scholarly journals Derived categories and the analytic approach to general reciprocity laws. Part I

2005 ◽  
Vol 2005 (13) ◽  
pp. 2133-2158 ◽  
Author(s):  
Michael Berg
Keyword(s):  
Open Problem ◽  
Derived Categories ◽  
Derived Category ◽  
Analytic Approach ◽  
Topological Spaces ◽  
Analytic Proof ◽  
Reciprocity Laws

We reformulate Hecke's open problem of 1923, regarding the Fourier-analytic proof of higher reciprocity laws, as a theorem about morphisms involving stratified topological spaces. We achieve this by placing Kubota's formulations ofn-Hilbert reciprocity in a new topological context, suited to the introduction of derived categories of sheaf complexes. Subsequently, we begin to investigate conditions on associated sheaves and a derived category of sheaf complexes specifically designed for an attack on Hecke's eighty-year-old challenge.

scholarly journals Derived Categories and the Analytic Approach to General Reciprocity Laws—Part II

2007 ◽  
Vol 2007 ◽  
pp. 1-27 ◽  
Author(s):  
Michael C. Berg
Keyword(s):  
Number Field ◽  
Open Problem ◽  
Special Linear Group ◽  
Derived Categories ◽  
Topological Spaces ◽  
Base Field ◽  
Analytic Proof ◽  
Reciprocity Laws ◽  
Fold Cover ◽  
Topological Form

Building on the topological foundations constructed in Part I, we now go on to address the homological algebra preparatory to the projected final arithmetical phase of our attack on the analytic proof of general reciprocity for a number field. In the present work, we develop two algebraic frameworks corresponding to two interpretations of Kubota'sn-Hilbert reciprocity formalism, presented in a quasi-dualized topological form in Part I, delineating two sheaf-theoretic routes toward resolving the aforementioned (open) problem. The first approach centers on factoring sheaf morphisms eventually to yield a splitting homomorphism for Kubota'sn-fold cover of the adelized special linear group over the base field. The second approach employs linked exact triples of derived sheaf categories and the yoga of gluingt-structures to evolve a means of establishing the vacuity of certain vertices in diagrams of underlying topological spaces from Part I. Upon assigning properly designedt-structures to three of seven specially chosen derived categories, the collapse just mentioned is enough to yieldn-Hilbert reciprocity.

scholarly journals Derived Categories and the Analytic Approach to General Reciprocity Laws: Part III

2010 ◽  
Vol 2010 ◽  
pp. 1-19
Author(s):  
Michael C. Berg
Keyword(s):  
Recent Work ◽  
Geometric Phase ◽  
Metric Spaces ◽  
Singularity Analysis ◽  
Derived Categories ◽  
Analytic Approach ◽  
Stability Conditions ◽  
Complex Manifolds ◽  
Reciprocity Laws ◽  
The Way

Building on the scaffolding constructed in the first two articles in this series, we now proceed to the geometric phase of our sheaf (-complex) theoretic quasidualization of Kubota's formalism forn-Hilbert reciprocity. Employing recent work by Bridgeland on stability conditions, we extend our yoga oft-structures situated above diagrams of specifically designed derived categories to arrangements of metric spaces or complex manifolds. This prepares the way for provingn-Hilbert reciprocity by means of singularity analysis.

scholarly journals Cohomology of groups and splendid equivalences of derived categories

2001 ◽  
Vol 131 (3) ◽  
pp. 459-472 ◽  
Author(s):  
ALEXANDER ZIMMERMANN
Keyword(s):  
Group Ring ◽  
Local Structure ◽  
Cohomology Ring ◽  
Derived Categories ◽  
Derived Category ◽  
Restriction Map ◽  
Group Rings ◽  
Cohomology Rings ◽  
The Impact ◽  
Cohomology Of Groups

In an earlier paper we studied the impact of equivalences between derived categories of group rings on their cohomology rings. Especially the group of auto-equivalences TrPic(RG) of the derived category of a group ring RG as introduced by Raphaël Rouquier and the author defines an action on the cohomology ring of this group. We study this action with respect to the restriction map, transfer, conjugation and the local structure of the group G.

scholarly journals A note on thick subcategories of stable derived categories

2013 ◽  
Vol 212 ◽  
pp. 87-96
Author(s):  
Henning Krause ◽  
Greg Stevenson
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Complete Intersections ◽  
Line Bundles ◽  
Finitely Generated ◽  
Exact Category ◽  
Coherent Sheaves ◽  
Noetherian Scheme ◽  
Finitely Generated Modules ◽  
Local Complete Intersections

