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Roy Joshua

Publications and source records attributed to Roy Joshua.

At least 19 recordsLinked to original sources
arXiv2024

Lambda-ring structures on the K-theory of algebraic stacks

In this paper we consider the K-theory of smooth algebraic stacks, establish lambda and gamma operations, and show that the higher K-theory of such stacks is always a pre-lambda-ring, and is a lambda-ring if every coherent sheaf is the quotient of a vector bundle. As a consequence, we are able to define Adams operations and absolute cohomology for smooth algebraic stacks satisfying this hypothesis. We also obtain a comparison of the absolute cohomology with the equivariant higher Chow groups in certain special cases.

math.KT↗
arXiv2024

Equivariant Algebraic K-Theory and Derived completions II: the case of Equivariant Homotopy K-Theory and Equivariant K-Theory

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known completion theorems for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem for equivariant Algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him. In an earlier work by the first two authors, we solved this conjecture by providing a derived completion theorem for equivariant G-theory. In the present paper, we provide a similar derived completion theorem for the homotopy Algebraic K-theory of equivariant perfect complexes, on schemes that need not be regular. Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any normal quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the Equivariant Homotopy Algebraic K-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). With finite coefficients invertible in the base fields, we are also able to obtain such derived completion theorems for equivariant algebraic K-theory but with respect to actions of diagonalizable group schemes. These enable us to obtain a wide range of applications, several of which are also explored.

math.AG↗
arXiv2024

Equivariant Algebraic K-Theory and Derived completions III: Applications

In the present paper, we discuss applications of the derived completion theorems proven in our previous two papers. One of the main applications is to Riemann-Roch problems for forms of higher equivariant K-theory, which we are able to establish in great generality both for equivariant G-theory and equivariant homotopy K-theory with respect to actions of linear algebraic groups on normal quasi-projective schemes over a given field. We show such Riemann-Roch theorems apply to all toric and spherical varieties. We also obtain Lefschetz-Riemann-Roch theorems involving the fixed point schemes with respect to actions of diagonalizable group schemes. We also show the existence of certain spectral sequences that compute the homotopy groups of the derived completions of equivariant G-theory starting with equivariant Borel-Moore motivic cohomology.

math.AG↗
arXiv2023

Additivity for the Motivic Trace and the Motivic Euler Characteristic

In this paper, we settle an open conjecture regarding the assertion that the Euler-characteristic of $\rmG/\NT$ for a split reductive group scheme $\rmG$ and the normalizer of a split maximal torus $\NT$ over a field is $1$ in the Grothendieck-Witt ring with the characteristic exponent of the field inverted, under the assumption that the base field contains a $\sqrt -1$. Numerous applications of this to splittings in the motivic stable homotopy category and to Algebraic K-Theory are worked out in several related papers by Gunnar Carlsson and the authors.

math.AG↗
arXiv2022

Approximating Absolute Galois Groups

This paper formulates a group condition which is enjoyed by absolute Galois groups, and which guarantees that profinite groups satisfying the condition can be approximated as an inverse limit of groups which are profinite analogues of Bieberbach groups. The condition is a key ingredient in the study of the "representational assembly" approach to descent problems in algebraic K-theory.

math.KT↗
arXiv2021

The Motivic Segal-Becker Theorem

The present paper is a continuation of earlier work by Gunnar Carlsson and the first author on a motivic variant of the classical Becker-Gottlieb transfer and an additivity theorem for such a transfer by the present authors. Here, we establish a motivic variant of the classical Segal-Becker theorem relating the classifying space of a 1-dimensional torus with the spectrum defining (algebraic) K-theory.

