Foundations Of Grothendieck Duality For Diagrams Of Schemes

Author : Joseph Lipman,Mitsuyasu Hashimoto
Publisher : Springer Science & Business Media
Page : 471 pages
File Size : 52,8 Mb
Release : 2009-02-05
Category : Mathematics
ISBN : 9783540854197

Get Book

Foundations of Grothendieck Duality for Diagrams of Schemes by Joseph Lipman,Mitsuyasu Hashimoto Pdf

The first part written by Joseph Lipman, accessible to mid-level graduate students, is a full exposition of the abstract foundations of Grothendieck duality theory for schemes (twisted inverse image, tor-independent base change,...), in part without noetherian hypotheses, and with some refinements for maps of finite tor-dimension. The ground is prepared by a lengthy treatment of the rich formalism of relations among the derived functors, for unbounded complexes over ringed spaces, of the sheaf functors tensor, hom, direct and inverse image. Included are enhancements, for quasi-compact quasi-separated schemes, of classical results such as the projection and Künneth isomorphisms. In the second part, written independently by Mitsuyasu Hashimoto, the theory is extended to the context of diagrams of schemes. This includes, as a special case, an equivariant theory for schemes with group actions. In particular, after various basic operations on sheaves such as (derived) direct images and inverse images are set up, Grothendieck duality and flat base change for diagrams of schemes are proved. Also, dualizing complexes are studied in this context. As an application to group actions, we generalize Watanabe's theorem on the Gorenstein property of invariant subrings.

Author : Allen Altman,Steven Kleiman
Publisher : Springer
Page : 188 pages
File Size : 53,9 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540363095

Get Book

Introduction to Grothendieck Duality Theory by Allen Altman,Steven Kleiman Pdf

Author : Leovigildo Alonso Tarrío,Ana Jeremías López,Joseph Lipman
Publisher : American Mathematical Soc.
Page : 138 pages
File Size : 47,6 Mb
Release : 1999
Category : Mathematics
ISBN : 9780821819425

Get Book

Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes by Leovigildo Alonso Tarrío,Ana Jeremías López,Joseph Lipman Pdf

This volume contains three papers on the foundations of Grothendieck duality on Noetherian formal schemes and on not-necessarily-Noetherian ordinary schemes. The first paper presents a self-contained treatment for formal schemes which synthesizes several duality-related topics, such as local duality, formal duality, residue theorems, dualizing complexes, etc. Included is an exposition of properties of torsion sheaves and of limits of coherent sheaves. A second paper extends Greenlees-May duality to complexes on formal schemes. This theorem has important applications to Grothendieck duality. The third paper outlines methods for eliminating the Noetherian hypotheses. A basic role is played by Kiehl's theorem affirming conservation of pseudo-coherence of complexes under proper pseudo-coherent maps. This work gives a detailed introduction to the subject of Grothendieck Duality. The approach is unique in its presentation of a complex series of special cases that build up to the main results.

Author : Brian Conrad
Publisher : Springer
Page : 300 pages
File Size : 46,8 Mb
Release : 2003-07-01
Category : Mathematics
ISBN : 9783540400158

Get Book

Grothendieck Duality and Base Change by Brian Conrad Pdf

Grothendieck's duality theory for coherent cohomology is a fundamental tool in algebraic geometry and number theory, in areas ranging from the moduli of curves to the arithmetic theory of modular forms. Presented is a systematic overview of the entire theory, including many basic definitions and a detailed study of duality on curves, dualizing sheaves, and Grothendieck's residue symbol. Along the way proofs are given of some widely used foundational results which are not proven in existing treatments of the subject, such as the general base change compatibility of the trace map for proper Cohen-Macaulay morphisms (e.g., semistable curves). This should be of interest to mathematicians who have some familiarity with Grothendieck's work and wish to understand the details of this theory.

Author : Guillermo Cortiñas,Charles A. Weibel
Publisher : American Mathematical Soc.
Page : 388 pages
File Size : 41,5 Mb
Release : 2024-07-03
Category : Education
ISBN : 9781470450267

Get Book

K-theory in Algebra, Analysis and Topology by Guillermo Cortiñas,Charles A. Weibel Pdf

This volume contains the proceedings of the ICM 2018 satellite school and workshop K-theory conference in Argentina. The school was held from July 16–20, 2018, in La Plata, Argentina, and the workshop was held from July 23–27, 2018, in Buenos Aires, Argentina. The volume showcases current developments in K-theory and related areas, including motives, homological algebra, index theory, operator algebras, and their applications and connections. Papers cover topics such as K-theory of group rings, Witt groups of real algebraic varieties, coarse homology theories, topological cyclic homology, negative K-groups of monoid algebras, Milnor K-theory and regulators, noncommutative motives, the classification of C∗-algebras via Kasparov's K-theory, the comparison between full and reduced C∗-crossed products, and a proof of Bott periodicity using almost commuting matrices.

