For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs’ result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable injective left modules and the isoclasses of indecomposable flat cotorsion right modules. This correspondence is an explicit realization of Herzog’s homeomorphism induced from elementary duality of Ziegler spectra.
{"title":"Flat cotorsion modules over Noether algebras","authors":"Ryo Kanda, Tsutomu Nakamura","doi":"10.4171/dm/893","DOIUrl":"https://doi.org/10.4171/dm/893","url":null,"abstract":"For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs’ result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable injective left modules and the isoclasses of indecomposable flat cotorsion right modules. This correspondence is an explicit realization of Herzog’s homeomorphism induced from elementary duality of Ziegler spectra.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85954753","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 : 2021-07-07DOI: 10.25537/dm.2022v27.1-16
A. Langer
A BSTRACT . We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolov’s inequality for semistable sheaves on a smooth projective variety of any dimension ≥ 2 without using any restriction theorems.
{"title":"On boundedness of semistable sheaves","authors":"A. Langer","doi":"10.25537/dm.2022v27.1-16","DOIUrl":"https://doi.org/10.25537/dm.2022v27.1-16","url":null,"abstract":"A BSTRACT . We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolov’s inequality for semistable sheaves on a smooth projective variety of any dimension ≥ 2 without using any restriction theorems.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74676680","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 prove global equivariant refinements of Miller’s stable splittings of the infinite orthogonal, unitary and symplectic groups, and more generally of the spaces O/O ( m ), U/U ( m ) and Sp/Sp ( m ). As such, our results encode compatible equivariant stable splittings, for all compact Lie groups, of specific equivariant refinements of these spaces. In the unitary and symplectic case, we also take the actions of the Galois groups into account. To properly formulate these Galois-global statements, we introduce a generalization of global stable homotopy theory in the presence of an extrinsic action of an additional topological group.
{"title":"Global stable splittings of Stiefel manifolds","authors":"S. Schwede","doi":"10.4171/dm/885","DOIUrl":"https://doi.org/10.4171/dm/885","url":null,"abstract":". We prove global equivariant refinements of Miller’s stable splittings of the infinite orthogonal, unitary and symplectic groups, and more generally of the spaces O/O ( m ), U/U ( m ) and Sp/Sp ( m ). As such, our results encode compatible equivariant stable splittings, for all compact Lie groups, of specific equivariant refinements of these spaces. In the unitary and symplectic case, we also take the actions of the Galois groups into account. To properly formulate these Galois-global statements, we introduce a generalization of global stable homotopy theory in the presence of an extrinsic action of an additional topological group.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84608464","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 prove that both local Galois representations and (φ,Γ)-modules can be recovered from prismatic F-crystals, from which we obtain a new proof of the equivalence of Galois representations and (φ,Γ)-modules.
{"title":"Galois representations, $(varphi, Gamma)$-modules and prismatic F-crystals","authors":"Z. Wu","doi":"10.4171/dm/855","DOIUrl":"https://doi.org/10.4171/dm/855","url":null,"abstract":"We prove that both local Galois representations and (φ,Γ)-modules can be recovered from prismatic F-crystals, from which we obtain a new proof of the equivalence of Galois representations and (φ,Γ)-modules.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86790340","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}
Consider an irreducible Markov chain which satisfies a ratio limit theorem, and let ρ be the spectral radius of the chain. We investigate the relation of the the ρ -Martin boundary with the boundary induced by the ρ -harmonic kernel which appears in the ratio limit. Special emphasis is on random walks on non-amenable groups, specifically, free groups and hyperbolic groups.
{"title":"Ratio limits and Martin boundary","authors":"W. Woess","doi":"10.4171/dm/847","DOIUrl":"https://doi.org/10.4171/dm/847","url":null,"abstract":"Consider an irreducible Markov chain which satisfies a ratio limit theorem, and let ρ be the spectral radius of the chain. We investigate the relation of the the ρ -Martin boundary with the boundary induced by the ρ -harmonic kernel which appears in the ratio limit. Special emphasis is on random walks on non-amenable groups, specifically, free groups and hyperbolic groups.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73522007","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 show that the central charge k reduction of the universal central extension of the elliptic Hall algebra is isomorphic to the trace, or zeroth Hochschild homology, of the quantum Heisenberg category of central charge k. As an application, we construct large families of representations of the universal extension of the elliptic Hall algebra.
