Algebraic Groups: Structure and Actions

306 Pages·2017·3.21 MB·English

by  Mahir Bilen Can

Volume 94
Algebraic Groups:
Structure and Actions
2015 Clifford Lectures
Algebraic Groups: Structure and Actions
March 2–5, 2015
Tulane University, New Orleans, Louisiana
Mahir Bilen Can
Editor
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
10.1090/pspum/094
Volume 94
Algebraic Groups:
Structure and Actions
2015 Clifford Lectures
Algebraic Groups: Structure and Actions
March 2–5, 2015
Tulane University, New Orleans, Louisiana
Mahir Bilen Can
Editor
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
Volume 94
Algebraic Groups:
Structure and Actions
2015 Clifford Lectures
Algebraic Groups: Structure and Actions
March 2–5, 2015
Tulane University, New Orleans, Louisiana
Mahir Bilen Can
Editor
Licensed to AMS.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
2010 Mathematics Subject Classification. Primary 12F10, 14C15, 14C35, 14C40, 14E07,
14E08,14J70, 14L15, 14L30, 14M17, 14M25, 20G15;Secondary 11G05, 11R34, 14G05,
14K05, 14K30, 14M27, 20G15.
Library of Congress Cataloging-in-Publication Data
Names: Can,Mahir,editor.
Title: Algebraic groups : structures and actions : 2015 Clifford lectures on algebraic groups,
structures and actions, March 2-5, 2015, Tulane University, New Orleans, LA / Mahir Bilen
Can,editor.
Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Pro-
ceedingsofsymposiainpuremathematics;volume94|Includesbibliographicalreferences.
Identifiers: LCCN2016021970|ISBN9781470426019(alk. paper)
Subjects: LCSH: Differential algebraic groups–Congresses. | Linear algebraic groups–
Congresses. | Group theory–Congresses. | Geometry, Algebraic–Congresses. | AMS: Field
theory and polynomials – Field extensions – Separable extensions, Galois theory. msc | Al-
gebraic geometry – Cycles and subschemes – (Equivariant) Chow groups and rings; motives.
msc | Algebraic geometry – Cycles and subschemes – Applications of methods of algebraic
K-theory. msc|Algebraicgeometry–Birationalgeometry–Birationalautomorphisms,Cre-
monagroupandgeneralizations. msc|Algebraicgeometry–Surfacesandhigher-dimensional
varieties – Hypersurfaces. msc | Algebraic geometry – Algebraic groups – Group schemes.
msc| Algebraicgeometry– Specialvarieties – Homogeneousspaces andgeneralizations. msc
|Algebraicgeometry–Specialvarieties–Toricvarieties,Newtonpolyhedra. msc|Algebraic
geometry–Birationalgeometry–Birationalautomorphisms,Cremonagroupandgeneraliza-
tions. msc | Group theory and generalizations– Linear algebraic groups and relatedtopics –
Linearalgebraicgroupsoverarbitraryfields. msc
Classification: LCCQA247.4.A472017|DDC512/.2–dc23LCrecordavailable athttps://lccn.
loc.gov/2016021970
DOI:http://dx.doi.org/10.1090/pspum/94
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Contents
Preface vii
Computing torus-equivariant K-theory of singular varieties
Dave Anderson 1
Algebraic structures of groups of birational transformations
J´er´emy Blanc 17
The Hermite-Joubert problem over p-closed fields
Matthew Brassil and Zinovy Reichstein 31
Some structure theorems for algebraic groups
Michel Brion 53
Structure and classification of pseudo-reductive groups
Brian Conrad and Gopal Prasad 127
Invariants of algebraic groups and retract rationality of classifying spaces
Alexander S. Merkurjev 277
v
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Preface
The prominent (semi)group theorist Alfred Hoblitzelle Clifford (1908–1992)
joinedTulaneUniversityin1955. Since1984,honoringhiscontributions,theMath-
ematicsDepartmentatTulane hashostedthe annual CliffordLectures, aweeklong
series of talks by a distinguished mathematician. A mini-conference is held in con-
junction with each of the Clifford Lecture series. The theme of the 2015 Clifford
