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De Rham cohomology of differential modules on algebraic varieties

Record Type:	Electronic resources : Monograph/item
Title/Author:	De Rham cohomology of differential modules on algebraic varieties/ by Yves Andre, Francesco Baldassarri, Maurizio Cailotto.
Author:	Andre, Yves.
other author:	Baldassarri, Francesco.
Published:	Cham :Springer International Publishing : : 2020.,
Description:	xiv, 241 p. :ill., digital ;24 cm.
[NT 15003449]:	1 Regularity in several variables -- §1 Geometric models of divisorially valued function fields -- §2 Logarithmic differential operators -- §3 Connections regular along a divisor -- §4 Extensions with logarithmic poles -- §5 Regular connections: the global case -- §6 Exponents -- Appendix A: A letter of Ph. Robba (Nov. 2, 1984) -- Appendix B: Models and log schemes -- 2 Irregularity in several variables -- §1 Spectral norms -- §2 The generalized Poincare-Katz rank of irregularity -- §3 Some consequences of the Turrittin-Levelt-Hukuhara theorem -- §4 Newton polygons -- §5 Stratification of the singular locus by Newton polygons -- §6 Formal decomposition of an integrable connection at a singular divisor -- §7 Cyclic vectors, indicial polynomials and tubular neighborhoods -- 3 Direct images (the Gauss-Manin connection) -- §1 Elementary fibrations -- §2 Review of connections and De Rham cohomology -- §3 Devissage -- §4 Generic finiteness of direct images -- §5 Generic base change for direct images -- §6 Coherence of the cokernel of a regular connection -- §7 Regularity and exponents of the cokernel of a regular connection -- §8 Proof of the main theorems: finiteness, regularity, monodromy, base change (in the regular case) -- Appendix C: Berthelot's comparison theorem on OXDX-linear duals -- Appendix D: Introduction to Dwork's algebraic dual theory -- 4 Complex and p-adic comparison theorems -- §1 Review of analytic connections and De Rham cohomology -- §2 Abstract comparison criteria -- §3 Comparison theorem for algebraic vs.complex-analytic cohomology -- §4 Comparison theorem for algebraic vs. rigid-analytic cohomology (regular coefficients) -- §5 Rigid-analytic comparison theorem in relative dimension one -- §6 Comparison theorem for algebraic vs. rigid-analytic cohomology (irregular coefficients) -- §7 The relative non-archimedean Turrittin theorem -- Appendix E: Riemann's "existence theorem" in higher dimension, an elementary approach -- References.
Contained By:	Springer Nature eBook
Subject:	Homology theory. -
Online resource:	https://doi.org/10.1007/978-3-030-39719-7
ISBN:	9783030397197

De Rham cohomology of differential modules on algebraic varieties
Andre, Yves.

De Rham cohomology of differential modules on algebraic varieties[electronic resource] /by Yves Andre, Francesco Baldassarri, Maurizio Cailotto. - Second edition. - Cham :Springer International Publishing :2020. - xiv, 241 p. :ill., digital ;24 cm. - Progress in mathematics,v.1890743-1643 ;. - Progress in mathematics ;v.189..

1 Regularity in several variables -- §1 Geometric models of divisorially valued function fields -- §2 Logarithmic differential operators -- §3 Connections regular along a divisor -- §4 Extensions with logarithmic poles -- §5 Regular connections: the global case -- §6 Exponents -- Appendix A: A letter of Ph. Robba (Nov. 2, 1984) -- Appendix B: Models and log schemes -- 2 Irregularity in several variables -- §1 Spectral norms -- §2 The generalized Poincare-Katz rank of irregularity -- §3 Some consequences of the Turrittin-Levelt-Hukuhara theorem -- §4 Newton polygons -- §5 Stratification of the singular locus by Newton polygons -- §6 Formal decomposition of an integrable connection at a singular divisor -- §7 Cyclic vectors, indicial polynomials and tubular neighborhoods -- 3 Direct images (the Gauss-Manin connection) -- §1 Elementary fibrations -- §2 Review of connections and De Rham cohomology -- §3 Devissage -- §4 Generic finiteness of direct images -- §5 Generic base change for direct images -- §6 Coherence of the cokernel of a regular connection -- §7 Regularity and exponents of the cokernel of a regular connection -- §8 Proof of the main theorems: finiteness, regularity, monodromy, base change (in the regular case) -- Appendix C: Berthelot's comparison theorem on OXDX-linear duals -- Appendix D: Introduction to Dwork's algebraic dual theory -- 4 Complex and p-adic comparison theorems -- §1 Review of analytic connections and De Rham cohomology -- §2 Abstract comparison criteria -- §3 Comparison theorem for algebraic vs.complex-analytic cohomology -- §4 Comparison theorem for algebraic vs. rigid-analytic cohomology (regular coefficients) -- §5 Rigid-analytic comparison theorem in relative dimension one -- §6 Comparison theorem for algebraic vs. rigid-analytic cohomology (irregular coefficients) -- §7 The relative non-archimedean Turrittin theorem -- Appendix E: Riemann's "existence theorem" in higher dimension, an elementary approach -- References.

