Annals of K-Theory Articles (Project Euclid)
https://projecteuclid.org/euclid.akt
The latest articles from Annals of K-Theory on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 19 Oct 2017 12:51 EDTThu, 19 Oct 2017 12:51 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
https://projecteuclid.org/
Real cohomology and the powers of the fundamental ideal in the Witt ring
https://projecteuclid.org/euclid.akt/1508431895
<strong>Jeremy Jacobson</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 3, 357--385.</p><p><strong>Abstract:</strong><br/> Let [math] be a local ring in which 2 is invertible. It is known that the localization of the cohomology ring [math] with respect to the class [math] is isomorphic to the ring [math] of continuous [math] -valued functions on the real spectrum of [math] . Let [math] denote the powers of the fundamental ideal in the Witt ring of symmetric bilinear forms over [math] . The starting point of this article is the “integral” version: the localization of the graded ring [math] with respect to the class [math] is isomorphic to the ring [math] of continuous [math] -valued functions on the real spectrum of [math] . This has interesting applications to schemes. For instance, for any algebraic variety [math] over the field of real numbers [math] and any integer [math] strictly greater than the Krull dimension of [math] , we obtain a bijection between the Zariski cohomology groups [math] with coefficients in the sheaf [math] associated to the [math] -th power of the fundamental ideal in the Witt ring [math] and the singular cohomology groups [math] . </p>projecteuclid.org/euclid.akt/1508431895_20171019125136Thu, 19 Oct 2017 12:51 EDTColocalising subcategories of modules over finite group schemes
https://projecteuclid.org/euclid.akt/1508431896
<strong>Dave Benson</strong>, <strong>Srikanth Iyengar</strong>, <strong>Henning Krause</strong>, <strong>Julia Pevtsova</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 3, 387--408.</p><p><strong>Abstract:</strong><br/>
The Hom closed colocalising subcategories of the stable module category of a finite group scheme are classified. This complements the classification of the tensor closed localising subcategories in our previous work. Both classifications involve [math] -points in the sense of Friedlander and Pevtsova. We identify for each [math] -point an endofinite module which both generates the corresponding minimal localising subcategory and cogenerates the corresponding minimal colocalising subcategory.
</p>projecteuclid.org/euclid.akt/1508431896_20171019125136Thu, 19 Oct 2017 12:51 EDTExterior power operations on higher $K$-groups via binary complexes
https://projecteuclid.org/euclid.akt/1508431897
<strong>Tom Harris</strong>, <strong>Bernhard Köck</strong>, <strong>Lenny Taelman</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 3, 409--450.</p><p><strong>Abstract:</strong><br/>
We use Grayson’s binary multicomplex presentation of algebraic [math] -theory to give a new construction of exterior power operations on the higher [math] -groups of a (quasicompact) scheme. We show that these operations satisfy the axioms of a [math] -ring, including the product and composition laws. To prove the latter we show that the Grothendieck group of the exact category of integral polynomial functors is the universal [math] -ring on one generator.
</p>projecteuclid.org/euclid.akt/1508431897_20171019125136Thu, 19 Oct 2017 12:51 EDT$\mathbb A^1$-homotopy invariance of algebraic $K$-theory with coefficients and du Val singularitieshttps://projecteuclid.org/euclid.akt/1510841622<strong>Gonçalo Tabuada</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 1, 1--25.</p><p><strong>Abstract:</strong><br/>
C. Weibel, and Thomason and Trobaugh, proved (under some assumptions) that algebraic [math] -theory with coefficients is [math] -homotopy invariant. We generalize this result from schemes to the broad setting of dg categories. Along the way, we extend the Bass–Quillen fundamental theorem as well as Stienstra’s foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above [math] -homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic [math] -theory with coefficients of dg cluster categories using solely the kernel and cokernel of the Coxeter matrix. This leads to a complete computation of the algebraic [math] -theory with coefficients of the du Val singularities parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain vanishing and divisibility properties of algebraic [math] -theory (without coefficients).
</p>projecteuclid.org/euclid.akt/1510841622_20171116091348Thu, 16 Nov 2017 09:13 ESTReciprocity laws and $K\mkern-2mu$-theoryhttps://projecteuclid.org/euclid.akt/1510841623<strong>Evgeny Musicantov</strong>, <strong>Alexander Yom Din</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 1, 27--46.</p><p><strong>Abstract:</strong><br/>
We associate to a full flag [math] in an [math] -dimensional variety [math] over a field [math] , a “symbol map” [math] . Here, [math] is the field of rational functions on [math] , and [math] is the [math] -theory spectrum. We prove a “reciprocity law” for these symbols: given a partial flag, the sum of all symbols of full flags refining it is [math] . Examining this result on the level of [math] -groups, we derive the following known reciprocity laws: the degree of a principal divisor is zero, the Weil reciprocity law, the residue theorem, the Contou-Carrère reciprocity law (when [math] is a smooth complete curve), as well as the Parshin reciprocity law and the higher residue reciprocity law (when [math] is higher-dimensional).
</p>projecteuclid.org/euclid.akt/1510841623_20171116091348Thu, 16 Nov 2017 09:13 ESTOn the cycle map of a finite grouphttps://projecteuclid.org/euclid.akt/1510841624<strong>Masaki Kameko</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 1, 47--72.</p><p><strong>Abstract:</strong><br/>
Let [math] be an odd prime number. We show that there exists a finite group of order [math] for which the mod [math] cycle map from the mod [math] Chow ring of its classifying space to its ordinary mod [math] cohomology is not injective.
