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MODULES OVER MULTIVARIATE RINGS

 
Acknowledgements
 
Introduction
 
Module Basics: Embedded and Reduced Modules
 
Monomial Orders
      Term Over Position: TOP
      Term Over Position (Weighted): TOPW
      Position Over Term: POT
      Position Over Term (Permutation): POT-PERM
      Block TOP-TOP: TOPTOP
      Block TOP-POT: TOPPOT
 
Basic Creation and Access
      Creation of Ambient Embedded Modules
      Creation of Reduced Modules
      Localization
      Basic Invariants
      Creation of Module Elements
      Element Operations
            Access
            Arithmetic
            Accessing the Underlying Representation
            Predicates
 
The Homomorphism Type
 
Submodules and Quotient Modules
      Creation
      Module Bases
 
Basic Module Constructions
 
Predicates
 
Module Operations
 
Changing Ring
 
Hilbert Series
 
Free Resolutions
      Constructing Free Resolutions
      Betti Numbers and Related Invariants
 
The Hom Module and Ext
 
Tensor Products and Tor
 
Cohomology Of Coherent Sheaves
 
Bibliography







DETAILS

 
Introduction

 
Module Basics: Embedded and Reduced Modules

 
Monomial Orders

      Term Over Position: TOP

      Term Over Position (Weighted): TOPW

      Position Over Term: POT

      Position Over Term (Permutation): POT-PERM

      Block TOP-TOP: TOPTOP

      Block TOP-POT: TOPPOT

 
Basic Creation and Access

      Creation of Ambient Embedded Modules
            EModule(R, k) : Rng, RngIntElt -> ModMPol
            EModule(R, k, order) : Rng, RngIntElt, MonStgElt, ... -> ModMPol
            EModule(R, W) : Rng, [ RngIntElt ] -> ModMPol
            EModule(R, W, order) : Rng, [ RngIntElt ], MonStgElt, ... -> ModMPol

      Creation of Reduced Modules
            RModule(R, k) : Rng, RngIntElt -> ModMPol
            RModule(R, W) : Rng, [ RngIntElt ] -> ModMPol
            GradedModule(R, k) : Rng, RngIntElt -> ModMPol
            GradedModule(R, W) : Rng, [ RngIntElt ] -> ModMPol

      Localization
            Localization(M) : ModMPol -> ModMPol

      Basic Invariants
            Ambient(M) : ModMPol -> ModMPol
            IsAmbient(M) : ModMPol -> BoolElt
            IsEmbedded(M) : ModMPol -> BoolElt
            IsReduced(M) : ModMPol -> BoolElt
            IsRoot(M) : ModMPol -> BoolElt
            CoefficientRing(M) : ModMPol -> ModMPol
            Degree(M) : ModMPol -> RngIntElt
            ColumnWeights(M) : ModMPol -> [ RngIntElt ]
            RelationModule(M) : ModMPol -> [ ModMPol ]
            Relations(M) : ModMPol -> [ ModMPol ]
            RelationMatrix(M) : ModMPol -> ModMatRngElt
            Presentation(M) : ModMPol -> [ ModMPol ]
            IsGraded(M) : ModMPol -> BoolElt

      Creation of Module Elements
            M ! Q : ModMPol, [ RngElt ] -> ModMPolElt
            M ! v : ModMPol, ModTupRngElt -> ModMPolElt
            M ! 0 : ModMPol, RngIntElt -> ModMPolElt
            UnitVector(M, i) : ModMPol, RngIntElt -> ModMPolElt

      Element Operations

            Access
                  Eltseq(f) : ModMPolElt -> [ RngMPolElt ]
                  Vector(f) : ModMPolElt -> ModTupRngElt
                  f[i] : ModMPolElt, RngIntElt -> RngMPolElt

            Arithmetic
                  f div s : ModMPolElt, RngMPolElt -> ModMPolElt
                  SPolynomial(f, g) : ModMPolElt, ModMPolElt -> ModMPolElt
                  Normalize(f) : ModMPolElt -> ModMPolElt
                  NormalForm(f, S) : ModMPolElt, ModMPol -> ModMPolElt
                  Coordinates(f, M) : ModMPolElt, ModMPol -> [ RngMPolElt ]

