arXiv subjects

Jakob Scholbach

Publications and source records attributed to Jakob Scholbach.

At least 19 recordsLinked to original sources

arXiv2026

Exponential motives on the affine Grassmannian

We develop a notion of exponential motives on general prestacks equipped with a $\mathbf{G}_a$-action, and compare them with Whittaker motives via Gaitsgory's Kirillov model. We then establish foundational results for exponential motives on affine flag varieties concerning Tate motives and t-structures. We use this to prove a motivic Casselman-Shalika equivalence, relating exponential Tate motives on the affine Grassmannian to ind-coherent sheaves on the classifying stack of the Langlands dual group. The decategorification of this equivalence provides a new construction of the Whittaker module for the spherical Hecke algebra which works for arbitrary coefficients, including a generic version.

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arXiv2025

Central motives on parahoric flag varieties

We construct a refinement of Gaitsgory's central functor for integral motivic sheaves, and show it preserves stratified Tate motives. Towards this end, we develop a reformulation of unipotent motivic nearby cycles, which also works over higher-dimensional bases. We moreover introduce Wakimoto motives and use them to show that our motivic central functor is t-exact. A decategorification of these functors yields a new approach to generic Hecke algebras for general parahorics.

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arXiv2025

The geometric Satake equivalence for integral motives

We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include Whitney--Tate stratifications of Beilinson--Drinfeld Grassmannians and cellular decompositions of semi-infinite orbits. With future global applications in mind, we also achieve an equivalence relative to a power of the affine line. Finally, we use our equivalence to give Tannakian constructions of Deligne's modification of the dual group and a modified form of Vinberg's monoid over the integers.

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arXiv2025

Categorical Künneth formulas for cohomological motives

The manuscript at hand systematically studies Künneth formulas at a categorical level. We give criteria for an abstract six functor formalism to satisfy the categorical Künneth formula, and use this to formulate conjectures for categories of étale motives. As supporting evidence for these conjectures, we prove categorical Künneth formulas for adic sheaves and for cohomological motives, i.e., étale motives modulo the kernel of the adic realization.

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arXiv2024

Frobenius rigidity in $\mathbb A^1$-homotopy theory

We study the homotopy fixed points under the Frobenius endomorphism on the stable $\mathbb A^1$-homotopy category of schemes in characteristic $p>0$ and prove a rigidity result for cellular objects in these categories after inverting $p$. As a consequence we determine the analogous fixed points on the $K$-theory of algebraically closed fields in positive characteristic. We also prove a rigidity result for the homotopy fixed points of the partial Frobenius pullback on motivic cohomology groups in weights at most $1$.

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arXiv2023

A categorical Künneth formula for constructible Weil sheaves

We prove a Künneth-type equivalence of derived categories of lisse and constructible Weil sheaves on schemes in characteristic $p > 0$ for various coefficients, including finite discrete rings, algebraic field extensions $E \supset \mathbf Q_\ell$, $\ell \ne p$ and their rings of integers $O_E$. We also consider a variant for ind-construtible sheaves which applies to the cohomology of moduli stacks of shtukas over global function fields.

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arXiv2023

Constructible sheaves on schemes

We present a uniform theory of constructible sheaves on arbitrary schemes with coefficients in topological or even condensed rings. This is accomplished by defining lisse sheaves to be the dualizable objects in the derived infinity-category of proétale sheaves, while constructible sheaves are those that are lisse on a stratification. We show that constructible sheaves satisfy proétale descent. We also establish a t-structure on constructible sheaves in a wide range of cases. We finally provide a toolset to manipulate categories of constructible sheaves with respect to the choices of coefficient rings, and use this to prove that our notions reproduce and extend the various approaches to, say, constructible ell-adic sheaves in the literature.

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arXiv2022

Admissibility and rectification of colored symmetric operads

We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, i.e., the category of algebras over any operad admits a model structure transferred from the original model category. We also give a necessary and sufficient criterion that ensures that a given weak equivalence of admissible operads admits rectification, i.e., the corresponding Quillen adjunction between the categories of algebras is a Quillen equivalence. In addition, we show that Quillen equivalences of underlying symmetric monoidal model categories yield Quillen equivalences of model categories of algebras over operads. Applications of these results include enriched categories, colored operads, prefactorization algebras, and commutative symmetric ring spectra.

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arXiv2022

Integral Motivic Sheaves And Geometric Representation Theory

With representation-theoretic applications in mind, we construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. We show that reduced stratified Tate motives satisfy favorable properties including weight and t-structures. We also prove that reduced motives on cellular (ind-)schemes unify various approaches to mixed sheaves in representation theory, such as Soergel-Wendt's semisimplified Hodge motives, Achar-Riche's complexes of parity sheaves, as well as Ho-Li's recent category of graded $\ell$-adic sheaves.