</p>projecteuclid.org/euclid.akt/1510841624_20171116091348Thu, 16 Nov 2017 09:13 ESTChern classes and compatible power operations in inertial K-theoryhttps://projecteuclid.org/euclid.akt/1510841626<strong>Dan Edidin</strong>, <strong>Tyler Jarvis</strong>, <strong>Takashi Kimura</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 1, 73--130.</p><p><strong>Abstract:</strong><br/> Let [math] be a smooth Deligne–Mumford quotient stack. In a previous paper we constructed a class of exotic products called inertial products on [math] , the Grothendieck group of vector bundles on the inertia stack [math] . In this paper we develop a theory of Chern classes and compatible power operations for inertial products. When [math] is diagonalizable these give rise to an augmented [math] -ring structure on inertial K-theory. One well-known inertial product is the virtual product . Our results show that for toric Deligne–Mumford stacks there is a [math] -ring structure on inertial K-theory. As an example, we compute the [math] -ring structure on the virtual K-theory of the weighted projective lines [math] and [math] . We prove that, after tensoring with [math] , the augmentation completion of this [math] -ring is isomorphic as a [math] -ring to the classical K-theory of the crepant resolutions of singularities of the coarse moduli spaces of the cotangent bundles [math] and [math] , respectively. We interpret this as a manifestation of mirror symmetry in the spirit of the hyper-Kähler resolution conjecture. </p>projecteuclid.org/euclid.akt/1510841626_20171116091348Thu, 16 Nov 2017 09:13 ESTOn the vanishing of Hochster's $\theta$ invarianthttps://projecteuclid.org/euclid.akt/1510841640<strong>Mark Walker</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 2, 131--174.</p><p><strong>Abstract:</strong><br/> Hochster’s theta invariant is defined for a pair of finitely generated modules on a hypersurface ring having only an isolated singularity. Up to a sign, it agrees with the Euler invariant of a pair of matrix factorizations. Working over the complex numbers, Buchweitz and van Straten have established an interesting connection between Hochster’s theta invariant and the classical linking form on the link of the singularity. In particular, they establish the vanishing of the theta invariant if the hypersurface is even-dimensional by exploiting the fact that the (reduced) cohomology of the Milnor fiber is concentrated in odd degrees in this situation. We give purely algebraic versions of some of these results. In particular, we establish the vanishing of the theta invariant for isolated hypersurface singularities of even dimension in characteristic [math] under some mild extra assumptions. This confirms, in a large number of cases, a conjecture of Hailong Dao. </p>projecteuclid.org/euclid.akt/1510841640_20171116091402Thu, 16 Nov 2017 09:14 ESTLow-dimensional Milnor–Witt stems over $\mathbb R$https://projecteuclid.org/euclid.akt/1510841641<strong>Daniel Dugger</strong>, <strong>Daniel Isaksen</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 2, 175--210.</p><p><strong>Abstract:</strong><br/>
We compute some motivic stable homotopy groups over [math] . For [math] , we describe the motivic stable homotopy groups [math] of a completion of the motivic sphere spectrum. These are the first four Milnor–Witt stems. We start with the known [math] groups over [math] and apply the [math] -Bockstein spectral sequence to obtain [math] groups over [math] . This is the input to an Adams spectral sequence, which collapses in our low-dimensional range.
</p>projecteuclid.org/euclid.akt/1510841641_20171116091402Thu, 16 Nov 2017 09:14 ESTLongitudes in $\mathrm{SL}_2$ representations of link groups and Milnor–Witt $K_2$-groups of fieldshttps://projecteuclid.org/euclid.akt/1510841642<strong>Takefumi Nosaka</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 2, 211--233.</p><p><strong>Abstract:</strong><br/>
We describe an arithmetic [math] -valued invariant for longitudes of a link [math] , obtained from an [math] representation of the link group. Furthermore, we show a nontriviality on the elements, and compute the elements for some links. As an application, we develop a method for computing longitudes in [math] representations for link groups, where [math] is the universal covering group of [math] .
</p>projecteuclid.org/euclid.akt/1510841642_20171116091402Thu, 16 Nov 2017 09:14 ESTEquivariant vector bundles, their derived category and $K$-theory on affine schemeshttps://projecteuclid.org/euclid.akt/1510841643<strong>Amalendu Krishna</strong>, <strong>Charanya Ravi</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 2, 235--275.</p><p><strong>Abstract:</strong><br/> Let [math] be an affine group scheme over a noetherian commutative ring [math] . We show that every [math] -equivariant vector bundle on an affine toric scheme over [math] with [math] -action is equivariantly extended from [math] for several cases of [math] and [math] . We show that, given two affine schemes with group scheme actions, an equivalence of the equivariant derived categories implies isomorphism of the equivariant [math] -theories as well as equivariant [math] -theories. </p>projecteuclid.org/euclid.akt/1510841643_20171116091402Thu, 16 Nov 2017 09:14 ESTMotivic complexes over nonperfect fieldshttps://projecteuclid.org/euclid.akt/1510841644<strong>Andrei Suslin</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 2, 277--302.</p><p><strong>Abstract:</strong><br/>
We show that the theory of motivic complexes developed by Voevodsky over perfect fields works over nonperfect fields as well provided that we work with sheaves with transfers of [math] -modules ( [math] ). In particular we show that every homotopy invariant sheaf with transfers of [math] -modules is strictly homotopy invariant.
