This, in particular, applies to the Frobenius morphism $F \colon X \to X$ so that we obtain a canonical isomorphism $ω^{\bullet}_{X} \xrightarrow{\cong} F^!ω^{\bullet}_{X}$. To prove this, we rely on a result of Gabber that every Noetherian $F$-finite ring is a quotient of a regular ring, from which it follows that every $F$-finite Noetherian scheme has a (potentially non-canonical) dualizing complex. To make this canonical, we identify the dualizing complex of any $F$-finite Noetherian scheme as a unit of an alternate symmetric monoidal structure on $D^b_{coh}(X)$ we call the $!$-tensor product. We also sketch an alternate approach to finding this canonical dualizing complex following the more classical approach to Grothendieck duality.

Applications of perverse sheaves in commutative algebra

Journal für die reine und angewandte Mathematik (Crelles Journal) · 2025-05-10

Abstract The goal of this paper is to explain how basic properties of perverse sheaves sometimes translate via Riemann–Hilbert correspondences (in both characteristic 0 and characteristic 𝑝) to highly non-trivial properties of singularities, especially their local cohomology. Along the way, we develop a theory of perverse <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="bold">F</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> \mathbf{F}_{p} -sheaves on varieties in characteristic 𝑝, expanding on previous work by various authors, and including a strong version of the Artin vanishing theorem.

Arithmetic Geometry

Oberwolfach Reports · 2025-02-14

Arithmetic geometry is at the interface between algebraic geometry and number theory, and studies schemes over the ring of integers of number fields, or their p -adic completions. The talks covered a wide range of topics including the categorical Langlands program, Shimura varieties, complex and p -adic Hodge theory, homotopy theory, and Diophantine geometry.

Test ideals in mixed characteristic: a unified theory up to perturbation

arXiv (Cornell University) · 2024-01-01 · 1 citations

Let $X$ be an integral scheme of finite type over a complete DVR of mixed characteristic. We provide a definition of a test ideal which agrees with the multiplier ideal after inverting $p$, is computed from a sufficiently large alteration, agrees with previous mixed characteristic BCM test ideals after completing at any point of residue characteristic $p$ (up to small perturbation), and which satisfies the full suite of expected properties of a multiplier or test ideal. This object is obtained via the $p$-adic Riemann-Hilbert functor.

Perfectoid pure singularities

arXiv (Cornell University) · 2024-09-26 · 1 citations

Fix a prime number $p$. Inspired by the notion of $F$-pure or $F$-split singularities, we study the condition that a Noetherian ring with $p$ in its Jacobson radical is pure inside some perfectoid (classical) ring, a condition we call \emph{perfectoid pure}. We also study a related a priori weaker condition which asks that $R$ is pure in its absolute perfectoidization, a condition we call \emph{lim-perfectoid pure}. We show that both these notions coincide when $R$ is LCI. Mixed characteristic analogs of $F$-injective and Du Bois singularities are also explored. We study these notions of singularity, proving that they are weakly normal and that they are Du Bois after inverting $p$. We also explore the behavior of perfectoid singularities under finite covers and their relation to log canonical singularities. Finally, we prove an inversion of adjunction result in the LCI setting, and use it to prove that many common examples are perfectoid pure.

Syntomic complexes and <i>p</i>-adic étale Tate twists

Forum of Mathematics Pi · 2023-01-01 · 3 citations

Abstract The primary goal of this paper is to identify syntomic complexes with the p -adic étale Tate twists of Geisser–Sato–Schneider on regular p -torsion-free schemes. Our methods apply naturally to a broader class of schemes that we call ‘ F -smooth’. The F -smoothness of regular schemes leads to new results on the absolute prismatic cohomology of regular schemes.

Algebraic geometry in mixed characteristic

EMS Press eBooks · 2023-12-15 · 2 citations

Fix a prime number $p$. We report on some recent developments in algebraic geometry (broadly construed) over $p$-adically complete commutative rings. These developments include foundational advances within the subject, as well as external applications.

Prismatic $F$-crystals and crystalline Galois representations

Cambridge Journal of Mathematics · 2023-01-01 · 20 citations

Let $K$ be a complete discretely valued field of mixed characteristic $(0,p)$ with perfect residue field. We prove that the category of prismatic $F$-crystals on $\mathcal O_K$ is equivalent to the category of lattices in crystalline $G_K$-representations.