Minimal Free Resolutions over Complete Intersections

113 Pages·2016·1.702 MB·English

by  David Eisenbud, Irena Peeva (auth.)

Lecture Notes in Mathematics 2152
David Eisenbud
Irena Peeva
Minimal Free
Resolutions
over Complete
Intersections
Lecture Notes in Mathematics 2152
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
CamilloDeLellis,Zurich
MariodiBernardo,Bristol
AlessioFigalli,Austin
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GaborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,ParisandNY
CatharinaStroppel,Bonn
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
David Eisenbud • Irena Peeva
Minimal Free Resolutions
over Complete Intersections
123
DavidEisenbud IrenaPeeva
MSRI&MathematicsDepartment MathematicsDepartment
UniversityofCalifornia CornellUniversity
Berkeley,CA,USA Ithaca,NY,USA
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-26436-3 ISBN978-3-319-26437-0 (eBook)
DOI10.1007/978-3-319-26437-0
LibraryofCongressControlNumber:2015958653
MathematicsSubjectClassification(2010):13D02,11-XX
SpringerChamHeidelbergNewYorkDordrechtLondon
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Preface
Thetheoryofhighermatrixfactorizationsofaregularsequencef1;:::;fcpresented
inthisbookisanextensionofthetheoryofmatrixfactorizationsofanelementin
a commutative ring. It allows us to give a description of the eventualstructure of
minimalfreeresolutionsoveracompleteintersectionRDS=.f1;:::;fc/ofarbitrary
codimension c (where S is a regular local ring). We construct the minimal free
resolutionsof high R-syzygymodules,both overthe regularlocalring S and over
thecompleteintersectionringR.
Berkeley,CA,USA DavidEisenbud
Ithaca,NY,USA IrenaPeeva
August2015
v

Acknowledgments
WearegratefultoJesseBurkeandHailongDaowhoreadadraftofthemanuscript
andmadehelpfulcomments.WewanttoexpressourthankstoDavidMorrisonfor
lecturing to us on the applications of matrix factorizations in physics. We thank
Luchezar Avramov and Ragnar Buchweitz for useful conversations. We profited
from examplescomputedin the system Macaulay2[M2], and we wish to express
ourthankstoitsprogrammersMikeStillmanandDanGrayson.
Eisenbud is partially supported by NSF grant DMS-1001867, and Peeva is
partially supported by NSF grants DMS-1100046 and DMS-1406062 and by a
SimonsFellowship;bothauthorsarepartiallysupportedbyNSFgrant0932078000
whileinresidenceatMSRI.
vii

Contents
1 IntroductionandSurvey .................................................... 1
1.1 HowWeGotHere....................................................... 1
1.2 WhatisaHigherMatrixFactorization?................................ 7
1.3 What’sinThisBook?................................................... 11
1.4 NotationandConventions.............................................. 15
2 MatrixFactorizationsofOneElement .................................... 19
2.1 MatrixFactorizationsandResolutionsoveraHypersurface.......... 19
3 FiniteResolutionsofHMFModules ...................................... 23
3.1 The MinimalS-FreeResolutionof a HigherMatrix
FactorizationModule ................................................... 23
3.2 Consequences ........................................................... 29
3.3 BuildingaKoszulExtension ........................................... 33
3.4 HigherHomotopies..................................................... 34
4 CIOperators ................................................................. 37
4.1 CIOperators............................................................. 37
4.2 TheActionoftheCIOperatorsonExt ................................ 40
4.3 ResolutionswithaSurjectiveCIOperator............................. 44
5 InfiniteResolutionsofHMFModules ..................................... 49
5.1 TheMinimalR-FreeResolutionofa HigherMatrix
FactorizationModule ................................................... 49
5.2 BettiNumbers........................................................... 56
5.3 StrongMatrixFactorizations........................................... 58
5.4 ResolutionsoverIntermediateRings................................... 60
6 Far-OutSyzygies ............................................................ 63
6.1 Pre-stableSyzygiesandGenericCIOperators........................ 63
6.2 TheGradedCase........................................................ 70
6.3 TheBoxComplex....................................................... 72
6.4 FromSyzygiestoHigherMatrixFactorizations....................... 76
6.5 BettiNumbersofPre-stableMatrixFactorizations.................... 82
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