</p>projecteuclid.org/euclid.akt/1510841644_20171116091402Thu, 16 Nov 2017 09:14 EST$K\mkern-2mu$-theory of derivators revisitedhttps://projecteuclid.org/euclid.akt/1510841645<strong>Fernando Muro</strong>, <strong>Georgios Raptis</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 2, 303--340.</p><p><strong>Abstract:</strong><br/>
We define a [math] -theory for pointed right derivators and show that it agrees with Waldhausen [math] -theory in the case where the derivator arises from a good Waldhausen category. This [math] -theory is not invariant under general equivalences of derivators, but only under a stronger notion of equivalence that is defined by considering a simplicial enrichment of the category of derivators. We show that derivator [math] -theory, as originally defined, is the best approximation to Waldhausen [math] -theory by a functor that is invariant under equivalences of derivators.
</p>projecteuclid.org/euclid.akt/1510841645_20171116091402Thu, 16 Nov 2017 09:14 ESTChow groups of some generically twisted flag varietieshttps://projecteuclid.org/euclid.akt/1510841646<strong>Nikita Karpenko</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 2, 341--356.</p><p><strong>Abstract:</strong><br/>
We classify the split simple affine algebraic groups [math] of types A and C over a field with the property that the Chow group of the quotient variety [math] is torsion-free, where [math] is a special parabolic subgroup (e.g., a Borel subgroup) and [math] is a generic [math] -torsor (over a field extension of the base field). Examples of [math] include the adjoint groups of type A. Examples of [math] include the Severi–Brauer varieties of generic central simple algebras.
</p>projecteuclid.org/euclid.akt/1510841646_20171116091402Thu, 16 Nov 2017 09:14 ESTRational mixed Tate motivic graphshttps://projecteuclid.org/euclid.akt/1510841679<strong>Susama Agarwala</strong>, <strong>Owen Patashnick</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 4, 451--515.</p><p><strong>Abstract:</strong><br/>
We study the combinatorics of a subcomplex of the Bloch–Kriz cycle complex that was used to construct the category of mixed Tate motives. The algebraic cycles we consider properly contain the subalgebra of cycles that correspond to multiple logarithms (as defined by Gangl, Goncharov and Levin). We associate an algebra of graphs to our subalgebra of algebraic cycles. We give a purely combinatorial criterion for admissibility. We show that sums of bivalent graphs correspond to coboundary elements of the algebraic cycle complex. Finally, we compute the Hodge realization for an infinite family of algebraic cycles represented by sums of graphs that are not describable in the combinatorial language of Gangl et al.
</p>projecteuclid.org/euclid.akt/1510841679_20171116091443Thu, 16 Nov 2017 09:14 ESTStable operations and cooperations in derived Witt theory with rational coefficientshttps://projecteuclid.org/euclid.akt/1510841680<strong>Alexey Ananyevskiy</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 2, Number 4, 517--560.</p><p><strong>Abstract:</strong><br/>
The algebras of stable operations and cooperations in derived Witt theory with rational coefficients are computed and an additive description of cooperations in derived Witt theory is given. The answer is parallel to the well-known case of K-theory of real vector bundles in topology. In particular, we show that stable operations in derived Witt theory with rational coefficients are given by the values on the powers of the Bott element.
</p>projecteuclid.org/euclid.akt/1510841680_20171116091443Thu, 16 Nov 2017 09:14 ESTHochschild homology, lax codescent, and duplicial structurehttps://projecteuclid.org/euclid.akt/1513774600<strong>Richard Garner</strong>, <strong>Stephen Lack</strong>, <strong>Paul Slevin</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 1, 1--31.</p><p><strong>Abstract:</strong><br/>
We study the duplicial objects of Dwyer and Kan, which generalize the cyclic objects of Connes. We describe duplicial objects in terms of the decalage comonads, and we give a conceptual account of the construction of duplicial objects due to Böhm and Ştefan. This is done in terms of a 2-categorical generalization of Hochschild homology. We also study duplicial structure on nerves of categories, bicategories, and monoidal categories.
</p>projecteuclid.org/euclid.akt/1513774600_20171220075645Wed, 20 Dec 2017 07:56 ESTLocalization, Whitehead groups and the Atiyah conjecturehttps://projecteuclid.org/euclid.akt/1513774601<strong>Wolfgang Lück</strong>, <strong>Peter Linnell</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 1, 33--53.</p><p><strong>Abstract:</strong><br/>
Let [math] be the [math] -group of square matrices over [math] which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let [math] be the division closure of [math] in the algebra [math] of operators affiliated to the group von Neumann algebra. Let [math] be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let [math] be a torsionfree group which belongs to [math] . Then we prove that [math] is isomorphic to [math] . Furthermore we show that [math] is a skew field and hence [math] is the abelianization of the multiplicative group of units in [math] .
</p>projecteuclid.org/euclid.akt/1513774601_20171220075645Wed, 20 Dec 2017 07:56 ESTSuslin's moving lemma with modulushttps://projecteuclid.org/euclid.akt/1513774602<strong>Wataru Kai</strong>, <strong>Hiroyasu Miyazaki</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 1, 55--70.</p><p><strong>Abstract:</strong><br/>
The moving lemma of Suslin (also known as the generic equidimensionality theorem) states that a cycle on [math] meeting all faces properly can be moved so that it becomes equidimensional over [math] . This leads to an isomorphism of motivic Borel–Moore homology and higher Chow groups.
