Pub Date : 2012-04-01DOI: 10.1017/IS011006015JKT160
A. Quéguiner-Mathieu, J. Tignol
Using triality, we define a relative Arason invariant for orthogonal involutions on a -possibly division- central simple algebra of degree 8. This invariant detects hyperbolicity, but it does not detect isomorphism. We produce explicit examples, in index 4 and 8, of nonisomorphic involutions with trivial relative Arason invariant.
{"title":"Cohomological invariants for orthogonal involutions on degree 8 algebras","authors":"A. Quéguiner-Mathieu, J. Tignol","doi":"10.1017/IS011006015JKT160","DOIUrl":"https://doi.org/10.1017/IS011006015JKT160","url":null,"abstract":"Using triality, we define a relative Arason invariant for orthogonal involutions on a -possibly division- central simple algebra of degree 8. This invariant detects hyperbolicity, but it does not detect isomorphism. We produce explicit examples, in index 4 and 8, of nonisomorphic involutions with trivial relative Arason invariant.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"9 1","pages":"333-358"},"PeriodicalIF":0.0,"publicationDate":"2012-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS011006015JKT160","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56666172","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}
Pub Date : 2012-04-01DOI: 10.1017/IS011006028JKT161
A. Krishna, P. A. Østvær
We show localization, excision and descent theorems for K -theory of Deligne-Mumford stacks. Our approach employs the Nisnevich site which is a complete, regular and bounded cd -structure on the category of such stacks and restricts to the usual Nisnevich site on schemes. By combining excision with a refinement of localization sequences due to Krishna and Toen, we show that K -theory of perfect complexes on tame Deligne-Mumford stacks satisfies Nisnevich descent.
{"title":"Nisnevich descent for K -theory of Deligne-Mumford stacks","authors":"A. Krishna, P. A. Østvær","doi":"10.1017/IS011006028JKT161","DOIUrl":"https://doi.org/10.1017/IS011006028JKT161","url":null,"abstract":"We show localization, excision and descent theorems for K -theory of Deligne-Mumford stacks. Our approach employs the Nisnevich site which is a complete, regular and bounded cd -structure on the category of such stacks and restricts to the usual Nisnevich site on schemes. By combining excision with a refinement of localization sequences due to Krishna and Toen, we show that K -theory of perfect complexes on tame Deligne-Mumford stacks satisfies Nisnevich descent.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"9 1","pages":"291-331"},"PeriodicalIF":0.0,"publicationDate":"2012-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS011006028JKT161","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56666247","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}
Pub Date : 2012-02-01DOI: 10.1017/IS011003005JKT142
N. Yagita
For a complex Grassmannian X , there is an isomorphism between Balmer's Witt group and the quotient of topological K -theories.
对于复格拉斯曼X，拓扑K -理论的商与Balmer's Witt群之间存在同构关系。
{"title":"A note on the Witt group and the KO -theory of complex Grassmannians","authors":"N. Yagita","doi":"10.1017/IS011003005JKT142","DOIUrl":"https://doi.org/10.1017/IS011003005JKT142","url":null,"abstract":"For a complex Grassmannian X , there is an isomorphism between Balmer's Witt group and the quotient of topological K -theories.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"51 1","pages":"161-175"},"PeriodicalIF":0.0,"publicationDate":"2012-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75954438","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}
Pub Date : 2012-02-01DOI: 10.1017/IS011001026JKT140
A. Legrand, David Poutriquet
Starting from the Karoubi multiplicative K -theory, we construct a Chern-Weil theory adapted to isolated conical singularities. The Chern character takes its values in the intersection cohomology of Goresky-MacPherson. We also propose an integer intersection K -theory for such singularities.
{"title":"Intersection K -theory for isolated conical singularities","authors":"A. Legrand, David Poutriquet","doi":"10.1017/IS011001026JKT140","DOIUrl":"https://doi.org/10.1017/IS011001026JKT140","url":null,"abstract":"Starting from the Karoubi multiplicative K -theory, we construct a Chern-Weil theory adapted to isolated conical singularities. The Chern character takes its values in the intersection cohomology of Goresky-MacPherson. We also propose an integer intersection K -theory for such singularities.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"9 1","pages":"177-200"},"PeriodicalIF":0.0,"publicationDate":"2012-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS011001026JKT140","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56666009","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}
Pub Date : 2012-01-30DOI: 10.1017/is013007003jkt236
O. Ben-Bassat, J. Block
We show that the functor from curved differential graded algebras to differential graded categories, defined by the second author in [B], sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.
{"title":"Milnor descent for cohesive dg-categories","authors":"O. Ben-Bassat, J. Block","doi":"10.1017/is013007003jkt236","DOIUrl":"https://doi.org/10.1017/is013007003jkt236","url":null,"abstract":"We show that the functor from curved differential graded algebras to differential graded categories, defined by the second author in [B], sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"12 1","pages":"433-459"},"PeriodicalIF":0.0,"publicationDate":"2012-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/is013007003jkt236","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56667583","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}
Pub Date : 2011-12-01DOI: 10.1017/IS010009021JKT130
M. Das
This paper examines the relation between the Euler class group of a Noetherian ring and the Euler class group of its polynomial extension. When the ring is a smooth affine domain, the two groups are canonically isomorphic. This is a consequence of a theorem of Bhatwadekar-Sridharan, which they proved in order to answer a question of Nori on sections of projective modules over such rings. If the smoothness assumption is removed, the result of Bhatwadekar-Sridharan is no longer valid and also the Euler class groups above are not in general isomorphic. In this paper we investigate a variant of Nori's question for arbitrary Noetherian rings and derive several consequences to understand the relation between various groups in the theory of Euler classes.