AbstractFor an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection, we classify thick subcategories of finitely generated modules over strict local complete intersections and produce generators for the category of coherent sheaves on a separated Noetherian scheme with an ample family of line bundles.

scholarly journals Pure derived and pure singularity categories

2019 ◽  
Vol 19 (06) ◽  
pp. 2050117
Author(s):  
Tianya Cao ◽  
Wei Ren
Keyword(s):  
Global Dimension ◽  
Derived Categories ◽  
Derived Category ◽  
Homotopy Category ◽  
Projective Modules ◽  
Strong Assumption ◽  
Singularity Category

Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study Verdier quotient of bounded pure derived category modulo the bounded homotopy category of pure-projective modules, which is called a pure singularity category since we show that it reflects the finiteness of pure-global dimension of rings. Moreover, invariance of pure singularity in a recollement of bounded pure derived categories is studied.

scholarly journals QUILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS AND THE RESOLUTION PROPERTY

2020 ◽  
pp. 1-19
Author(s):  
SERGIO ESTRADA ◽  
ALEXANDER SLÁVIK
Keyword(s):  
Vector Bundles ◽  
Derived Categories ◽  
Derived Category ◽  
Projective Modules ◽  
Dimensional Vector ◽  
Infinite Dimensional ◽  
Equivalent Models ◽  
Flat Modules ◽  
Quillen Equivalent ◽  
Resolution Property

We investigate the assumptions under which a subclass of flat quasicoherent sheaves on a quasicompact and semiseparated scheme allows us to ‘mock’ the homotopy category of projective modules. Our methods are based on module-theoretic properties of the subclass of flat modules involved as well as their behaviour with respect to Zariski localizations. As a consequence we get that, for such schemes, the derived category of flat quasicoherent sheaves is equivalent to the derived category of very flat quasicoherent sheaves. If, in addition, the scheme satisfies the resolution property then both derived categories are equivalent to the derived category of infinite-dimensional vector bundles. The equivalences are inferred from a Quillen equivalence between the corresponding models.

scholarly journals Completing perfect complexes

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1387-1427 ◽  
Author(s):  
Henning Krause
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Direct Construction ◽  
Coherent Sheaves ◽  
Perfect Complexes ◽  
Coherent Ring ◽  
Abelian Categories ◽  
Finitely Presented

Abstract This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

scholarly journals A Study of Frontier and Semifrontier in Intuitionistic Fuzzy Topological Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Athar Kharal
Keyword(s):  
Open Problem ◽  
Topological Spaces ◽  
Fuzzy Topology ◽  
Classical Form ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Topological Spaces

Notions of frontier and semifrontier in intuitionistic fuzzy topology have been studied and several of their properties, characterizations, and examples established. Many counter-examples have been presented to point divergences between the IF topology and its classical form. The paper also presents an open problem and one of its weaker forms.

The sequential topology on is not regular

2009 ◽  
Vol 19 (5) ◽  
pp. 943-957 ◽  
Author(s):  
MATTHIAS SCHRÖDER
Keyword(s):  
Open Subset ◽  
Open Problem ◽  
Polish Space ◽  
Topological Spaces ◽  
Computable Analysis ◽  
Topological Properties ◽  
Compact Open Topology ◽  
Cartesian Closed Categories ◽  
Cartesian Closed ◽  
Continuous Functionals

The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of that violates regularity. The topological properties of are known to be closely related to an open problem in Computable Analysis. We also show that the sequential topology on the space of continuous real-valued functions on a Polish space need not be regular.

Where to Go from Here

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Mirror Symmetry ◽  
Derived Categories ◽  
Derived Category ◽  
Vector Spaces ◽  
Crepant Resolution ◽  
Hilbert Schemes ◽  
Mckay Correspondence ◽  
Homological Mirror Symmetry ◽  
Twisted Sheaves ◽  
A Finite Group

This chapter gives pointers for more advanced topics, which require prerequisites that are beyond standard introductions to algebraic geometry. The Mckay correspondence relates the equivariant-derived category of a variety endowed with the action of a finite group and the derived category of a crepant resolution of the quotient. This chapter gives the results from Bridgeland, King, and Reid for a special crepant resolution provided by Hilbert schemes and of Bezrukavnikov and Kaledin for symplectic vector spaces. A brief discussion of Kontsevich's homological mirror symmetry is included, as well as a discussion of stability conditions on triangulated categories. Twisted sheaves and their derived categories can be dealt with in a similar way, and some of the results in particular for K3 surfaces are presented.

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