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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
给出了具有对合$R$环上厄米向量束自同构群的分类空间的几何模型，使得$frac{1}{2} in R$;这推广了schlicht - tripathi的结果cite{SchTri}。然后证明了厄米特$K$ -理论的一个周期性定理，并用它构造了一个表示同伦厄米特$K$ -理论的$E_infty$动力环谱$mathbf{KR}^{mathrm{alg}}$。从这些结果中，我们证明了$mathbf{KR}^{mathrm{alg}}$在碱基变化下是稳定的，并且对合环的同伦厄米$K$ -理论的cdh下降是一个形式推论。
{"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":null,"pages":null},"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}
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 $mathsf{Mod}(R)$.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
Pub Date : 2020-08-03DOI: 10.25537/dm.2022v27.719-764
A. Aleman, Michael Hartz, John E. McCarthy, S. Richter
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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