In this short paper we formulate and prove a variant of this. It leads to a modulus version of the isomorphism, in an appropriate pro setting.
</p>projecteuclid.org/euclid.akt/1513774602_20171220075645Wed, 20 Dec 2017 07:56 ESTAbstract tilting theory for quivers and related categorieshttps://projecteuclid.org/euclid.akt/1513774603<strong>Moritz Groth</strong>, <strong>Jan Šťovíček</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 1, 71--124.</p><p><strong>Abstract:</strong><br/>
We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between the corresponding representation theories with values in arbitrary stable homotopy theories, including representations over fields, rings or schemes as well as differential-graded and spectral representations.
Specializing to representations over a field and to specific shapes, this recovers derived equivalences of Happel for finite, acyclic quivers. However, even over a field our main result leads to new derived equivalences, for example, for not necessarily finite or acyclic quivers.
Our results rely on a careful analysis of the compatibility of gluing constructions for small categories with homotopy Kan extensions and homotopical epimorphisms, and on a study of the combinatorics of amalgamations of categories.
</p>projecteuclid.org/euclid.akt/1513774603_20171220075645Wed, 20 Dec 2017 07:56 ESTEquivariant noncommutative motiveshttps://projecteuclid.org/euclid.akt/1513774604<strong>Gonçalo Tabuada</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 1, 125--156.</p><p><strong>Abstract:</strong><br/>
Given a finite group [math] , we develop a theory of [math] -equivariant noncommutative motives. This theory provides a well-adapted framework for the study of [math] -schemes, Picard groups of schemes, [math] -algebras, [math] -cocycles, [math] -equivariant algebraic [math] -theory, etc. Among other results, we relate our theory with its commutative counterpart as well as with Panin’s theory. As a first application, we extend Panin’s computations, concerning twisted projective homogeneous varieties, to a large class of invariants. As a second application, we prove that whenever the category of perfect complexes of a [math] -scheme [math] admits a full exceptional collection of [math] -invariant ( [math] -equivariant) objects, the [math] -equivariant Chow motive of [math] is of Lefschetz type. Finally, we construct a [math] -equivariant motivic measure with values in the Grothendieck ring of [math] -equivariant noncommutative Chow motives.
</p>projecteuclid.org/euclid.akt/1513774604_20171220075645Wed, 20 Dec 2017 07:56 ESTCohomologie non ramifiée de degré 3 : variétés cellulaires et surfaces de del Pezzo de degré au moins 5https://projecteuclid.org/euclid.akt/1513774605<strong>Yang Cao</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 1, 157--171.</p><p><strong>Abstract:</strong><br/>
Dans cet article, où le corps de base est un corps de caractéristique zéro quelconque, pour [math] une variété géométriquement cellulaire, on étudie le quotient du troisième groupe de cohomologie non ramifiée [math] par sa partie constante. Pour [math] une compactification lisse d’un torseur universel sur une surface géométriquement rationnelle, on montre que ce quotient est fini. Pour [math] une surface de del Pezzo de degré [math] , on montre que ce quotient est trivial, sauf si [math] est une surface de del Pezzo de degré 8 d’un type particulier.
We consider geometrically cellular varieties [math] over an arbitrary field of characteristic zero. We study the quotient of the third unramified cohomology group [math] by its constant part. For [math] a smooth compactification of a universal torsor over a geometrically rational surface, we show that this quotient is finite. For [math] a del Pezzo surface of degree [math] , we show that this quotient is zero, unless [math] is a del Pezzo surface of degree 8 of a special type.
</p>projecteuclid.org/euclid.akt/1513774605_20171220075645Wed, 20 Dec 2017 07:56 ESTAn explicit basis for the rational higher Chow groups of abelian number fieldshttps://projecteuclid.org/euclid.akt/1522807253<strong>Matthew Kerr</strong>, <strong>Yu Yang</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 2, 173--191.</p><p><strong>Abstract:</strong><br/>
We review and simplify A. Beĭlinson’s construction of a basis for the motivic cohomology of a point over a cyclotomic field, then promote the basis elements to higher Chow cycles and evaluate the KLM regulator map on them.
</p>projecteuclid.org/euclid.akt/1522807253_20180403220109Tue, 03 Apr 2018 22:01 EDTAlgebraic $K$-theory and a semifinite Fuglede–Kadison determinanthttps://projecteuclid.org/euclid.akt/1522807257<strong>Peter Hochs</strong>, <strong>Jens Kaad</strong>, <strong>André Schemaitat</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 2, 193--206.</p><p><strong>Abstract:</strong><br/>
In this paper we apply algebraic [math] -theory techniques to construct a Fuglede–Kadison type determinant for a semifinite von Neumann algebra equipped with a fixed trace. Our construction is based on the approach to determinants for Banach algebras developed by Skandalis and de la Harpe. This approach can be extended to the semifinite case since the first topological [math] -group of the trace ideal in a semifinite von Neumann algebra is trivial. Along the way we also improve the methods of Skandalis and de la Harpe by considering relative [math] -groups with respect to an ideal instead of the usual absolute [math] -groups. Our construction recovers the determinant homomorphism introduced by Brown, but all the relevant algebraic properties are automatic due to the algebraic [math] -theory framework.