Lectureseries wasAlgebraic Groups: StructureandActions, andthe mainspeaker
was Michel Brion. This volume presents the proceedings of the associated mini-
conference.
The theory of algebraic groups forms a very active research area in contempo-
rary mathematics. It has rich relations to many other areas, including algebraic
geometry, number theory, and representation theory. The topics that were cov-
ered in the Clifford Lectures contributed widely to this spectrum. They included
pseudo-reductivegroups,structuretheoryforalgebraicgroups,groupsofbirational
transformations, the Tschirnhaus transformations and applications, algebraic the-
ory of quadratic forms, geometry of classifying spaces, and G-torsors, as well as
operational K-theory and its applications.
The papers in this volume not only present new results on the aforementioned
themes, but also provide much awaited expos´es of the foundational results of alge-
braic group theory recast in the language of schemes at the desired generality.
WegratefullyacknowledgeNationalScienceFoundationWorkshopgrantDMS–
1522969.
Mahir Bilen Can
vii
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10.1090/pspum/094/01
ProceedingsofSymposiainPureMathematics
Volume94,2017
http://dx.doi.org/10.1090/pspum/094/01619
Computing torus-equivariant K-theory of singular varieties
Dave Anderson
1. Introduction
Vector bundles on algebraic varieties are basic objects of study. Among the
manyquestionsonecanask,somefundamentalonesarethese: Whataretheglobal
sections of a vector bundle E on a variety X? How can they be computed? Does
X carry any nontrivial vector bundles at all?
Somewhat more tr(cid:2)actable than the space of global sections is the Euler char-
acteristic χ(X,E) := (−1)idim(Hi(X,E)), which makes sense whenever these
dimensions are finite—e.g., when X is complete. This function is additive on short
exactsequences, so one isled toconsider the Grothendieck group of vector bundles,
(cid:3) (cid:4) (cid:5)
(cid:4)
K◦ (X):= [E](cid:4)[E]=[E(cid:4)]+[E(cid:4)(cid:4)] whenever 0→E(cid:4) →E →E(cid:4)(cid:4) →0 ,
vb
i.e., the free abelian group on isomorphism classes of vector bundles, modulo the
given relation for each short exact sequence. The Euler characteristic thus defines
a function K◦ (X)→Z, when X is complete.
vb
IfX isnonsingularandcomeswithanactionofanalgebraicgroup—say,atorus
T = (G )n—then one can often take advantage of the group action to simplify
m
many calculations, including Euler characteristics. Indeed, there is a version of the
Atiyah-Bott localization formula in this context: assuming for simplicity that X
has finitely many T-fixed points,
(cid:6)
[E ]
(1) χ (X,E)= p .
T (1−[L (p)∗])···(1−[L (p)∗])
p∈XT 1 d
Here, for any finite-dimensional T-representation V, [V] denotes its graded charac-
ter, or equivalently, its class in the representation ring R(T). In the numerator on
the right-hand side, E is the fiber of the equivariant vector bundle E at the fixed
p
point p; in the denominator, the L ’s form a decomposition of the tangent space
i
TpX into one-dimensional weight spaces for the T-action. On (cid:2)the left-hand side,
the equivariant Euler characteristic is defined as χ (X,E) := (−1)i[Hi(X,E)]
T
in R(T). By forgetting the T-action and remembering only dimension, one gets a
homomorphism R(T)→Z which takes χ to χ.
T
The localization formula (1) thus reduces the computation of Euler character-
istics to a finite calculation, and one which is often quite easy. As a toy example,
2010 MathematicsSubjectClassification. Primary14C35,14C40,14C15,14M25,14L30.
ThisworkwaspartiallysupportedbyNSFDMS-1502201.
(cid:2)c2017 American Mathematical Society
1
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