{"title":"Categorification of the elliptic Hall algebra","authors":"Youssef Mousaaid, Alistair Savage","doi":"10.4171/dm/896","DOIUrl":"https://doi.org/10.4171/dm/896","url":null,"abstract":"We show that the central charge k reduction of the universal central extension of the elliptic Hall algebra is isomorphic to the trace, or zeroth Hochschild homology, of the quantum Heisenberg category of central charge k. As an application, we construct large families of representations of the universal extension of the elliptic Hall algebra.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77395248","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 prove an approximation result for Lipschitz functions on the quantum sphere S q , from which we deduce that the two natural quantum metric structures on S q have quantum Gromov-Hausdorff distance zero.
{"title":"Polynomial approximation of quantum Lipschitz functions","authors":"Konrad Aguilar, Jens Kaad, David Kyed","doi":"10.4171/dm/884","DOIUrl":"https://doi.org/10.4171/dm/884","url":null,"abstract":"We prove an approximation result for Lipschitz functions on the quantum sphere S q , from which we deduce that the two natural quantum metric structures on S q have quantum Gromov-Hausdorff distance zero.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79651954","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 prove general adjoint functor theorems for weakly (co)complete n -categories. This class of n -categories includes the homotopy n -categories of (co)complete ∞ -categories, so these n -categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) n -categories and prove a Brown representability theorem for localizations of compactly generated n -categories. This class of n -categories includes the homotopy n -categories of presentable ∞ -categories if n ≥ 2, and the homotopy n -categories of presentable stable ∞ -categories for any n ≥ 1.
． 我们保证对所有的细则都给予共同的保证。这届n -categories includes homotopy n -categories》(co)完整∞-categories,所以这些n -categories不要承认所有小(co)限制在将军。我们还介绍了布朗的特点这届n -categories homotopy n -categories》includes presentable∞-categories如果n≥2,和《presentable homotopy n -categories马厩∞-categories对于任何n≥1。
{"title":"Higher weak (co)limits, adjoint functor theorems, and higher Brown representability","authors":"Hoang Kim Nguyen, G. Raptis, Christoph Schrade","doi":"10.4171/dm/900","DOIUrl":"https://doi.org/10.4171/dm/900","url":null,"abstract":". We prove general adjoint functor theorems for weakly (co)complete n -categories. This class of n -categories includes the homotopy n -categories of (co)complete ∞ -categories, so these n -categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) n -categories and prove a Brown representability theorem for localizations of compactly generated n -categories. This class of n -categories includes the homotopy n -categories of presentable ∞ -categories if n ≥ 2, and the homotopy n -categories of presentable stable ∞ -categories for any n ≥ 1.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81979557","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}
The notion of commutator width of a group, defined as the smallest number of commutators needed to represent each element of the derived group as their product, has been extensively studied over the past decades. In particular, in 1992 Barge and Ghys discovered the first example of a simple group of commutator width greater than one among groups of diffeomorphisms of smooth manifolds. We consider a parallel notion of bracket width of a Lie algebra and present the first examples of simple Lie algebras of bracket width greater than one. They are found among the algebras of polynomial vector fields on smooth affine varieties.
{"title":"Bracket width of simple Lie algebras","authors":"A. Dubouloz, B. Kunyavskii, A. Regeta","doi":"10.4171/dm/850","DOIUrl":"https://doi.org/10.4171/dm/850","url":null,"abstract":"The notion of commutator width of a group, defined as the smallest number of commutators needed to represent each element of the derived group as their product, has been extensively studied over the past decades. In particular, in 1992 Barge and Ghys discovered the first example of a simple group of commutator width greater than one among groups of diffeomorphisms of smooth manifolds. We consider a parallel notion of bracket width of a Lie algebra and present the first examples of simple Lie algebras of bracket width greater than one. They are found among the algebras of polynomial vector fields on smooth affine varieties.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90163654","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}
In this paper we study families of representations of the outer automorphism groups indexed on a collection of finite groups U . We encode this large amount of data into a convenient abelian category which generalizes the category of VI-modules appearing in the representation theory of the finite general linear groups. Inspired by work of Church–Ellenberg–Farb, we investigate for which choices of U the abelian category is locally noetherian and deduce analogues of central stability and representation stability results in this setting. Finally, we show that some invariants coming from rational global homotopy theory exhibit representation stability.
{"title":"Representation stability and outer automorphism groups","authors":"Luca Pol, N. Strickland","doi":"10.4171/dm/866","DOIUrl":"https://doi.org/10.4171/dm/866","url":null,"abstract":"In this paper we study families of representations of the outer automorphism groups indexed on a collection of finite groups U . We encode this large amount of data into a convenient abelian category which generalizes the category of VI-modules appearing in the representation theory of the finite general linear groups. Inspired by work of Church–Ellenberg–Farb, we investigate for which choices of U the abelian category is locally noetherian and deduce analogues of central stability and representation stability results in this setting. Finally, we show that some invariants coming from rational global homotopy theory exhibit representation stability.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76277229","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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