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Algebra and Algebraic Geometry

an algebraic variety; Residues for algebraic varieties. Local duality; Duality and residue theorems for projective varieties; Complete duality; Applications of residues and duality; Toric residues; Bibliography; Index.

University Lecture Series, Volume 47

Residues and Duality for Projective Algebraic Varieties

December 2008, 158 pages, Softcover, ISBN: 978-0-8218-4760-2, LC 2008038860, 2000 Mathematics Subject Classification: 14Fxx, 14F10, 14B15; 32A27, 14M10, 14M25, AMS members US$31, List US$39, Order code ULECT/47

Ernst Kunz, University of Regensburg, Germany with the assistance of and contributions by David A. Cox, Amherst College, MA, and Alicia Dickenstein, University of Buenos Aires, Argentina

This book, which grew out of lectures by E. Kunz for students with a background in algebra and algebraic geometry, develops local and global duality theory in the special case of (possibly singular) algebraic varieties over algebraically closed base fields. It describes duality and residue theorems in terms of Kähler di erential forms and their residues. The properties of residues are introduced via local cohomology. Special emphasis is given to the relation between residues to classical results of algebraic geometry and their generalizations. The contribution by A. Dickenstein gives applications of residues and duality to polynomial solutions of constant coefficient partial di erential equations and to problems in interpolation and ideal membership. D. A. Cox explains toric residues and relates them to the earlier text.

The book is intended as an introduction to more advanced treatments and further applications of the subject, to which numerous bibliographical hints are given.

This item will also be of interest to those working in analysis.

Contents: Local cohomology functors; Local cohomology of noetherian affine schemes; ňáCech cohomology; Koszul complexes and local cohomology; Residues and local cohomology for power series rings; The cohomology of projective schemes; Duality and residue theorems for projective space; Traces, complementary modules, and di erents; The sheaf of regular di erential forms on

Representation Theory

Zongzhu Lin, Kansas State University, Manhattan, KS, and Jianpan Wang, East China Normal University, Shanghai, People’s Republic of China, Editors

Articles in this volume cover topics related to representation theory of various algebraic objects such as algebraic groups, quantum groups, Lie algebras, (finite- and infinite-dimensional) finite groups, and quivers. Collected in one book, these articles show deep relations between all these aspects of representation theory, as well as the diversity of algebraic, geometric, topological, and categorical techniques used in studying representations.

Contents: H. H. Andersen, Sum formulas and Ext-groups; S. Doty, Schur-Weyl duality in positive characteristic; A. Francis and W. Wang, The centers of Iwahori-Hecke algebras are filtered; University of Georgia Vigre Algebra Group, On Kostant’s theorem for Lie algebra cohomology; X. He, G-stable pieces and partial flag varieties; L. Ji, Steinberg representations and duality properties of arithmetic groups, mapping class groups, and outer automorphism groups of free groups; S. Kumar, G. Lusztig, and D. Prasad, Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups; G. Liu, Classification of finite-dimensional basic Hopf algebras according to their representation type; G. Lusztig, Twelve bridges from a reductive group to its Langlands dual; B. J. Parshall and

January 2009

Notices of the AMS

75

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