Local cohomology modules with nonclosed support
We construct noetherian rings admitting local cohomology modules with nonclosed support, or equivalently with infinitely many minimal primes; this answers a question of Huneke--Lyubeznik.
math.AC↗
arXiv subjects
Bhargav Bhatt
Publications and source records attributed to Bhargav Bhatt.
Reduction modulo p of crystalline Galois representations via μ_p-equivariance
For a crystalline representation of the absolute Galois group of Q_p, with given Hodge-Tate weights, we obtain new constraints on the inertial weights of its mod p reduction. This allows us to formulate an explicit Serre weight conjecture, in the generality of L-parameters for unramified connected reductive groups over Q_p, and to prove the elimination direction of this conjecture. The proof uses prismatic techniques to show that the reductions modulo p of the Breuil-Kisin modules attached to crystalline Galois representations acquire a natural μ_p-equivariant structure. Combining this with results on the geometry of the μ_p-fixed points of affine Grassmannians leads to our new constraint.
math.NT↗
Perfectoid pure singularities
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 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 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 \claperfdpure 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.
math.AG↗
Sheared Witt Vectors
V. Drinfeld and E. Lau introduced a ``decompletion'' of the ring of $p$-typical Witt vectors, following earlier work of T. Zink. The goal of this paper is to offer an exposition of this construction, which we call the sheared Witt vectors, on the category of rings $R$ whose reduction is a perfect $\mathbb{F}_p$-algebra.
math.AG↗
Vanishing of cohomology in infinitely ramified towers
We give a new proof of vanishing result of Esnault for the cohomology of constructible sheaves in the tower of ``mock'' Frobenius covers of projective space. The key idea is to use (a global form of) the perversity of nearby cycles.
math.AG↗
$F$-finite schemes have a dualizing complex
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.
math.AG↗
Diophantine sets
Diophantine subsets of $\mathbb{Z}$ play a key role in the negative answer to Hilbert's tenth problem. The definition of diophantine set generalizes in several ways to other commutative rings. We compare these definitions. Along the way, we prove that for every finitely presented scheme $Y$ over a ring $R$, there exists an affine $R$-scheme $X$ with a finitely presented $R$-morphism $X \to Y$ such that $X(R') \to Y(R')$ is surjective for every $R$-algebra $R'$.
math.NT↗
Test ideals in mixed characteristic: a unified theory up to perturbation
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.
math.AG↗
Applications of perverse sheaves in commutative algebra
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 $\mathbf{F}_p$-sheaves on varieties in characteristic $p$, expanding on previous work by various authors, and including a strong version of the Artin vanishing theorem.
math.AG↗
Lim Cohen-Macaulay sequences of modules
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 $p>0$, and all of them have the property of capturing colon ideals. In fact they satisfy axioms formulated by G.~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.
math.AC↗
Crystals and Chern classes
The goal of this paper is to study the Chern classes of coherent sheaves (and more generally perfect complexes) that admit crystal structures in the setting of crystalline cohomology and more generally relative prismatic cohomology. In the former theory, we show that the Chern classes vanish on the nose; in the latter theory, we show the classes are torsion with uniformly bounded exponents determined by suitable Bernoulli numbers. We also formulate some questions about syntomic Chern classes of such sheaves.
math.AG↗
Prismatic $F$-crystals and crystalline Galois representations
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.
math.NT↗
Frobenius on the cohomology of thickenings
We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds -- i.e., independent of the characteristic -- on the thickening that ensures an injective Frobenius map when the projective variety is a smooth complete intersection or an arbitrary projective embedding of an elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the case of elliptic curves.
math.AG↗
Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic
We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global $F$-regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita's conjecture to mixed characteristic.
math.AG↗
Syntomic complexes and $p$-adic étale Tate twists
The primary goal of this paper is to identify syntomic complexes with the $p$-adic étale Tate twists of Geisser--Schneider--Sato on regular $p$-torsionfree 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.
math.AG↗
Absolute prismatic cohomology
The goal of this paper is to study the absolute prismatic cohomology of $p$-adic formal schemes. We do so by recasting the notion of a prismatic crystal on $\mathrm{Spf}(\mathbf{Z}_p)$ in terms of quasicoherent sheaves on a geometric object we call the Cartier-Witt stack.
math.AG↗
The prismatization of $p$-adic formal schemes
In this note, we introduce and study the Cartier--Witt stack $\mathrm{WCart}_X$ attached to a $p$-adic formal scheme $X$ as well as some variants. In particular, we reinterpret the notion of prismatic crystals on $X$ and their cohomology in terms of quasicoherent sheaf theory on $\mathrm{WCart}_X$ in favorable situations.
math.AG↗
Prisms and Prismatic Cohomology
We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site -- the prismatic site -- to a $p$-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral $p$-adic cohomology theories. As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free description of $q$-de Rham cohomology as conjectured by the second author, and settle a vanishing conjecture for the $p$-adic Tate twists $\mathbf{Z}_p(n)$ introduced in previous joint work with Morrow.
math.AG↗