{"title":"Revisiting Nori's question and homotopy invariance of Euler class groups","authors":"M. Das","doi":"10.1017/IS010009021JKT130","DOIUrl":"https://doi.org/10.1017/IS010009021JKT130","url":null,"abstract":"This paper examines the relation between the Euler class group of a Noetherian ring and the Euler class group of its polynomial extension. When the ring is a smooth affine domain, the two groups are canonically isomorphic. This is a consequence of a theorem of Bhatwadekar-Sridharan, which they proved in order to answer a question of Nori on sections of projective modules over such rings. If the smoothness assumption is removed, the result of Bhatwadekar-Sridharan is no longer valid and also the Euler class groups above are not in general isomorphic. In this paper we investigate a variant of Nori's question for arbitrary Noetherian rings and derive several consequences to understand the relation between various groups in the theory of Euler classes.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"8 1","pages":"451-480"},"PeriodicalIF":0.0,"publicationDate":"2011-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS010009021JKT130","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56665301","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}
Pub Date : 2011-12-01DOI: 10.1017/IS010011013JKT134
Mark Blunk, S. J. Sierra, S. P. Smith
Let S be a degree six del Pezzo surface over an arbitrary field F. Motivated by the first author's classification of all such S up to isomorphism (3) in terms of a separable F-algebra B×Q×F, and by his K-theory isomorphism Kn(S) � Kn(B × Q × F) for n � 0, we prove an equivalence of derived categories D b (cohS) � D b (modA) where A is an explicitly given finite dimensional F-algebra whose semisimple part is B × Q × F.
{"title":"A derived equivalence for a degree 6 del Pezzo surface over an arbitrary field","authors":"Mark Blunk, S. J. Sierra, S. P. Smith","doi":"10.1017/IS010011013JKT134","DOIUrl":"https://doi.org/10.1017/IS010011013JKT134","url":null,"abstract":"Let S be a degree six del Pezzo surface over an arbitrary field F. Motivated by the first author's classification of all such S up to isomorphism (3) in terms of a separable F-algebra B×Q×F, and by his K-theory isomorphism Kn(S) � Kn(B × Q × F) for n � 0, we prove an equivalence of derived categories D b (cohS) � D b (modA) where A is an explicitly given finite dimensional F-algebra whose semisimple part is B × Q × F.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"8 1","pages":"481-492"},"PeriodicalIF":0.0,"publicationDate":"2011-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS010011013JKT134","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56665412","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}
Pub Date : 2011-10-01DOI: 10.1017/IS010006009JKT121
Carla Farsi, C. Seaton
{"title":"Algebraic Structures Associated to Orbifold Wreath Products","authors":"Carla Farsi, C. Seaton","doi":"10.1017/IS010006009JKT121","DOIUrl":"https://doi.org/10.1017/IS010006009JKT121","url":null,"abstract":"","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"8 1","pages":"323-338"},"PeriodicalIF":0.0,"publicationDate":"2011-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS010006009JKT121","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56663797","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}
Pub Date : 2011-10-01DOI: 10.1017/IS010010004JKT123
Gael Collinet
We prove that the homology of unitary groups over rings of S-integers in number fields stabilizes. Results of this kind are well known to follow from the high acyclicity of ad-hoc polyhedra. Given this, we exhibit two simple conditions on the arithmetic of hermitian forms over a ring A relatively to an antiautomorphism which, if they are satisfied, imply the stabilization of the homology of the corresponding unitary groups. When R is a ring of S-integers in a number field K, and A is a maximal R-order in an associative composition algebra F over K, we use the strong approximation theorem to show that both of these properties are satisfied. Finally we take a closer look at the case of On(Z[ 1 2 ]).
{"title":"Homology stability for unitary groups over S-arithmetic rings","authors":"Gael Collinet","doi":"10.1017/IS010010004JKT123","DOIUrl":"https://doi.org/10.1017/IS010010004JKT123","url":null,"abstract":"We prove that the homology of unitary groups over rings of S-integers in number fields stabilizes. Results of this kind are well known to follow from the high acyclicity of ad-hoc polyhedra. Given this, we exhibit two simple conditions on the arithmetic of hermitian forms over a ring A relatively to an antiautomorphism which, if they are satisfied, imply the stabilization of the homology of the corresponding unitary groups. When R is a ring of S-integers in a number field K, and A is a maximal R-order in an associative composition algebra F over K, we use the strong approximation theorem to show that both of these properties are satisfied. Finally we take a closer look at the case of On(Z[ 1 2 ]).","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"8 1","pages":"293-322"},"PeriodicalIF":0.0,"publicationDate":"2011-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS010010004JKT123","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56665335","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}
Pub Date : 2011-08-18DOI: 10.1017/is014003008jkt259
U. Pennig
We decompose θ(M), the twisted index obstruction to a positive scalar curvature metric for closed oriented manifolds with spin universal cover, into a pairing of a twisted K-homology with a twisted K-theory class and prove that θ(M) does not vanish if M is a closed orientable enlargeable manifold with spin universal cover.
{"title":"Twisted K-theory and obstructions against positive scalar curvature metrics","authors":"U. Pennig","doi":"10.1017/is014003008jkt259","DOIUrl":"https://doi.org/10.1017/is014003008jkt259","url":null,"abstract":"We decompose θ(M), the twisted index obstruction to a positive scalar curvature metric for closed oriented manifolds with spin universal cover, into a pairing of a twisted K-homology with a twisted K-theory class and prove that θ(M) does not vanish if M is a closed orientable enlargeable manifold with spin universal cover.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"14 1","pages":"47-71"},"PeriodicalIF":0.0,"publicationDate":"2011-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/is014003008jkt259","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56668196","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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