</p>projecteuclid.org/euclid.akt/1522807257_20180403220109Tue, 03 Apr 2018 22:01 EDTAlgebraic $K$-theory of quotient stackshttps://projecteuclid.org/euclid.akt/1522807258<strong>Amalendu Krishna</strong>, <strong>Charanya Ravi</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 2, 207--233.</p><p><strong>Abstract:</strong><br/>
We prove some fundamental results like localization, excision, Nisnevich descent, and the regular blow-up formula for the algebraic [math] -theory of certain stack quotients of schemes with affine group scheme actions. We show that the homotopy [math] -theory of such stacks is homotopy invariant. This implies a similar homotopy invariance property of the algebraic [math] -theory with coefficients.
</p>projecteuclid.org/euclid.akt/1522807258_20180403220109Tue, 03 Apr 2018 22:01 EDTA fixed point theorem on noncompact manifoldshttps://projecteuclid.org/euclid.akt/1522807259<strong>Peter Hochs</strong>, <strong>Hang Wang</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 2, 235--286.</p><p><strong>Abstract:</strong><br/>
We generalise the Atiyah–Segal–Singer fixed point theorem to noncompact manifolds. Using [math] -theory, we extend the equivariant index to the noncompact setting, and obtain a fixed point formula for it. The fixed point formula is the explicit cohomological expression from Atiyah–Segal–Singer’s result. In the noncompact case, however, we show in examples that this expression yields characters of infinite-dimensional representations. In one example, we realise characters of discrete series representations on the regular elements of a maximal torus, in terms of the index we define. Further results are a fixed point formula for the index pairing between equivariant [math] -theory and [math] -homology, and a nonlocalised expression for the index we use, in terms of deformations of principal symbols. The latter result is one of several links we find to indices of deformed symbols and operators studied by various authors.
</p>projecteuclid.org/euclid.akt/1522807259_20180403220109Tue, 03 Apr 2018 22:01 EDTConnectedness of cup products for polynomial representations of $\mathrm{GL}_n$ and applicationshttps://projecteuclid.org/euclid.akt/1522807261<strong>Antoine Touzé</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 2, 287--329.</p><p><strong>Abstract:</strong><br/>
We find conditions such that cup products induce isomorphisms in low degrees for extensions between stable polynomial representations of the general linear group. We apply this result to prove generalizations and variants of the Steinberg tensor product theorem. Our connectedness bounds for cup product maps depend on numerical invariants which seem also relevant to other problems, such as the cohomological behavior of the Schur functor.
</p>projecteuclid.org/euclid.akt/1522807261_20180403220109Tue, 03 Apr 2018 22:01 EDTStable $\mathbb{A}^1$-connectivity over Dedekind schemeshttps://projecteuclid.org/euclid.akt/1522807265<strong>Johannes Schmidt</strong>, <strong>Florian Strunk</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 2, 331--367.</p><p><strong>Abstract:</strong><br/>
We show that [math] -localization decreases the stable connectivity by at most one over a Dedekind scheme with infinite residue fields. For the proof, we establish a version of Gabber’s geometric presentation lemma over a henselian discrete valuation ring with infinite residue field.
</p>projecteuclid.org/euclid.akt/1522807265_20180403220109Tue, 03 Apr 2018 22:01 EDTTriple linkagehttps://projecteuclid.org/euclid.akt/1532397762<strong>Karim Johannes Becher</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 3, 369--378.</p><p><strong>Abstract:</strong><br/>
We study the condition on a field that any triple of (bilinear) Pfister forms of a given dimension are linked. This is a strengthening of the condition of linkage investigated by Elman and Lam, which asks the same for pairs of Pfister forms. In characteristic different from two this condition for triples of [math] -fold Pfister forms is related to the Hasse number.
</p>projecteuclid.org/euclid.akt/1532397762_20180723220301Mon, 23 Jul 2018 22:03 EDT$\mathbb A^1$-equivalence of zero cycles on surfaces, IIhttps://projecteuclid.org/euclid.akt/1532397763<strong>Qizheng Yin</strong>, <strong>Yi Zhu</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 3, 379--393.</p><p><strong>Abstract:</strong><br/>
Using recent developments in the theory of mixed motives, we prove that the log Bloch conjecture holds for an open smooth complex surface if the Bloch conjecture holds for its compactification. This verifies the log Bloch conjecture for all [math] -homology planes and for open smooth surfaces which are not of log general type.
</p>projecteuclid.org/euclid.akt/1532397763_20180723220301Mon, 23 Jul 2018 22:03 EDTTopological K-theory of affine Hecke algebrashttps://projecteuclid.org/euclid.akt/1532397764<strong>Maarten Solleveld</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 3, 395--460.</p><p><strong>Abstract:</strong><br/>
Let [math] be an affine Hecke algebra with a positive parameter function [math] . We are interested in the topological K-theory of its [math] -completion [math] . We prove that [math] does not depend on the parameter [math] , solving a long-standing conjecture of Higson and Plymen. For this we use representation-theoretic methods, in particular elliptic representations of Weyl groups and Hecke algebras.
Thus, for the computation of these K-groups it suffices to work out the case [math] . These algebras are considerably simpler than for [math] , just crossed products of commutative algebras with finite Weyl groups. We explicitly determine [math] for all classical root data [math] . This will be useful for analyzing the K-theory of the reduced [math] -algebra of any classical [math] -adic group.
For the computations in the case [math] , we study the more general situation of a finite group [math] acting on a smooth manifold [math] . We develop a method to calculate the K-theory of the crossed product [math] . In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.
