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On Suslin matrices and their connection to spin groups 关于Suslin矩阵及其与自旋群的联系
IF 0.9 3区 数学
Documenta Mathematica Pub Date : 2020-09-25 DOI: 10.4171/dm/498
Vineeth Chintala
{"title":"On Suslin matrices and their connection to spin groups","authors":"Vineeth Chintala","doi":"10.4171/dm/498","DOIUrl":"https://doi.org/10.4171/dm/498","url":null,"abstract":"A concrete representation of the Clifford algebra (for any hyperbolic quadratic space) is given using what are called Suslin matrices. This explicit construction is used to analyze the corresponding Spin groups and the involution and might be of interest in low dimensional computations. Conversely, this connection to Clifford algebras gives a conceptual foundation to some (seemingly accidental) properties of Suslin matrices.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"54 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85738297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Virtual equivariant Grothendieck-Riemann-Roch formula 虚等变Grothendieck-Riemann-Roch公式
IF 0.9 3区 数学
Documenta Mathematica Pub Date : 2020-09-21 DOI: 10.4171/dm/864
Charanya Ravi, Bhamidi Sreedhar
{"title":"Virtual equivariant Grothendieck-Riemann-Roch formula","authors":"Charanya Ravi, Bhamidi Sreedhar","doi":"10.4171/dm/864","DOIUrl":"https://doi.org/10.4171/dm/864","url":null,"abstract":"For a $G$-scheme $X$ with a given equivariant perfect obstruction theory, we prove a virtual equivariant Grothendieck-Riemann-Roch formula, this is an extension of a result of Fantechi-Gottsche to the equivariant context. We also prove a virtual non-abelian localization theorem for schemes over $mathbb{C}$ with proper actions.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"68 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75938646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Cdh descent for homotopy Hermitian $K$-theory of rings with involution 对合环的同伦厄密K理论的Cdh下降
IF 0.9 3区 数学
Documenta Mathematica Pub Date : 2020-09-19 DOI: 10.4171/dm/842
D. Carmody
{"title":"Cdh descent for homotopy Hermitian $K$-theory of rings with involution","authors":"D. Carmody","doi":"10.4171/dm/842","DOIUrl":"https://doi.org/10.4171/dm/842","url":null,"abstract":"We provide a geometric model for the classifying space of automorphism groups of Hermitian vector bundles over a ring with involution $R$ such that $frac{1}{2} in R$; this generalizes a result of Schlichting-Tripathi cite{SchTri}. We then prove a periodicity theorem for Hermitian $K$-theory and use it to construct an $E_infty$ motivic ring spectrum $mathbf{KR}^{mathrm{alg}}$ representing homotopy Hermitian $K$-theory. From these results, we show that $mathbf{KR}^{mathrm{alg}}$ is stable under base change, and cdh descent for homotopy Hermitian $K$-theory of rings with involution is a formal consequence.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"110 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72865881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Hearts for commutative Noetherian rings: torsion pairs and derived equivalences 交换诺瑟环的心:扭转对和派生等价
IF 0.9 3区 数学
Documenta Mathematica Pub Date : 2020-09-18 DOI: 10.4171/dm/831
Sergio Pavon, Jorge Vit'oria
{"title":"Hearts for commutative Noetherian rings: torsion pairs and derived equivalences","authors":"Sergio Pavon, Jorge Vit'oria","doi":"10.4171/dm/831","DOIUrl":"https://doi.org/10.4171/dm/831","url":null,"abstract":"Over a commutative noetherian ring $R$, the prime spectrum controls, via the assignment of support, the structure of both $mathsf{Mod}(R)$ and $mathsf{D}(R)$. We show that, just like in $mathsf{Mod}(R)$, the assignment of support classifies hereditary torsion pairs in the heart of any nondegenerate compactly generated $t$-structure of $mathsf{D}(R)$. Moreover, we investigate whether these $t$-structures induce derived equivalences, obtaining a new source of Grothendieck categories which are derived equivalent to $mathsf{Mod}(R)$.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"31 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74169185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Spectral theory of regular sequences 正则序列的谱理论
IF 0.9 3区 数学
Documenta Mathematica Pub Date : 2020-09-03 DOI: 10.4171/dm/880
M. Coons, James Evans, Neil Mañibo
{"title":"Spectral theory of regular sequences","authors":"M. Coons, James Evans, Neil Mañibo","doi":"10.4171/dm/880","DOIUrl":"https://doi.org/10.4171/dm/880","url":null,"abstract":"Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we study the limiting asymptotics of regular sequences by constructing a systematic measure-theoretic framework surrounding them. The constructed measures are generalisations of mass distributions supported on attractors of iterated function systems.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"186 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72566041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
On a torsion analogue of the weight-monodromy conjecture 权单猜想的一个扭转类似
IF 0.9 3区 数学
Documenta Mathematica Pub Date : 2020-08-27 DOI: 10.4171/dm/854