math.AG↗
arXiv2021

Brauer Groups of Algebraic Stacks and GIT-Quotients

In this paper we consider the Brauer groups of algebraic stacks and GIT quotients: the only algebraic stacks we consider in this paper are quotient stacks [X/G], where X is a smooth scheme of finite type over a field k, and G is a linear algebraic group over k and acting on X, as well as various moduli stacks of principal G-bundles on a smooth projective curve X, associated to a reductive group G. We also consider the Brauer groups of the corresponding coarse moduli spaces, which most often identify with the corresponding GIT-quotients. One conclusion that we seem to draw then is that the Brauer groups (or their $\ell$-primary torsion parts, for a fixed prime $\ell$ different from char(k)) of the corresponding stacks and coarse moduli spaces depend strongly on the Brauer group of the given scheme X.

math.AG↗
arXiv2020

Motivic and \'Etale Spanier-Whitehead duality and the Becker-Gottlieb transfer

In this paper, we develop a theory of Becker-Gottlieb transfer based on Spanier-Whitehead duality that holds in both the motivic and \'etale settings for smooth quasi-projective varieties in as broad a context as possible: for example, for varieties over non-separably closed fields in all characteristics, and also for both the \'etale and motivic settings. In view of the fact that the most promising applications of the traditional Becker-Gottlieb transfer has been to torsors and Borel-style equivariant cohomology theories, we focus our applications to motivic cohomology theories for torsors as well as Borel-style equivariant motivic cohomology theories, both defined with respect to motivic spectra. We obtain several results in this direction, including a stable splitting in generalized motivic cohomology theories. Various further applications will be discussed in forthcoming papers.

math.AG↗
arXiv2020

Additivity and Double Coset formulae for the Motivic and \'Etale Becker-Gottlieb transfer

In this paper, which is a continuation of earlier work by the first author and Gunnar Carlsson, one of the first results we establish is the additivity of the motivic Becker-Gottlieb transfer, as well as their \'etale realizations. This extends the additivity results the authors already established for the corresponding traces. We then apply this to derive several important consequences: for example, in addition to obtaining the analogues of various double coset formulae known in the classical setting of algebraic topology, we also obtain applications to Brauer groups of homogeneous spaces associated to reductive groups over separably closed fields. We also consider the relationship between the transfer on schemes provided with a compatible action by a $1$-parameter subgroup and the transfer associated to the fixed point scheme of the $1$-parameter subgroup.

math.AG↗
arXiv2020

Equivariant Derived Categories for Toroidal Group Imbeddings

Let $\text{X}$ denote a projective variety over an algebraically closed field on which a linear algebraic group acts with finitely many orbits. Then, a conjecture of Soergel and Lunts in the setting of Koszul duality and Langlands' philosophy, postulates that the equivariant derived category of bounded complexes with constructible equivariant cohomology sheaves on $\text{X}$ is equivalent to a full subcategory of the derived category of modules over a graded ring defined as a suitable graded $Ext$. Only special cases of this conjecture have been proven so far. {\it The purpose of this paper is to provide a proof of this conjecture for all projective toroidal imbeddings of complex reductive groups.} In fact, we show that the methods used by Lunts for a proof in the case of toric varieties can be extended with suitable modifications to handle the toroidal imbedding case. {\it Since every equivariant imbedding of a complex reductive group is dominated by a toroidal imbedding, the class of varieties for which our proof applies is quite large.} We also show that, in general, there exist a countable number of obstructions for this conjecture to be true and that half of these vanish when the odd dimensional equivariant intersection cohomology sheaves on the orbit closures vanish. This last vanishing condition had been proven to be true in many cases of spherical varieties by Michel Brion and the author in prior work.

math.AG↗
arXiv2019

Atiyah-Segal Derived Completions for Equivariant Algebraic G-Theory and K-Theory

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K- Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant Algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. The main goal of the present paper is to provide a proof of this conjecture in as broad a context as possible, making use of the technique of derived completion, and to consider several of the applications. Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the Equivariant Algebraic G-Theory of large classes of varieties like all Toric varieties (for the action of a torus) and all Spherical varieties (for the action of a reductive group). Restricting to actions by split tori, we are also able to consider actions on Algebraic Spaces. These enable us to obtain a wide range of applications, some of which are briefly sketched and which we plan to explore in detail in the future. A comparison of our results with previously known results, none of which made use of derived completions, shows that without the use of derived completions, one can only obtain results which are indeed very restrictive.

