Commutative Algebra and its Interactions to Algebraic Geometry

265 Pages·2018·3.47 MB·English

by  Nguyen Tu CUONG

Lecture Notes in Mathematics 2210
Nguyen Tu CUONG · Le Tuan HOA 
Ngo Viet TRUNG Editors
Commutative
Algebra and
its Interactions
to Algebraic
Geometry
VIASM 2013–2014
Lecture Notes in Mathematics 2210
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryBoard:
MichelBrion,Grenoble
CamilloDeLellis,Princeton
AlessioFigalli,Zurich
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GáborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304

Nguyen Tu CUONG (cid:129) Le Tuan HOA (cid:129)
Ngo Viet TRUNG
Editors
Commutative Algebra
and its Interactions
to Algebraic Geometry
VIASM 2013–2014
123
Editors
NguyenTuCUONG LeTuanHOA
InstituteofMathematics InstituteofMathematics
VietnamAcademyofScience VietnamAcademyofScience
andTechnology andTechnology
Hanoi,Vietnam Hanoi,Vietnam
NgoVietTRUNG
InstituteofMathematics
VietnamAcademyofScience
andTechnology
Hanoi,Vietnam
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-75564-9 ISBN978-3-319-75565-6 (eBook)
https://doi.org/10.1007/978-3-319-75565-6
LibraryofCongressControlNumber:2018942527
MathematicsSubjectClassification(2010):13-XX,14-XX
©SpringerInternationalPublishingAG,partofSpringerNature2018
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAGpart
ofSpringerNature.
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
This collection of notes is based on four lectures given during the programme
CommutativeAlgebraattheVietnamInstitute ofAdvancedStudyin Mathematics
in the winter semester 2013–2014. The lectures provide introductions to recent
researchtopicsinCommutativeAlgebra,whicharerelatedtoAlgebraicGeometry
and other fields. The topics were chosen to represent different aspects of the use
of the basic tools of Commutative Algebra. The notes are mainly self-contained,
withthehopethatstudentswithadvancedbackgroundsinalgebracangetthrough
andabsorbdifferenttechniquesand ideasin CommutativeAlgebrabeforesettling
on concrete research problems. They can also be used separately as courses for
graduatestudents,dependingonthelevelandinterestofthestudents.
The first lecture, by M. Brodmann, offers an introduction to the theory of
rings of differential operators and their modules, also known as Weyl algebras
andD-modules.TheseconceptsrelateNon-commutativeAlgebraandCommutative
AlgebrawithAlgebraicGeometryandAnalysisinaveryappealingway.Thelecture
presents this theory from the viewpoint of Commutative Algebra and is aimed at
an audience having only a basic background in Commutative Algebra. The main
featureisthereforenottoexplaineverythingaboutWeylalgebrasandD-modules,
but only the relevant aspects which are directly related to Commutative Algebra,
suchasthecharacteristicvarietyviathetheoryoffilteredalgebrasandmodules.The
lastpartalsocontainssomerecentresultsonthestability,deformationanddefining
equations of the characteristic variety. The material is developed systematically
andis accompaniedbyexamplesandexercises.Thesenotesarewellsuitedforan
undergraduatecourse.
The second lecture, by J. Elias, is a short introductionto the theory of inverse
systems and its application in the classification of Artinian Gorenstein rings. The
classification of Artinian rings (rings of finite length) up to analytic isomorphism
isabasicprobleminCommutativeAlgebraandAlgebraicGeometry.Thisproblem
is even open for Artinian Gorenstein rings, when the ring is an injective module
over itself. Inverse systems provide an important tool in Commutative Algebra,
establishingabeautifulcorrespondencebetweenArtinianGorensteinquotientrings
andcertainpolynomialsviaderivations.Thenotesgiveathoroughintroductionto
v
vi Preface
the theory of injective modules and inverse systems and show how to use these
toolstoclassifyArtinianGorensteinringsandtocomputetheirBettinumbers.The
presentedmaterialcombinesseveralbasictechniquesofCommutativeAlgebraand
couldbeusedforagraduatecourse.
The third lecture, by R.M. Miró-Roig, is on the complexity of the structure of
projective varieties. This complexity can be measured by the representation type,
whichisthedimensionandthenumberoffamiliesofindecomposablearithmetically
