Let $F$ be a local field with residue field $k$. The classifying space of $GL_n(F)$ comes canonically equipped with a map to the delooping of the $K$-theory space of $k$. Passing to loop spaces, such a map abstractly encodes a homotopy coherently associative map of A-infinity-spaces $GL_n(F)to K_k$. Using a generalized Waldhausen construction, we construct an explicit model built for the $A_infty$-structure of this map, built from nested systems of lattices in $F^n$. 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.
{"title":"The A∞-structure of the index map","authors":"O. Braunling, M. Groechenig, J. Wolfson","doi":"10.2140/AKT.2018.3.581","DOIUrl":"https://doi.org/10.2140/AKT.2018.3.581","url":null,"abstract":"Let $F$ be a local field with residue field $k$. The classifying space of $GL_n(F)$ comes canonically equipped with a map to the delooping of the $K$-theory space of $k$. Passing to loop spaces, such a map abstractly encodes a homotopy coherently associative map of A-infinity-spaces $GL_n(F)to K_k$. Using a generalized Waldhausen construction, we construct an explicit model built for the $A_infty$-structure of this map, built from nested systems of lattices in $F^n$. 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.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/AKT.2018.3.581","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45274698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We use techniques of relative algebraic K-theory to develop a common refinement of the existing theories of metrized and hermitian Galois structures in arithmetic. As a first application of this very general approach, we then use it to prove several new results, and to formulate a framework of new conjectures, concerning the detailed arithmetic properties of wildly ramified Galois-Gauss sums.
{"title":"On refined metric and hermitian structures in\u0000arithmetic, I : Galois–Gauss sums and weak ramification","authors":"W. Bley, D. Burns, Carl Hahn","doi":"10.2140/akt.2020.5.79","DOIUrl":"https://doi.org/10.2140/akt.2020.5.79","url":null,"abstract":"We use techniques of relative algebraic K-theory to develop a common refinement of the existing theories of metrized and hermitian Galois structures in arithmetic. As a first application of this very general approach, we then use it to prove several new results, and to formulate a framework of new conjectures, concerning the detailed arithmetic properties of wildly ramified Galois-Gauss sums.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2020.5.79","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48743209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using the theory of framed correspondences developed by Voevodsky [24] and the machinery of framed motives introduced and developed in [6], various explicit fibrant resolutions for a motivic Thom spectrum $E$ are constructed in this paper. It is shown that the bispectrum $$M_E^{mathbb G}(X)=(M_{E}(X),M_{E}(X)(1),M_{E}(X)(2),ldots),$$ each term of which is a twisted $E$-framed motive of $X$, introduced in the paper, represents $X_+wedge E$ in the category of bispectra. As a topological application, it is proved that the $E$-framed motive with finite coefficients $M_E(pt)(pt)/N$, $N>0$, of the point $pt=Spec (k)$ evaluated at $pt$ is a quasi-fibrant model of the topological $S^2$-spectrum $Re^epsilon(E)/N$ whenever the base field $k$ is algebraically closed of characteristic zero with an embedding $epsilon:khookrightarrowmathbb C$. Furthermore, the algebraic cobordism spectrum $MGL$ is computed in terms of $Omega$-correspondences in the sense of [15]. It is also proved that $MGL$ is represented by a bispectrum each term of which is a sequential colimit of simplicial smooth quasi-projective varieties.
{"title":"Fibrant resolutions for motivic Thom spectra","authors":"G. Garkusha, A. Neshitov","doi":"10.2140/akt.2023.8.421","DOIUrl":"https://doi.org/10.2140/akt.2023.8.421","url":null,"abstract":"Using the theory of framed correspondences developed by Voevodsky [24] and the machinery of framed motives introduced and developed in [6], various explicit fibrant resolutions for a motivic Thom spectrum $E$ are constructed in this paper. It is shown that the bispectrum \u0000$$M_E^{mathbb G}(X)=(M_{E}(X),M_{E}(X)(1),M_{E}(X)(2),ldots),$$ each term of which is a twisted $E$-framed motive of $X$, introduced in the paper, represents $X_+wedge E$ in the category of bispectra. As a topological application, it is proved that the $E$-framed motive with finite coefficients $M_E(pt)(pt)/N$, $N>0$, of the point $pt=Spec (k)$ evaluated at $pt$ is a quasi-fibrant model of the topological $S^2$-spectrum $Re^epsilon(E)/N$ whenever the base field $k$ is algebraically closed of characteristic zero with an embedding $epsilon:khookrightarrowmathbb C$. Furthermore, the algebraic cobordism spectrum $MGL$ is computed in terms of $Omega$-correspondences in the sense of [15]. It is also proved that $MGL$ is represented by a bispectrum each term of which is a sequential colimit of simplicial smooth quasi-projective varieties.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42522433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $G$ be a semisimple Lie group with discrete series. We use maps $K_0(C^*_rG)to mathbb{C}$ defined by orbital integrals to recover group theoretic information about $G$, including information contained in $K$-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 $K_0(C^*_rG)$, the (known) injectivity of Dirac induction, versions of Selberg's principle in $K$-theory and for matrix coefficients of the discrete series, a Tannaka-type duality, and a way to extract characters of representations from $K$-theory. Finally, we obtain a continuity property near the identity element of $G$ of families of maps $K_0(C^*_rG)to mathbb{C}$, parametrised by semisimple elements of $G$, defined by stable orbital integrals. This implies a continuity property for $L$-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.