Author : Ulrich Görtz,Torsten Wedhorn
Publisher : Springer Nature
Page : 877 pages
File Size : 49,5 Mb
Release : 2023-11-22
Category : Mathematics
ISBN : 9783658430313

Get Book

Algebraic Geometry II: Cohomology of Schemes by Ulrich Görtz,Torsten Wedhorn Pdf

This book completes the comprehensive introduction to modern algebraic geometry which was started with the introductory volume Algebraic Geometry I: Schemes. It begins by discussing in detail the notions of smooth, unramified and étale morphisms including the étale fundamental group. The main part is dedicated to the cohomology of quasi-coherent sheaves. The treatment is based on the formalism of derived categories which allows an efficient and conceptual treatment of the theory, which is of crucial importance in all areas of algebraic geometry. After the foundations are set up, several more advanced topics are studied, such as numerical intersection theory, an abstract version of the Theorem of Grothendieck-Riemann-Roch, the Theorem on Formal Functions, Grothendieck's algebraization results and a very general version of Grothendieck duality. The book concludes with chapters on curves and on abelian schemes, which serve to develop the basics of the theory of these two important classes of schemes on an advanced level, and at the same time to illustrate the power of the techniques introduced previously. The text contains many exercises that allow the reader to check their comprehension of the text, present further examples or give an outlook on further results.

Author : Amnon Yekutieli
Publisher : Cambridge University Press
Page : 621 pages
File Size : 49,7 Mb
Release : 2019-12-19
Category : Mathematics
ISBN : 9781108419338

Get Book

Derived Categories by Amnon Yekutieli Pdf

The first systematic exposition of the theory of derived categories, with key applications in commutative and noncommutative algebra.

Author : David Eisenbud,Srikanth B. Iyengar,Anurag K. Singh,Michel Van den Bergh,J. Toby Stafford
Publisher : Cambridge University Press
Page : 303 pages
File Size : 45,8 Mb
Release : 2015-11-19
Category : Mathematics
ISBN : 9781107149724

Get Book

Commutative Algebra and Noncommutative Algebraic Geometry by David Eisenbud,Srikanth B. Iyengar,Anurag K. Singh,Michel Van den Bergh,J. Toby Stafford Pdf

This book surveys fundamental current topics in these two areas of research, emphasising the lively interaction between them. Volume 2 focuses on the most recent research.

Author : Takeo Ohsawa,Norihiko Minami
Publisher : Springer Nature
Page : 438 pages
File Size : 45,5 Mb
Release : 2020-03-18
Category : Mathematics
ISBN : 9789811515880

Get Book

Bousfield Classes and Ohkawa's Theorem by Takeo Ohsawa,Norihiko Minami Pdf

This volume originated in the workshop held at Nagoya University, August 28–30, 2015, focusing on the surprising and mysterious Ohkawa's theorem: the Bousfield classes in the stable homotopy category SH form a set. An inspiring, extensive mathematical story can be narrated starting with Ohkawa's theorem, evolving naturally with a chain of motivational questions: Ohkawa's theorem states that the Bousfield classes of the stable homotopy category SH surprisingly forms a set, which is still very mysterious. Are there any toy models where analogous Bousfield classes form a set with a clear meaning? The fundamental theorem of Hopkins, Neeman, Thomason, and others states that the analogue of the Bousfield classes in the derived category of quasi-coherent sheaves Dqc(X) form a set with a clear algebro-geometric description. However, Hopkins was actually motivated not by Ohkawa's theorem but by his own theorem with Smith in the triangulated subcategory SHc, consisting of compact objects in SH. Now the following questions naturally occur: (1) Having theorems of Ohkawa and Hopkins-Smith in SH, are there analogues for the Morel-Voevodsky A1-stable homotopy category SH(k), which subsumes SH when k is a subfield of C?, (2) Was it not natural for Hopkins to have considered Dqc(X)c instead of Dqc(X)? However, whereas there is a conceptually simple algebro-geometrical interpretation Dqc(X)c = Dperf(X), it is its close relative Dbcoh(X) that traditionally, ever since Oka and Cartan, has been intensively studied because of its rich geometric and physical information. This book contains developments for the rest of the story and much more, including the chromatics homotopy theory, which the Hopkins–Smith theorem is based upon, and applications of Lurie's higher algebra, all by distinguished contributors.

