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. 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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