Kazuhiro Ito
{"title":"On a torsion analogue of the weight-monodromy conjecture","authors":"Kazuhiro Ito","doi":"10.4171/dm/854","DOIUrl":"https://doi.org/10.4171/dm/854","url":null,"abstract":"We formulate and study a torsion analogue of the weight-monodromy conjecture for a proper smooth scheme over a non-archimedean local field. We prove it for proper smooth schemes over equal characteristic non-archimedean local fields, abelian varieties, surfaces, varieties uniformized by Drinfeld upper half spaces, and set-theoretic complete intersections in toric varieties. In the equal characteristic case, our methods rely on an ultraproduct variant of Weil II established by Cadoret.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"7 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90752370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
A Finiteness Theorem for Special Unitary Groups of Quaternionic Skew-Hermitian Forms with Good Reduction 具有良好约化的四元数斜厄米形式特殊酉群的有限性定理
IF 0.9 3区 数学
Documenta Mathematica Pub Date : 2020-08-25 DOI: 10.4171/dm/773
Srimathy Srinivasan
{"title":"A Finiteness Theorem for Special Unitary Groups of Quaternionic Skew-Hermitian Forms with Good Reduction","authors":"Srimathy Srinivasan","doi":"10.4171/dm/773","DOIUrl":"https://doi.org/10.4171/dm/773","url":null,"abstract":"Given a field $K$ equipped with a set of discrete valuations $V$, we develop a general theory to relate reduction properties of skew-hermitian forms over a quaternion $K$-algebra $Q$ to quadratic forms over the function field $K(Q)$ obtained via Morita equivalence. Using this we show that if $(K,V)$ satisfies certain conditions, then the number of $K$-isomorphism classes of the universal coverings of the special unitary groups of quaternionic skew-hermitian forms that have good reduction at all valuations in $V$ is finite and bounded by a value that depends on size of a quotient of the Picard group of $V$ and the size of the kernel and cokernel of residue maps in Galois cohomology of $K$ with finite coefficients. As a corollary we prove a conjecture of Chernousov, Rapinchuk, Rapinchuk for groups of this type.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"4 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87564516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
An index formula for groups of isometric linear canonical transformations 等距线性正则变换群的一个指标公式
IF 0.9 3区 数学
Documenta Mathematica Pub Date : 2020-08-03 DOI: 10.4171/dm/890
A. Savin, E. Schrohe
{"title":"An index formula for groups of isometric linear canonical transformations","authors":"A. Savin, E. Schrohe","doi":"10.4171/dm/890","DOIUrl":"https://doi.org/10.4171/dm/890","url":null,"abstract":"We define a representation of the unitary group $U(n)$ by metaplectic operators acting on $L^2(mathbb{R}^n)$ and consider the operator algebra generated by the operators of the representation and pseudodifferential operators of Shubin class. Under suitable conditions, we prove the Fredholm property for elements in this algebra and obtain an index formula.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"66 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80378315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
Multiplier tests and subhomogeneity of multiplier algebras 乘数代数的乘数检验与次齐性
IF 0.9 3区 数学
Documenta Mathematica Pub Date : 2020-08-03 DOI: 10.25537/dm.2022v27.719-764
A. Aleman, Michael Hartz, John E. McCarthy, S. Richter
{"title":"Multiplier tests and subhomogeneity of multiplier algebras","authors":"A. Aleman, Michael Hartz, John E. McCarthy, S. Richter","doi":"10.25537/dm.2022v27.719-764","DOIUrl":"https://doi.org/10.25537/dm.2022v27.719-764","url":null,"abstract":"Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"25 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81986710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
Algebraic slice spectral sequences 代数切片谱序列
IF 0.9 3区 数学
Documenta Mathematica Pub Date : 2020-07-16 DOI: 10.4171/dm/836
D. Culver, Hana Jia Kong, J. Quigley
{"title":"Algebraic slice spectral sequences","authors":"D. Culver, Hana Jia Kong, J. Quigley","doi":"10.4171/dm/836","DOIUrl":"https://doi.org/10.4171/dm/836","url":null,"abstract":"For certain motivic spectra, we construct a square of spectral sequences relating the effective slice spectral sequence and the motivic Adams spectral sequence. We show the square can be constructed for connective algebraic K-theory, motivic Morava K-theory, and truncated motivic Brown-Peterson spectra. In these cases, we show that the $mathbb{R}$-motivic effective slice spectral sequence is completely determined by the $rho$-Bockstein spectral sequence. Using results of Heard, we also obtain applications to the Hill-Hopkins-Ravenel slice spectral sequences for connective Real K-theory, Real Morava K-theory, and truncated Real Brown-Peterson spectra.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"15 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87334625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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