{"title":"Orbital integrals and K-theory classes","authors":"P. Hochs, Han Wang","doi":"10.2140/akt.2019.4.185","DOIUrl":"https://doi.org/10.2140/akt.2019.4.185","url":null,"abstract":"Let $G$ be a semisimple Lie group with discrete series. We use maps $K_0(C^*_rG)to mathbb{C}$ defined by orbital integrals to recover group theoretic information about $G$, including information contained in $K$-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 $K_0(C^*_rG)$, the (known) injectivity of Dirac induction, versions of Selberg's principle in $K$-theory and for matrix coefficients of the discrete series, a Tannaka-type duality, and a way to extract characters of representations from $K$-theory. Finally, we obtain a continuity property near the identity element of $G$ of families of maps $K_0(C^*_rG)to mathbb{C}$, parametrised by semisimple elements of $G$, defined by stable orbital integrals. This implies a continuity property for $L$-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.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2019.4.185","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45261556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study the Witt groups of symmetric and anti-symmetric forms on perverse sheaves on a finite-dimensional topologically stratified space with even dimensional strata. We show that the Witt group has a canonical decomposition as a direct sum of the Witt groups of shifted local systems on strata. We compare this with another `splitting decomposition' for Witt classes of perverse sheaves obtained inductively from our main new tool, a `splitting relation' which is a generalisation of isotropic reduction. The Witt groups we study are identified with the (non-trivial) Balmer-Witt groups of the constructible derived category of sheaves on the stratified space, and also with the corresponding cobordism groups defined by Youssin. Our methods are primarily algebraic and apply more widely. The general context in which we work is that of a triangulated category with duality, equipped with a self-dual t-structure with noetherian heart, glued from self-dual t-structures on a thick subcategory and its quotient.
{"title":"Witt groups of abelian categories and perverse sheaves","authors":"Jorg Schurmann, J. Woolf","doi":"10.2140/akt.2019.4.621","DOIUrl":"https://doi.org/10.2140/akt.2019.4.621","url":null,"abstract":"In this paper we study the Witt groups of symmetric and anti-symmetric forms on perverse sheaves on a finite-dimensional topologically stratified space with even dimensional strata. We show that the Witt group has a canonical decomposition as a direct sum of the Witt groups of shifted local systems on strata. We compare this with another `splitting decomposition' for Witt classes of perverse sheaves obtained inductively from our main new tool, a `splitting relation' which is a generalisation of isotropic reduction. \u0000The Witt groups we study are identified with the (non-trivial) Balmer-Witt groups of the constructible derived category of sheaves on the stratified space, and also with the corresponding cobordism groups defined by Youssin. \u0000Our methods are primarily algebraic and apply more widely. The general context in which we work is that of a triangulated category with duality, equipped with a self-dual t-structure with noetherian heart, glued from self-dual t-structures on a thick subcategory and its quotient.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2019.4.621","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41745908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Topological Period-Index Conjecture for\u0000spinc6-manifolds","authors":"D. Crowley, Mark Grant","doi":"10.2140/AKT.2020.5.605","DOIUrl":"https://doi.org/10.2140/AKT.2020.5.605","url":null,"abstract":"**On publication email publisher and request a copy of PDF for repository - see https://msp.org/publications/policies/**","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/AKT.2020.5.605","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49607883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be a Lie group with finitely many connected components satisfying the rapid decay (RD) property. As an example, we can take G to be a connected semisimple Lie group. Let M be a G-proper manifold with compact quotient M/G. In this paper we establish index formulae for the C^*-higher indices of a G-equivariant Dirac-type operator on M. We use these formulae to investigate geometric properties of suitably defined higher genera on M. In particular, we establish the G-proper homotopy invariance of the higher signatures of a G-proper manifold and the vanishing of the A-hat genera of a G-spin, G-proper manifold admitting a G-invariant metric of positive scalar curvature.