Author : Arnaud Beauville,Brendan Hassett,Alexander Kuznetsov,Alessandro Verra
Publisher : Springer
Page : 170 pages
File Size : 40,6 Mb
Release : 2016-12-06
Category : Mathematics
ISBN : 9783319462097

Get Book

Rationality Problems in Algebraic Geometry by Arnaud Beauville,Brendan Hassett,Alexander Kuznetsov,Alessandro Verra Pdf

Providing an overview of the state of the art on rationality questions in algebraic geometry, this volume gives an update on the most recent developments. It offers a comprehensive introduction to this fascinating topic, and will certainly become an essential reference for anybody working in the field. Rationality problems are of fundamental importance both in algebra and algebraic geometry. Historically, rationality problems motivated significant developments in the theory of abelian integrals, Riemann surfaces and the Abel–Jacobi map, among other areas, and they have strong links with modern notions such as moduli spaces, Hodge theory, algebraic cycles and derived categories. This text is aimed at researchers and graduate students in algebraic geometry.

Author : Peter Schenzel,Anne-Marie Simon
Publisher : Springer
Page : 346 pages
File Size : 50,9 Mb
Release : 2018-09-15
Category : Mathematics
ISBN : 9783319965178

Get Book

Completion, Čech and Local Homology and Cohomology by Peter Schenzel,Anne-Marie Simon Pdf

The aim of the present monograph is a thorough study of the adic-completion, its left derived functors and their relations to the local cohomology functors, as well as several completeness criteria, related questions and various dualities formulas. A basic construction is the Čech complex with respect to a system of elements and its free resolution. The study of its homology and cohomology will play a crucial role in order to understand left derived functors of completion and right derived functors of torsion. This is useful for the extension and refinement of results known for modules to unbounded complexes in the more general setting of not necessarily Noetherian rings. The book is divided into three parts. The first one is devoted to modules, where the adic-completion functor is presented in full details with generalizations of some previous completeness criteria for modules. Part II is devoted to the study of complexes. Part III is mainly concerned with duality, starting with those between completion and torsion and leading to new aspects of various dualizing complexes. The Appendix covers various additional and complementary aspects of the previous investigations and also provides examples showing the necessity of the assumptions. The book is directed to readers interested in recent progress in Homological and Commutative Algebra. Necessary prerequisites include some knowledge of Commutative Algebra and a familiarity with basic Homological Algebra. The book could be used as base for seminars with graduate students interested in Homological Algebra with a view towards recent research.

Author : Greg Friedman
Publisher : Cambridge University Press
Page : 491 pages
File Size : 44,6 Mb
Release : 2011-03-28
Category : Mathematics
ISBN : 9780521191678

Get Book

Topology of Stratified Spaces by Greg Friedman Pdf

This book explores the study of singular spaces using techniques from areas within geometry and topology and the interactions among them.

Author : Irena Peeva
Publisher : Springer Science & Business Media
Page : 705 pages
File Size : 48,6 Mb
Release : 2013-02-01
Category : Mathematics
ISBN : 9781461452928

Get Book

Commutative Algebra by Irena Peeva Pdf

This contributed volume brings together the highest quality expository papers written by leaders and talented junior mathematicians in the field of Commutative Algebra. Contributions cover a very wide range of topics, including core areas in Commutative Algebra and also relations to Algebraic Geometry, Algebraic Combinatorics, Hyperplane Arrangements, Homological Algebra, and String Theory. The book aims to showcase the area, especially for the benefit of junior mathematicians and researchers who are new to the field; it will aid them in broadening their background and to gain a deeper understanding of the current research in this area. Exciting developments are surveyed and many open problems are discussed with the aspiration to inspire the readers and foster further research.

Author : Raf Bocklandt
Publisher : Cambridge University Press
Page : 403 pages
File Size : 49,5 Mb
Release : 2021-08-19
Category : Mathematics
ISBN : 9781108483506

Get Book

A Gentle Introduction to Homological Mirror Symmetry by Raf Bocklandt Pdf

Introduction to homological mirror symmetry from the point of view of representation theory, suitable for graduate students.

Author : André 1906- Weil
Publisher : Hassell Street Press
Page : 392 pages
File Size : 44,7 Mb
Release : 2021-09-10
Category : Electronic
ISBN : 1015107672

Get Book

Foundations of Algebraic Geometry. --; 29 by André 1906- Weil Pdf

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.