</p>projecteuclid.org/euclid.akt/1532397764_20180723220301Mon, 23 Jul 2018 22:03 EDTOn a localization formula of epsilon factors via microlocal geometryhttps://projecteuclid.org/euclid.akt/1532397765<strong>Tomoyuki Abe</strong>, <strong>Deepam Patel</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 3, 461--490.</p><p><strong>Abstract:</strong><br/>
Given a lisse [math] -adic sheaf [math] on a smooth proper variety [math] and a lisse sheaf [math] on an open dense [math] in [math] , Kato and Saito conjectured a localization formula for the global [math] -adic epsilon factor [math] in terms of the global epsilon factor of [math] and a certain intersection number associated to [math] and the Swan class of [math] . In this article, we prove an analog of this conjecture for global de Rham epsilon factors in the classical setting of [math] -modules on smooth projective varieties over a field of characteristic zero.
</p>projecteuclid.org/euclid.akt/1532397765_20180723220301Mon, 23 Jul 2018 22:03 EDTPoincaré duality and Langlands duality for extended affine Weyl groupshttps://projecteuclid.org/euclid.akt/1532397766<strong>Graham A. Niblo</strong>, <strong>Roger Plymen</strong>, <strong>Nick Wright</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 3, 491--522.</p><p><strong>Abstract:</strong><br/>
In this paper we construct an equivariant Poincaré duality between dual tori equipped with finite group actions. We use this to demonstrate that Langlands duality induces a rational isomorphism between the group [math] -algebras of extended affine Weyl groups at the level of [math] -theory.
</p>projecteuclid.org/euclid.akt/1532397766_20180723220301Mon, 23 Jul 2018 22:03 EDTGeometric obstructions for Fredholm boundary conditions for manifolds with cornershttps://projecteuclid.org/euclid.akt/1532397767<strong>Paulo Carrillo Rouse</strong>, <strong>Jean-Marie Lescure</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 3, 523--563.</p><p><strong>Abstract:</strong><br/>
For every connected manifold with corners there is a homology theory called conormal homology, defined in terms of faces and orientation of their conormal bundle and whose cycles correspond geometrically to corner cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space), [math] , is given by the alternating sum of the number of (open) faces of a given codimension.
The main result of the present paper is that for a compact connected manifold with corners [math] , given as a finite product of manifolds with corners of codimension less or equal to three, we have that:
1) If [math] satisfies the Fredholm perturbation property (every elliptic pseudodifferential b-operator on [math] can be perturbed by a b-regularizing operator so it becomes Fredholm) then the even Euler corner character of [math] vanishes, i.e., [math] .
2) If the even periodic conormal homology group vanishes, i.e., [math] , then [math] satisfies the stably homotopic Fredholm perturbation property (i.e., every elliptic pseudodifferential b-operator on [math] satisfies the same named property up to stable homotopy among elliptic operators).
3) If [math] is torsion free and if the even Euler corner character of [math] vanishes, i.e., [math] , then [math] satisfies the stably homotopic Fredholm perturbation property. For example, for every finite product of manifolds with corners of codimension at most two the conormal homology groups are torsion free.
The main theorem behind the above result is the explicit computation in terms of conormal homology of the K-theory groups of the algebra [math] of b-compact operators for [math] as above. Our computation unifies the known cases of codimension zero (smooth manifolds) and of codimension one (smooth manifolds with boundary).
</p>projecteuclid.org/euclid.akt/1532397767_20180723220301Mon, 23 Jul 2018 22:03 EDTPositive scalar curvature and low-degree group homologyhttps://projecteuclid.org/euclid.akt/1532397768<strong>Noé Bárcenas</strong>, <strong>Rudolf Zeidler</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 3, 565--579.</p><p><strong>Abstract:</strong><br/>
Let [math] be a discrete group. Assuming rational injectivity of the Baum–Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz’ positive scalar curvature sequence for [math] . The lower bounds are formulated in terms of the part of degree up to [math] in the group homology of [math] with coefficients in the [math] -module generated by finite order elements. Our results use and extend work of Botvinnik and Gilkey which treated the case of finite groups. Further crucial ingredients are a real counterpart to the delocalized equivariant Chern character and Matthey’s work on explicitly inverting this Chern character in low homological degrees.
</p>projecteuclid.org/euclid.akt/1532397768_20180723220301Mon, 23 Jul 2018 22:03 EDTThe $A_\infty$-structure of the index maphttps://projecteuclid.org/euclid.akt/1546657257<strong>Oliver Bräunling</strong>, <strong>Michael Groechenig</strong>, <strong>Jesse Wolfson</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 4, 581--614.</p><p><strong>Abstract:</strong><br/>
Let [math] be a local field with residue field [math] . The classifying space of [math] comes canonically equipped with a map to the delooping of the [math] -theory space of [math] . Passing to loop spaces, such a map abstractly encodes a homotopy coherently associative map of [math] -spaces [math] . Using a generalized Waldhausen construction, we construct an explicit model built for the [math] -structure of this map, built from nested systems of lattices in [math] . More generally, we construct this model in the framework of Tate objects in exact categories, with finite dimensional vector spaces over local fields as a motivating example.
</p>projecteuclid.org/euclid.akt/1546657257_20190104220104Fri, 04 Jan 2019 22:01 ESTLocalization $C^*$-algebras and $K$-theoretic dualityhttps://projecteuclid.org/euclid.akt/1546657258<strong>Marius Dadarlat</strong>, <strong>Rufus Willett</strong>, <strong>Jianchao Wu</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 4, 615--630.</p><p><strong>Abstract:</strong><br/>
Based on the localization algebras of Yu, and their subsequent analysis by Qiao and Roe, we give a new picture of [math] -theory in terms of time-parametrized families of (locally) compact operators that asymptotically commute with appropriate representations.
