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Purity for flat cohomology

Cesnavicius Kestutis, Scholze Peter
23 Dec 2019 math.AG math.NT arxiv.org/abs/1912.10932

We establish the flat cohomology version of the Gabber-Thomason purity for \'etale cohomology: for a complete intersection Noetherian local ring $(R, \mathfrak{m})$ and a commutative, finite, flat $R$-group $G$, the flat cohomology $H^i_{\mathfrak{m}}(R, G)$ vanishes for $i < \mathrm{dim}(R)$. For small $i$, this settles conjectures of Gabber that extend the Grothendieck-Lefschetz theorem and give purity for the Brauer group for schemes with complete intersection singularities. For the proof, we reduce to a flat purity statement for perfectoid rings, establish $p$-complete arc descent for flat cohomology of perfectoids, and then relate to coherent cohomology of $\mathbb{A}_{\mathrm{inf}}$ via prismatic Dieudonn\'e theory. We also present an algebraic version of tilting for \'etale cohomology, use it reprove the Gabber-Thomason purity, and exhibit general properties of fppf cohomology of (animated) rings with finite, locally free group scheme coefficients, such as excision, agreement with fpqc cohomology, and $p$-adic continuity.

On the generic part of the cohomology of non-compact unitary Shimura varieties

Caraiani Ana, Scholze Peter
04 Sep 2019 math.NT math.AG arxiv.org/abs/1909.01898

We prove that the generic part of the mod l cohomology of Shimura varieties associated to quasi-split unitary groups of even dimension is concentrated above the middle degree, extending previous work to a non-compact case. The result applies even to Eisenstein cohomology classes coming from the locally symmetric space of the general linear group, and has been used in [ACC+18] to get good control on these classes and deduce potential automorphy theorems without any self-duality hypothesis. Our main geometric result is a computation of the fibers of the Hodge--Tate period map on compactified Shimura varieties, in terms of similarly compactified Igusa varieties.

Prisms and Prismatic Cohomology

Bhatt Bhargav, Scholze Peter
20 May 2019 math.AG math.NT arxiv.org/abs/1905.08229

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.

Potential automorphy over CM fields

Allen Patrick B., Calegari Frank, Caraiani Ana, Gee Toby, Helm David, Hung Bao V. Le, Newton James, Scholze Peter, Taylor Richard, Thorne Jack A.
24 Dec 2018 math.NT arxiv.org/abs/1812.09999

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm{GL}_2(\mathbf{A}_F)$.

Topological Hochschild homology and integral $p$-adic Hodge theory

Bhatt Bhargav, Morrow Matthew, Scholze Peter
09 Feb 2018 math.AG math.KT math.NT arxiv.org/abs/1802.03261

In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$-theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex $A\Omega$ constructed in our previous work, and in equal characteristic $p$ to crystalline cohomology. Our construction of the filtration on $\mathrm{THH}$ is via flat descent to semiperfectoid rings. As one application, we refine the construction of the $A\Omega$-complex by giving a cohomological construction of Breuil--Kisin modules for proper smooth formal schemes over $\mathcal O_K$, where $K$ is a discretely valued extension of $\mathbb Q_p$ with perfect residue field. As another application, we define syntomic sheaves $\mathbb Z_p(n)$ for all $n\geq 0$ on a large class of $\mathbb Z_p$-algebras, and identify them in terms of $p$-adic nearby cycles in mixed characteristic, and in terms of logarithmic de~Rham-Witt sheaves in equal characteristic $p$.

$p$-adic geometry

Scholze Peter
11 Dec 2017 math.AG math.NT arxiv.org/abs/1712.03708

We discuss recent developments in $p$-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for "compact $p$-adic manifolds" over new period maps on moduli spaces of abelian varieties to applications to the local and global Langlands conjectures, and the construction of "universal" $p$-adic cohomology theories. We finish with some speculations on how a theory that combines all primes $p$, including the archimedean prime, might look like.

Etale cohomology of diamonds

Scholze Peter
21 Sep 2017 math.AG math.NT arxiv.org/abs/1709.07343

Motivated by problems on the \'etale cohomology of Rapoport--Zink spaces and their generalizations, as well as Fargues's geometrization conjecture for the local Langlands correspondence, we develop a six functor formalism for the \'etale cohomology of diamonds, and more generally small v-stacks on the category of perfectoid spaces of characteristic $p$. Using a natural functor from analytic adic spaces over $\mathbb Z_p$ to diamonds which identifies \'etale sites, this induces a similar formalism in that setting, which in the noetherian setting recovers the formalism from Huber's book.

On topological cyclic homology

Nikolaus Thomas, Scholze Peter
06 Jul 2017 math.AT math.KT arxiv.org/abs/1707.01799

Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by B\"okstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas--Goodwillie--McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing $K$-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum. The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the $\infty$-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum $X$ with $S^1$-action (in the most naive sense) together with $S^1$-equivariant maps $\varphi_p: X\to X^{tC_p}$ for all primes $p$. Here $X^{tC_p}=\mathrm{cofib}(\mathrm{Nm}: X_{hC_p}\to X^{hC_p})$ is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology. In order to construct the maps $\varphi_p: X\to X^{tC_p}$ in the example of topological Hochschild homology we introduce and study Tate diagonals for spectra and Frobenius homomorphisms of commutative ring spectra. In particular we prove a version of the Segal conjecture for the Tate diagonals and relate these Frobenius homomorphisms to power operations.

Vanishing theorems for perverse sheaves on abelian varieties, revisited

Bhatt Bhargav, Schnell Christian, Scholze Peter
21 Feb 2017 math.AG arxiv.org/abs/1702.06395

We revisit some of the basic results of generic vanishing theory, as pioneered by Green and Lazarsfeld, in the context of constructible sheaves. Using the language of perverse sheaves, we give new proofs of some of the basic results of this theory. Our approach is topological/arithmetic, and avoids Hodge theory.

Topological realisations of absolute Galois groups

Kucharczyk Robert A., Scholze Peter
15 Sep 2016 math.AT math.AG math.NT arxiv.org/abs/1609.04717

Let $F$ be a field of characteristic $0$ containing all roots of unity. We construct a functorial compact Hausdorff space $X_F$ whose profinite fundamental group agrees with the absolute Galois group of $F$, i.e. the category of finite covering spaces of $X_F$ is equivalent to the category of finite extensions of $F$. The construction is based on the ring of rational Witt vectors of $F$. In the case of the cyclotomic extension of $\mathbb{Q}$, the classical fundamental group of $X_F$ is a (proper) dense subgroup of the absolute Galois group of $F$. We also discuss a variant of this construction when the field is not required to contain all roots of unity, in which case there are natural Frobenius-type automorphisms which encode the descent along the cyclotomic extension.