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arXiv2021

Tate motives on Witt vector affine flag varieties

Relying on recent advances in the theory of motives we develope a general formalism for derived categories of motives with Q-coefficients on perfect (ind-)schemes. As an application we give a motivic refinement of Zhu's geometric Satake equivalence for Witt vector affine Grassmannians in this set-up.

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arXiv2021

The motivic Satake equivalence

We refine the geometric Satake equivalence due to Ginzburg, Beilinson-Drinfeld, and Mirković-Vilonen to an equivalence between mixed Tate motives on the double quotient $L^+ G \backslash LG / L^+ G$ and representations of Deligne's modification of the Langlands dual group $\hat G$.

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arXiv2020

Symmetric operads in abstract symmetric spectra

This paper sets up the foundations for derived algebraic geometry, Goerss--Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën--Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith's stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and E-infinity ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of E-infinity rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent.

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arXiv2020

Homotopy theory of symmetric powers

We introduce the symmetricity notions of symmetric h-monoidality, symmetroidality, and symmetric flatness. As shown in our paper arXiv:1410.5675, these properties lie at the heart of the homotopy theory of colored symmetric operads and their algebras. In particular, the former property can be seen as the analog of Schwede and Shipley's monoid axiom for algebras over symmetric operads and allows one to equip categories of such algebras with model structures, whereas the latter ensures that weak equivalences of operads induce Quillen equivalences of categories of algebras. We discuss these properties for elementary model categories such as simplicial sets, simplicial presheaves, and chain complexes. Moreover, we provide powerful tools to promote these properties from such basic model categories to more involved ones, such as the stable model structure on symmetric spectra.

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arXiv2019

The intersection motive of the moduli stack of shtukas

For a split reductive group G over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated G-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic t-structures on triangulated categories of motives. This is in accordance with general expectations on the independence of l in the Langlands correspondence for function fields.

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arXiv2016

Modules over the de Rham cohomology spectrum

We show that the bounded derived category of regular holonomic D-modules on a smooth variety is equivalent to the homotopy catgory of compact (or constructible) modules over the motivic ring spectrum $H_{dR}$ representing algebraic de Rham cohomology. This equivalence is compatible with the six functors on both sides. This way, the classical functors in the world of D-modules, $f_! := D_Y f_* D_X, f^* := D_X f^! D_Y$ ($f: X \to Y$), are conceptually explained and embedded into a larger and more flexible framework. We also apply this equivalence to obtain a motivic t-structure on $H_{dR}$-modules on not necessarily smooth schemes.

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arXiv2015

Special L-values of geometric motives

This paper proposes a conjecture on special values of L-functions of geometric motives over Z. This includes L-functions of mixed motives over Q and Hasse-Weil zeta-functions of schemes over Z. We conjecture the following: the order of L(M, s) at s=0 is given by the negative Euler characteristic of motivic cohomology of $M^\vee(-1)$. Up to a nonzero rational factor, the L-value at s=0 is given by the determinant of the pairing of Arakelov motivic cohomology of M with the motivic homology of M. Under standard assumptions concerning mixed motives over Q, $F_p$, and Z, this conjecture is essentially equivalent to the conjunction of Soulé's conjecture about pole orders of zeta-functions of schemes over Z, Beilinson's conjecture about special L-values for motives over Q and the Tate conjecture over $F_p$.

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arXiv2015

Arakelov motivic cohomology I

This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta functions formulated by the second author. Taking advantage of the six functors formalism in motivic stable homotopy theory, we establish a number of formal properties, including pullbacks for arbitrary morphisms, pushforwards for projective morphisms between regular schemes, localization sequences, $h$-descent. We round off the picture with a purity result and a higher arithmetic Riemann-Roch theorem.

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arXiv2015

f-cohomology and motives over number rings

This paper is concerned with an interpretation of f-cohomology, a modification of motivic cohomology of motives over number fields, in terms of motives over number rings. Under standard assumptions on mixed motives over finite fields, number fields and number rings, we show that the two extant definitions of f-cohomology of mixed motives $M_η$ over F--one via ramification conditions on $\ell$-adic realizations, another one via the K-theory of proper regular models--both agree with motivic cohomology of $η_{!*} M_η[1]$. Here $η_{!*}$ is constructed by a limiting process in terms of intermediate extension functors $j_{!*}$ defined in analogy to perverse sheaves.

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