Cohen–Macaulay sheaves (i.e. sheaves without intermediate cohomology) on the
underlyingvariety.ThisisafascinatingtopicofAlgebraicGeometry,whichrequires
anadvancedbackgroundinCommutativeAlgebra.Thenotescoverthebasicfacts
on this and related subjects such as moduli spaces of sheaves, liaison theory,
minimal resolutions and Hilbert schemes of points. Many interesting results are
presented on arithmetically Cohen–Macaulay sheaves and bundles having natural
extremal algebraic properties, and several examples of varieties of wild represen-
tationtypearegiven.Theexpositionisself-containedandfeaturesnumerousopen
problemsandpromisingideasforfurtherinvestigation.Itmayserveasa graduate
courseinAlgebraicGeometry.
Thelastlecture,byM.Morales,addressesaclassicalproblemofbothCommu-
tative Algebra and Algebraic Geometry, namely, how many equations are needed
to define an algebraic variety set-theoretically. This seemingly simple problem is
wide open even for toric varieties, which are given parametricallyby monomials.
Thenotesprovideanextensivesurveyonthisprobleminthecaseofsimplicialtoric
varieties,whicharedefinedbythepropertythattheexponentsoftheparametrizing
monomialsspanasimplicialcomplex.Onecanusearithmeticalandcombinatorial
tools (semigroups, lattices) to obtain satisfactory results for large classes of sim-
plicialtoricvarieties.Thematerialispresentedinasystematicwayandcaneasily
be followed by any reader with some basic backgroundin CommutativeAlgebra.
These notes are recommended as a first course for anyone who wants to see the
interaction between algebra, combinatorics and geometry. They can be used as a
startingpointforgraduatestudiesinCommutativeAlgebra.
Hanoi,Vietnam NguyenTuCUONG
14October2017 LeTuanHOA
NgoVietTRUNG
Contents
1 NotesonWeylAlgebraandD-Modules................................... 1
MarkusBrodmann
2 InverseSystemsofLocalRings............................................. 119
JuanElias
3 LecturesontheRepresentationTypeofaProjectiveVariety........... 165
RosaM.Miró-Roig
4 SimplicialToricVarietiesWhichAreSet-TheoreticComplete
Intersections.................................................................. 217
MarcelMorales
vii
Contributors
Markus Brodmann Universität Zürich, Institut für Mathematik, Zürich,
Switzerland
JuanElias DepartmentdeMatemàtiquesiInformàtica,UniversitatdeBarcelona,
Barcelona,Spain
Rosa M. Miró-Roig Department de Matemàtiques i Informàtica, Universitat de
Barcelona,Barcelona,Spain
Marcel Morales Université Grenoble Alpes, Institut Fourier, UMR 5582, Saint-
MartinD’HèresCedex,France
ix
Chapter 1
D
Notes on Weyl Algebra and -Modules
MarkusBrodmann
Abstract Weylalgebras,sometimescalledalgebrasofdifferentialoperators,area
fascinatingandimportantsubject,whichrelatesNon-CommutativeandCommuta-
tiveAlgebra,AlgebraicGeometryandAnalysisinveryappealingway.Thetheory
ofmodulesoverWeylalgebras,sometimescalled D-modules,findsapplicationin
thetheoryofpartialdifferentialequations,andthushasagreatimpacttomanyfields
of Mathematics. In our course, we shall give a short introduction to the subject,
using only prerequisites from Linear Algebra, Basic Abstract Algebra, and Basic
CommutativeAlgebra.Inaddition,inthelasttwosections,wepresentafewrecent
results.
1.1 Introduction
Thepresentnotesbaseontwoshortcourses:
(1) Introductionto Weyl Algebras:five Twin Lessons, Thai NguyenUniversityof
ScienceTNUS(ThaiNguyen,Vietnam),November1–10,2013.
(2) Weyl Algebras, Universal Gröbner Bases, Filtration Deformations and Char-
acteristic Varieties of D-Modules: four Twin Lessons, Vietnam Institute for
AdvancedStudyinMathematicsVIASM(Hanoi,Vietnam),November12–26,
2013.
Theywerealsothebaseforathirdcourse:
(3) Introduction to Weyl Algebras and D-Modules: four Lessons and two Tuto-
rial Sessions, “Workshop on Local Cohomology”, St. Joseph’s College Irin-
jalakuda,Kerala(India),June20–July2,2016.
M.Brodmann((cid:2))
UniversitätZürich,InstitutfürMathematik,Zürich,Switzerland
e-mail:[email protected]
©SpringerInternationalPublishingAG,partofSpringerNature2018 1
N.TuCUONGetal.(eds.),CommutativeAlgebraanditsInteractions
toAlgebraicGeometry,LectureNotesinMathematics2210,
https://doi.org/10.1007/978-3-319-75565-6_1