</p>projecteuclid.org/euclid.akt/1546657258_20190104220104Fri, 04 Jan 2019 22:01 ESTHecke modules for arithmetic groups via bivariant $K$-theoryhttps://projecteuclid.org/euclid.akt/1546657259<strong>Bram Mesland</strong>, <strong>Mehmet Haluk Şengün</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 4, 631--656.</p><p><strong>Abstract:</strong><br/>
Let [math] be a lattice in a locally compact group [math] . In another work, we used [math] -theory to equip with Hecke operators the [math] -groups of any [math] - [math] -algebra on which the commensurator of [math] acts. When [math] is arithmetic, this gives Hecke operators on the [math] -theory of certain [math] -algebras that are naturally associated with [math] . In this paper, we first study the topological [math] -theory of the arithmetic manifold associated to [math] . We prove that the Chern character commutes with Hecke operators. Afterwards, we show that the Shimura product of double cosets naturally corresponds to the Kasparov product and thus that the [math] -groups associated to an arithmetic group [math] become true Hecke modules. We conclude by discussing Hecke equivariant maps in [math] -theory in great generality and apply this to the Borel–Serre compactification as well as various noncommutative compactifications associated with [math] . Along the way we discuss the relation between the [math] -theory and the integral cohomology of low-dimensional manifolds as Hecke modules.
</p>projecteuclid.org/euclid.akt/1546657259_20190104220104Fri, 04 Jan 2019 22:01 ESTThe slice spectral sequence for singular schemes and applicationshttps://projecteuclid.org/euclid.akt/1546657260<strong>Amalendu Krishna</strong>, <strong>Pablo Pelaez</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 4, 657--708.</p><p><strong>Abstract:</strong><br/>
We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic [math] -spectra, especially the motivic cobordism spectrum. When the base field [math] admits resolution of singularities and [math] is a scheme of finite type over [math] , we show that Voevodsky’s slice filtration leads to a spectral sequence for [math] whose terms are the motivic cohomology groups of [math] defined using the cdh-hypercohomology. As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of [math] .
A similar spectral sequence for the connective [math] -theory leads to a cycle class map from the motivic cohomology to the homotopy invariant [math] -theory of [math] . We show that this cycle class map is injective for a large class of projective schemes. We also deduce applications to the torsion in the motivic cohomology of singular schemes.
</p>projecteuclid.org/euclid.akt/1546657260_20190104220104Fri, 04 Jan 2019 22:01 ESTK-theory, local cohomology and tangent spaces to Hilbert schemeshttps://projecteuclid.org/euclid.akt/1546657261<strong>Sen Yang</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 4, 709--722.</p><p><strong>Abstract:</strong><br/>
Using K-theory, we construct a map [math] from the tangent space to the Hilbert scheme at a point [math] to the local cohomology group. We use this map [math] to answer (after slight modification) a question by Mark Green and Phillip Griffiths on constructing a map from the tangent space [math] to the Hilbert scheme at a point [math] to the tangent space to the cycle group [math] .
</p>projecteuclid.org/euclid.akt/1546657261_20190104220104Fri, 04 Jan 2019 22:01 ESTDroites sur les hypersurfaces cubiqueshttps://projecteuclid.org/euclid.akt/1546657262<strong>Jean-Louis Colliot-Thélène</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 3, Number 4, 723--728.</p><p><strong>Abstract:</strong><br/>
Over any complex cubic hypersurface of dimension at least 2, the Chow group of 1-dimensional cycles is spanned by the lines lying on the hypersurface. The smooth case had already been given several other proofs.
</p>projecteuclid.org/euclid.akt/1546657262_20190104220104Fri, 04 Jan 2019 22:01 ESTSegal operations in the algebraic $K$-theory of topological spaceshttps://projecteuclid.org/euclid.akt/1554775254<strong>Thomas Gunnarsson</strong>, <strong>Ross Staffeldt</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 4, Number 1, 1--56.</p><p><strong>Abstract:</strong><br/>
We extend earlier work of Waldhausen which defines operations on the algebraic [math] -theory of the one-point space. For a connected simplicial abelian group [math] and symmetric groups [math] , we define operations [math] in the algebraic [math] -theory of spaces. We show that our operations can be given the structure of [math] -maps. Let [math] be the [math] -transfer. We also develop an inductive procedure to compute the compositions [math] , and outline some applications.
</p>projecteuclid.org/euclid.akt/1554775254_20190408220056Mon, 08 Apr 2019 22:00 EDTOn the Farrell–Jones conjecture for algebraic $K$-theory of spaces: the Farrell–Hsiang methodhttps://projecteuclid.org/euclid.akt/1554775255<strong>Mark Ullmann</strong>, <strong>Christoph Winges</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 4, Number 1, 57--138.</p><p><strong>Abstract:</strong><br/>
We prove the Farrell–Jones conjecture for algebraic [math] -theory of spaces for virtually poly- [math] -groups. For this, we transfer the “Farrell–Hsiang method” from the linear case to categories of equivariant, controlled retractive spaces.
