Computational algebraic geometry

208 Pages·2003·0.988 MB·English

by  Hal Schenck

COMPUTATIONAL ALGEBRAIC GEOMETRY

LONDON MATHEMATICAL SOCIETY STUDENT TEXTS
Managingeditor:ProfessorJ.W.Bruce,DepartmentofMathematics,
UniversityofLiverpool,UK
3 Localfields,J.W.S.CASSELS
4 Anintroductiontotwistortheory:Secondedition,S.A.HUGGETT&K.P.TOD
5 Introductiontogeneralrelativity,L.P.HUGHSTON&K.P.TOD
7 Thetheoryofevolutionanddynamicalsystems,J.HOFBAUER&K.SIGMUND
8 SummingandnuclearnormsinBanachspacetheory,G.J.O.JAMESON
9 AutomorphismsofsurfacesafterNielsenandThurston,A.CASSON&S.BLEILER
11 Spacetimeandsingularities,G.NABER
12 Undergraduatealgebraicgeometry,MILESREID
13 AnintroductiontoHankeloperators,J.R.PARTINGTON
15 Presentationsofgroups:Secondedition,D.L.JOHNSON
17 Aspectsofquantumfieldtheoryincurvedspacetime,S.A.FULLING
18 Braidsandcoverings:Selectedtopics,VAGNLUNDSGAARDHANSEN
19 Stepsincommutativealgebra,R.Y.SHARP
20 Communicationtheory,C.M.GOLDIE&R.G.E.PINCH
21 RepresentationsoffinitegroupsofLietype,FRANC¸OISDIGNE&JEANMICHEL
22 Designs,graphs,codes,andtheirlinks,P.J.CAMERON&J.H.VANLINT
23 Complexalgebraiccurves,FRANCESKIRWAN
24 Lecturesonellipticcurves,J.W.S.CASSELS
25 Hyperbolicgeometry,BIRGERIVERSEN
26 AnintroductiontothetheoryofL-functionsandEisensteinseries,H.HIDA
27 Hilbertspace:Compactoperatorsandthetracetheorem,J.R.RETHERFORD
28 Potentialtheoryinthecomplexplane,T.RANSFORD
29 Undergraduatecommutativealgebra,M.REID
31 TheLaplacianonaRiemannianmanifold,S.ROSENBERG
32 LecturesonLiegroupsandLiealgebras,R.CARTER,G.SEGAL,&
I.MACDONALD
33 AprimerofalgebraicD-modules,S.C.COUTINHO
34 Complexalgebraicsurfaces,A.BEAUVILLE
35 Youngtableaux,W.FULTON
37 Amathematicalintroductiontowavelets,P.WOJTASZCZYK
38 Harmonicmaps,loopgroups,andintegrablesystems,M.GUEST
39 Settheoryfortheworkingmathematician,K.CIESIELSKI
40 Ergodictheoryanddynamicalsystems,M.POLLICOTT&M.YURI
41 Thealgorithmicresolutionofdiophantineequations,N.P.SMART
42 Equilibriumstatesinergodictheory,G.KELLER
43 Fourieranalysisonfinitegroupsandapplications,AUDREYTERRAS
44 Classicalinvarianttheory,PETERJ.OLVER
45 Permutationgroups,P.J.CAMERON
46 Riemannsurfaces:APrimer,A.BEARDON
47 Intoductorylecturesonringsandmodules,J.BEACHY
48 Settheory,A.HAJNA´L&P.HAMBURGER
49 K-theoryforC∗-algebras,M.RORDAM,F.LARSEN,&N.LAUSTSEN
50 Abriefguidetoalgebraicnumbertheory,H.P.F.SWINNERTON-DYER
51 Stepsincommutativealgebra:Secondedition,R.Y.SHARP
52 FiniteMarkovchainsandalgorithmicapplications,O.HAGGSTROM
53 Theprimenumbertheorem,G.J.O.JAMESON
54 Topicsingraphautomorphismsandreconstruction,J.LAURI&R.SCAPELLATO
55 Elementarynumbertheory,grouptheory,andRamanujangraphs,G.DAVIDOFF,
P.SARNAK,&A.VALETTE

COMPUTATIONAL ALGEBRAIC
GEOMETRY
HAL SCHENCK
TexasA&MUniversity
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521829649
© Hal Schenck 2003
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2003
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ToMomandDad

Contents
Preface pagexi
1 BasicsofCommutativeAlgebra 1
1.1 IdealsandVarieties 2
1.2 NoetherianRingsandtheHilbertBasisTheorem 4
1.3 AssociatedPrimesandPrimaryDecomposition 6
1.4 TheNullstellensatzandZariskiTopology 12
2 ProjectiveSpaceandGradedObjects 18
2.1 ProjectiveSpaceandProjectiveVarieties 18
2.2 GradedRingsandModules,HilbertFunctionandSeries 21
2.3 LinearAlgebraFlashback,HilbertPolynomial 26
3 FreeResolutionsandRegularSequences 34
3.1 FreeModulesandProjectiveModules 35
3.2 FreeResolutions 36
3.3 RegularSequences,MappingCone 42
4 Gro¨bnerBasesandtheBuchbergerAlgorithm 50
4.1 Gro¨bnerBases 51
4.2 MonomialIdealsandApplications 55
4.3 SyzygiesandGro¨bnerBasesforModules 58
4.4 ProjectionandElimination 60
5 Combinatorics,TopologyandtheStanley–ReisnerRing 64
5.1 SimplicialComplexesandSimplicialHomology 65
5.2 TheStanley–ReisnerRing 72
5.3 AssociatedPrimesandPrimaryDecomposition 77
6 Functors:Localization,Hom,andTensor 80
6.1 Localization 81
6.2 TheHomFunctor 84
6.3 TensorProduct 88
7 GeometryofPointsandtheHilbertFunction 92
7.1 HilbertFunctionsofPoints,Regularity 92
ix