{"title":"Higher genera for proper actions of Lie groups","authors":"P. Piazza, H. Posthuma","doi":"10.2140/akt.2019.4.473","DOIUrl":"https://doi.org/10.2140/akt.2019.4.473","url":null,"abstract":"Let G be a Lie group with finitely many connected components satisfying the rapid decay (RD) property. As an example, we can take G to be a connected semisimple Lie group. Let M be a G-proper manifold with compact quotient M/G. In this paper we establish index formulae for the C^*-higher indices of a G-equivariant Dirac-type operator on M. We use these formulae to investigate geometric properties of suitably defined higher genera on M. In particular, we establish the G-proper homotopy invariance of the higher signatures of a G-proper manifold and the vanishing of the A-hat genera of a G-spin, G-proper manifold admitting a G-invariant metric of positive scalar curvature.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48297750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We classify all Witt invariants of the functor $I^n$ (powers of the fundamental ideal of the Witt ring), that is functions $I^n(K)rightarrow W(K)$ compatible with field extensions, and all mod 2 cohomological invariants, that is functions $I^n(K)rightarrow H^*(K,mu_2)$. This is done in both cases in terms of certain operations (denoted $pi_n^{d}$ and $u_{nd}^{(n)}$ respectively) looking like divided powers, which are shown to be independent and generate all invariants. This can be seen as a lifting of operations defined on mod 2 Milnor K-theory (or equivalently mod 2 Galois cohomology). We also study various properties of these invariants, including behaviour under similitudes, residues for discrete valuations, and restriction from $I^n$ to $I^{n+1}$. The goal is to use this to study invariants of algebras with involutions in future articles.
{"title":"Witt and cohomological invariants of Witt classes","authors":"N. Garrel","doi":"10.2140/AKT.2020.5.213","DOIUrl":"https://doi.org/10.2140/AKT.2020.5.213","url":null,"abstract":"We classify all Witt invariants of the functor $I^n$ (powers of the fundamental ideal of the Witt ring), that is functions $I^n(K)rightarrow W(K)$ compatible with field extensions, and all mod 2 cohomological invariants, that is functions $I^n(K)rightarrow H^*(K,mu_2)$. This is done in both cases in terms of certain operations (denoted $pi_n^{d}$ and $u_{nd}^{(n)}$ respectively) looking like divided powers, which are shown to be independent and generate all invariants. This can be seen as a lifting of operations defined on mod 2 Milnor K-theory (or equivalently mod 2 Galois cohomology). \u0000We also study various properties of these invariants, including behaviour under similitudes, residues for discrete valuations, and restriction from $I^n$ to $I^{n+1}$. The goal is to use this to study invariants of algebras with involutions in future articles.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/AKT.2020.5.213","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48060003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we construct an equivariant Poincar'e duality between dual tori equipped with finite group actions. We use this to demonstrate that Langlands duality induces a rational isomorphism between the group $C^*$-algebras of extended affine Weyl groups at the level of $K$-theory.
{"title":"Poincaré duality and Langlands duality for\u0000extended affine Weyl groups","authors":"Graham A. Niblo, R. Plymen, N. Wright","doi":"10.2140/AKT.2018.3.491","DOIUrl":"https://doi.org/10.2140/AKT.2018.3.491","url":null,"abstract":"In this paper we construct an equivariant Poincar'e duality between dual tori equipped with finite group actions. We use this to demonstrate that Langlands duality induces a rational isomorphism between the group $C^*$-algebras of extended affine Weyl groups at the level of $K$-theory.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/AKT.2018.3.491","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41471720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $R$ be a ring with $2 in R^{times}$. Then the usual Vaserstein symbol is a map from the orbit space of unimodular rows of length $3$ under the action of the group $E_3 (R)$ to the elementary symplectic Witt group. Now let $P_0$ be a projective module of rank $2$ with trivial determinant. Then we provide a generalized symbol map which is defined on the orbit space of the set of epimorphisms $P_0 oplus R rightarrow R$ under the action of the group of elementary automorphisms of $P_0 oplus R$. We also generalize results by Vaserstein and Suslin on the surjectivity and injectivity of the Vaserstein symbol. Finally, we use local-global principles for transvection groups in order to deduce that the generalized Vaserstein symbol is an isomorphism if $R$ is a regular Noetherian ring of dimension $2$ or a regular affine algebra of dimension $3$ over a field $k$ with $c.d.(k) leq 1$ and $6 in k^{times}$.
{"title":"A generalized Vaserstein symbol","authors":"T. Syed","doi":"10.2140/akt.2019.4.671","DOIUrl":"https://doi.org/10.2140/akt.2019.4.671","url":null,"abstract":"Let $R$ be a ring with $2 in R^{times}$. Then the usual Vaserstein symbol is a map from the orbit space of unimodular rows of length $3$ under the action of the group $E_3 (R)$ to the elementary symplectic Witt group. Now let $P_0$ be a projective module of rank $2$ with trivial determinant. Then we provide a generalized symbol map which is defined on the orbit space of the set of epimorphisms $P_0 oplus R rightarrow R$ under the action of the group of elementary automorphisms of $P_0 oplus R$. We also generalize results by Vaserstein and Suslin on the surjectivity and injectivity of the Vaserstein symbol. Finally, we use local-global principles for transvection groups in order to deduce that the generalized Vaserstein symbol is an isomorphism if $R$ is a regular Noetherian ring of dimension $2$ or a regular affine algebra of dimension $3$ over a field $k$ with $c.d.(k) leq 1$ and $6 in k^{times}$.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2019.4.671","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45801291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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