</p>projecteuclid.org/euclid.akt/1554775255_20190408220056Mon, 08 Apr 2019 22:00 EDTAlmost complex structures on connected sums of complex projective spaceshttps://projecteuclid.org/euclid.akt/1554775256<strong>Oliver Goertsches</strong>, <strong>Panagiotis Konstantis</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 4, Number 1, 139--149.</p><p><strong>Abstract:</strong><br/>
We show that the [math] -fold connected sum [math] admits an almost complex structure if and only if [math] is odd.
</p>projecteuclid.org/euclid.akt/1554775256_20190408220056Mon, 08 Apr 2019 22:00 EDT$G$-theory of root stacks and equivariant $K$-theoryhttps://projecteuclid.org/euclid.akt/1565661790<strong>Ajneet Dhillon</strong>, <strong>Ivan Kobyzev</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 4, Number 2, 151--183.</p><p><strong>Abstract:</strong><br/>
Using the description of the category of quasicoherent sheaves on a root stack, we compute the [math] -theory of root stacks via localization methods. We apply our results to the study of equivariant [math] -theory of algebraic varieties under certain conditions.
</p>projecteuclid.org/euclid.akt/1565661790_20190812220315Mon, 12 Aug 2019 22:03 EDTOrbital integrals and $K$-theory classeshttps://projecteuclid.org/euclid.akt/1565661791<strong>Peter Hochs</strong>, <strong>Hang Wang</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 4, Number 2, 185--209.</p><p><strong>Abstract:</strong><br/>
Let [math] be a semisimple Lie group with discrete series. We use maps [math] defined by orbital integrals to recover group theoretic information about [math] , including information contained in [math] -theory classes not associated to the discrete series. An important tool is a fixed point formula for equivariant indices obtained by the authors in an earlier paper. Applications include a tool to distinguish classes in [math] , the (known) injectivity of Dirac induction, versions of Selberg’s principle in [math] -theory and for matrix coefficients of the discrete series, a Tannaka-type duality, and a way to extract characters of representations from [math] -theory. Finally, we obtain a continuity property near the identity element of [math] of families of maps [math] , parametrised by semisimple elements of [math] , defined by stable orbital integrals. This implies a continuity property for [math] -packets of discrete series characters, which in turn can be used to deduce a (well-known) expression for formal degrees of discrete series representations from Harish-Chandra’s character formula.
</p>projecteuclid.org/euclid.akt/1565661791_20190812220315Mon, 12 Aug 2019 22:03 EDTOn derived categories of arithmetic toric varietieshttps://projecteuclid.org/euclid.akt/1565661792<strong>Matthew Ballard</strong>, <strong>Alexander Duncan</strong>, <strong>Patrick McFaddin</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 4, Number 2, 211--242.</p><p><strong>Abstract:</strong><br/>
We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections, making it possible to give concrete descriptions of their derived categories. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskiĭ and Klyachko, and toric varieties associated to Weyl fans of type [math] . Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly nontoric) varieties over nonclosed fields.
</p>projecteuclid.org/euclid.akt/1565661792_20190812220315Mon, 12 Aug 2019 22:03 EDT$K$-theory of Hermitian Mackey functors, real traces, and assemblyhttps://projecteuclid.org/euclid.akt/1565661793<strong>Emanuele Dotto</strong>, <strong>Crichton Ogle</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 4, Number 2, 243--316.</p><p><strong>Abstract:</strong><br/>
We define a [math] -equivariant real algebraic [math] -theory spectrum [math] , for every [math] -equivariant spectrum [math] equipped with a compatible multiplicative structure. This construction extends the real algebraic [math] -theory of Hesselholt and Madsen for discrete rings, and the Hermitian [math] -theory of Burghelea and Fiedorowicz for simplicial rings. It supports a trace map of [math] -spectra [math] to the real topological Hochschild homology spectrum, which extends the [math] -theoretic trace of Bökstedt, Hsiang and Madsen.
We show that the trace provides a splitting of the real [math] -theory of the spherical group-ring. We use the splitting induced on the geometric fixed points of [math] , which we regard as an [math] -theory of [math] -equivariant ring spectra, to give a purely homotopy theoretic reformulation of the Novikov conjecture on the homotopy invariance of the higher signatures, in terms of the module structure of the rational [math] -theory of the “Burnside group-ring”.
</p>projecteuclid.org/euclid.akt/1565661793_20190812220315Mon, 12 Aug 2019 22:03 EDTOn the K-theory coniveau epimorphism for products of Severi–Brauer varietieshttps://projecteuclid.org/euclid.akt/1565661794<strong>Nikita Karpenko</strong>, <strong>Eoin Mackall</strong>. <p><strong>Source: </strong>Annals of K-Theory, Volume 4, Number 2, 317--344.</p><p><strong>Abstract:</strong><br/>
For [math] a product of Severi–Brauer varieties, we conjecture that if the Chow ring of [math] is generated by Chern classes, then the canonical epimorphism from the Chow ring of [math] to the graded ring associated to the coniveau filtration of the Grothendieck ring of [math] is an isomorphism. We show this conjecture is equivalent to the condition that if [math] is a split semisimple algebraic group of type [math] , [math] is a Borel subgroup of [math] and [math] is a standard generic [math] -torsor, then the canonical epimorphism from the Chow ring of [math] to the graded ring associated with the coniveau filtration of the Grothendieck ring of [math] is an isomorphism. In certain cases we verify this conjecture.
</p>projecteuclid.org/euclid.akt/1565661794_20190812220315Mon, 12 Aug 2019 22:03 EDT