math.AG↗
arXiv2017

Brauer groups of schemes associated to symmetric powers of smooth projective curves in arbitrary characteristics

In this paper we show that the l^n-torsion part of the cohomological Brauer groups of certain schemes associated to symmetric powers of a projective smooth curve over a separably closed field k are isomorphic, when `l is invertible in k. The schemes considered are the Symmetric powers themselves, then the corresponding Picard schemes and also certain Quot-schemes. We also obtain similar results for Prym varieties associated to certain finite covers of such curves: we prove such results only for curves defined over the field of complex numbers.

math.AG↗
arXiv2014

Motivic E_{infinity} algebras and the motivic dga

In this paper we define an explicit E_{infinity}-structure, i.e. a coherently homotopy associative and commutative product on chain complexes defining (integral and mod-l) motivic cohomology as well as mod -l étale cohomology. We also discuss several applications. In addition, our constructions show that the source of the E_{\infinity}-structure on the motivic complexes provided with the pairing defined by Suslin and Voevodsky is not chain-theoretic as is the case for the singular co-chain complexes for topological spaces.

math.AG↗
arXiv2014

Equivariant motivic homotopy theory

In this paper, we develop the theory of equivariant motivic homotopy theory, both unstable and stable. While our original interest was in the case of profinite group actions on smooth schemes, we discuss our results in as broad a setting as possible so as to be applicable in a variety of contexts, for example to the case of smooth group scheme actions on schemes that are not necessarily smooth. We also discuss how ${\mathbb A}^1$-localization behaves with respect to mod-$\ell$-completion, where $\ell$ is a fixed prime.

math.AT↗
arXiv2013

Galois descent for completed Algebraic K-theory

In this paper we consider the problem of Galois descent for suitably completed algebraic K-theory of fields. One of the main results is a suitable form of rigidity for Borel-style generalized equivariant cohomology with respect to certain spectra. In order to apply this to the problem at hand, we need to invoke a derived Atiyah-Segal completion theorem for pro-groups. In the present paper, the authors apply such a derived completion theorem proven by the first author elsewhere. These two results provide a proof of the Galois descent problem for equivariant algebraic K-theory as formulated by the first author, at least when restricted to the case where the absolute Galois groups are pro-$l$ groups for some prime $l$ different from the characteristic of the base field and the K-theory spectrum is completed at the same prime $l$. Work in progress hopes to remove these restrictions.

math.KT↗
arXiv2013

Explicit Chow-Lefschetz Decomposition for Kummer manifolds

Let X be the quotient of a smooth projective variety over a field by a finite group action (in which case we say X is pseudo-smooth), such that the singularities of X are isolated k-rational points. Let Y be obtained by blowing up these points on X. Assume further that Y is pseudo-smooth, and that the components of the exceptional divisor are projective spaces. Then, without invoking the theory of finite-dimensional motives or assuming any of the standard conjectures, we show that a Chow-Kunneth decomposition on either X or Y gives rise, by means of an explicit construction, to a Chow-Kunneth decomposition on the other. We use these constructions to show that various properties (among them, Conjectures of Murre, and being of Lefschetz type) hold for X if and only if they hold for Y. The main examples of interest to us are Kummer manifolds. We give several further applications of our construction to this particular class of examples.

math.AG↗
arXiv2012

Higher K-theory of Toric stacks

In this paper, we develop several techniques for computing the higher G-theory and K-theory of quotient stacks. Our main results for computing these groups are in terms of spectral sequences. We show that these spectral sequences degenerate in the case of many toric stacks, thereby providing an efficient computation of their higher K-theory. We apply our main results to give explicit description for the higher K-theory for many smooth toric stacks. As another application, we describe the higher K-theory of toric stack bundles over smooth base schemes.

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