This is the revised second edition of the well-received book by the first two authors. It offers a systematic treatment of the theory of vector bundles with integrable connection on smooth algebraic varieties over a field of characteristic 0. Special attention is paid to singularities along divisors at infinity, and to the corresponding distinction between regular and irregular singularities. The topic is first discussed in detail in dimension 1, with a wealth of examples, and then in higher dimension using the method of restriction to transversal curves. The authors develop a new approach to classical algebraic/analytic comparison theorems in De Rham cohomology, and provide a unified discussion of the complex and the p-adic situations while avoiding the resolution of singularities. They conclude with a proof of a conjecture by Baldassarri to the effect that algebraic and p-adic analytic De Rham cohomologies coincide, under an arithmetic condition on exponents. As used in this text, the term "De Rham cohomology" refers to the hypercohomology of the De Rham complex of a connection with respect to a smooth morphism of algebraic varieties, equipped with the Gauss-Manin connection. This simplified approach suffices to establish the stability of crucial properties of connections based on higher direct images. The main technical tools used include: Artin local decomposition of a smooth morphism in towers of elementary fibrations, and spectral sequences associated with affine coverings and with composite functors.

ISBN: 9783030397197

Standard No.: 10.1007/978-3-030-39719-7doiSubjects--Topical Terms:

555733
Homology theory.

LC Class. No.: QA612.3 / .A537 2020

Dewey Class. No.: 514.23

De Rham cohomology of differential modules on algebraic varieties
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245 1 0 $a De Rham cohomology of differential modules on algebraic varieties $h [electronic resource] / $c by Yves Andre, Francesco Baldassarri, Maurizio Cailotto.
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505 0 $a 1 Regularity in several variables -- §1 Geometric models of divisorially valued function fields -- §2 Logarithmic differential operators -- §3 Connections regular along a divisor -- §4 Extensions with logarithmic poles -- §5 Regular connections: the global case -- §6 Exponents -- Appendix A: A letter of Ph. Robba (Nov. 2, 1984) -- Appendix B: Models and log schemes -- 2 Irregularity in several variables -- §1 Spectral norms -- §2 The generalized Poincare-Katz rank of irregularity -- §3 Some consequences of the Turrittin-Levelt-Hukuhara theorem -- §4 Newton polygons -- §5 Stratification of the singular locus by Newton polygons -- §6 Formal decomposition of an integrable connection at a singular divisor -- §7 Cyclic vectors, indicial polynomials and tubular neighborhoods -- 3 Direct images (the Gauss-Manin connection) -- §1 Elementary fibrations -- §2 Review of connections and De Rham cohomology -- §3 Devissage -- §4 Generic finiteness of direct images -- §5 Generic base change for direct images -- §6 Coherence of the cokernel of a regular connection -- §7 Regularity and exponents of the cokernel of a regular connection -- §8 Proof of the main theorems: finiteness, regularity, monodromy, base change (in the regular case) -- Appendix C: Berthelot's comparison theorem on OXDX-linear duals -- Appendix D: Introduction to Dwork's algebraic dual theory -- 4 Complex and p-adic comparison theorems -- §1 Review of analytic connections and De Rham cohomology -- §2 Abstract comparison criteria -- §3 Comparison theorem for algebraic vs.complex-analytic cohomology -- §4 Comparison theorem for algebraic vs. rigid-analytic cohomology (regular coefficients) -- §5 Rigid-analytic comparison theorem in relative dimension one -- §6 Comparison theorem for algebraic vs. rigid-analytic cohomology (irregular coefficients) -- §7 The relative non-archimedean Turrittin theorem -- Appendix E: Riemann's "existence theorem" in higher dimension, an elementary approach -- References.
520 $a This is the revised second edition of the well-received book by the first two authors. It offers a systematic treatment of the theory of vector bundles with integrable connection on smooth algebraic varieties over a field of characteristic 0. Special attention is paid to singularities along divisors at infinity, and to the corresponding distinction between regular and irregular singularities. The topic is first discussed in detail in dimension 1, with a wealth of examples, and then in higher dimension using the method of restriction to transversal curves. The authors develop a new approach to classical algebraic/analytic comparison theorems in De Rham cohomology, and provide a unified discussion of the complex and the p-adic situations while avoiding the resolution of singularities. They conclude with a proof of a conjecture by Baldassarri to the effect that algebraic and p-adic analytic De Rham cohomologies coincide, under an arithmetic condition on exponents. As used in this text, the term "De Rham cohomology" refers to the hypercohomology of the De Rham complex of a connection with respect to a smooth morphism of algebraic varieties, equipped with the Gauss-Manin connection. This simplified approach suffices to establish the stability of crucial properties of connections based on higher direct images. The main technical tools used include: Artin local decomposition of a smooth morphism in towers of elementary fibrations, and spectral sequences associated with affine coverings and with composite functors.
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650 0 $a Modules (Algebra) $3 594721
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650 2 4 $a Several Complex Variables and Analytic Spaces. $3 893953
650 2 4 $a Commutative Rings and Algebras. $3 891076
700 1 $a Baldassarri, Francesco. $3 3461612
700 1 $a Cailotto, Maurizio. $3 3461613
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830 0 $a Progress in mathematics ; $v v.189. $3 3461614
856 4 0 $u https://doi.org/10.1007/978-3-030-39719-7
950 $a Mathematics and Statistics (SpringerNature-11649)