            Accessing the Underlying Representation
                  Column(f) : ModMPolElt -> RngIntElt
                  Degree(f) : ModMPolElt -> RngIntElt
                  IsHomogeneous(f) : ModMPolElt -> BoolElt

            Predicates
                  IsZero(f) : ModMPolElt -> BoolElt
                  f eq g : ModMPolElt, ModMPolElt -> BoolElt
                  f lt g : ModMPolElt, ModMPolElt -> BoolElt
                  f in M : ModMPolElt, ModMPol -> BoolElt
                  Example PMod_CreateEmbedded (H109E1)

 
The Homomorphism Type
      Homomorphism(M, N, A): ModMPol, ModMPol, Mtrx -> ModMPolHom
      Domain(f) : ModMPolHom -> ModMPol
      Codomain(f) : ModMPolHom -> ModMPol
      PresentationMatrix(f) : ModMPolHom -> ModMatRngElt
      AmbientMatrix(f) : ModMPolHom -> ModMatRngElt
      f(v) : ModMPolHom, RngMPolElt -> RngMPolElt
      f[i] : ModMPolHom, RngIntElt -> RngMPolElt
      Image(f) : ModMPolHom -> ModMPol
      Kernel(f) : ModMPolHom -> ModMPol
      Cokernel(f) : ModMPolHom -> ModMPol
      IsZero(f) : ModMPolHom -> BoolElt
      IsInjective(f) : ModMPolHom -> BoolElt
      IsSurjective(f) : ModMPolHom -> BoolElt
      IsBijective(f) : ModMPolHom -> BoolElt
      IsGraded(f) : ModMPolHom -> BoolElt
      Degree(f) : ModMPolHom -> RngIntElt
      Example PMod_HomomorphismEmbedded (H109E2)

 
Submodules and Quotient Modules

      Creation
            sub<M | L> : ModMPol, List -> ModMPol
            quo<M | L> : ModMPol, List -> ModMPol
            Morphism(M, N) : ModMPol, ModMPol -> ModMPolHom
            Submodule(I) : RngMPol -> ModMPol
            QuotientModule(I) : RngMPol -> ModMPol
            GradedModule(I) : RngMPol -> ModMPol

      Module Bases
            Basis(M) : ModMPol -> RngMPolElt
            BasisElement(M, i) : ModMPol, RngIntElt -> RngMPolElt
            BasisMatrix(M) : ModMPol -> ModMatRngElt
            Groebner(M) : ModMPol ->
            Example PMod_SubQuoEmbedded (H109E3)
            Example PMod_SubQuoReduced (H109E4)

 
Basic Module Constructions
      M + N : ModMPol, ModMPol -> ModMPol
      M meet N : ModMPol, ModMPol -> ModMPol
      f * M : ModMPol, RngElt -> ModMPol
      I * M : RngMPol, ModMPol -> ModMPol
      M / N : ModMPol, ModMPol -> ModMPol
      DirectSum(M, N) : ModMPol, ModMPol -> ModMPol, [ModMPolHom], [ModMPolHom]
      DirectSum(S) : [ModMPol] -> ModMPol, [ModMPolHom], [ModMPolHom]
      Twist(M, d) : ModMPol, RngIntElt -> [ ModMPolElt ], ModMPolHom

 
Predicates
      IsZero(M) : ModMPol -> ModMPol
      M subset N : ModMPol, ModMPol -> BoolElt
      M eq N : ModMPol, ModMPol -> BoolElt
      IsFree(M) : ModMPol -> BoolElt

 
Module Operations
      MinimalBasis(M) : ModMPol -> [ ModMPolElt ]
      MinimalBasis(S) : [ ModMPolElt ] -> [ ModMPolElt ]
      Rank(M) : ModMPol -> RngIntElt
      ColonModule(M, J) : ModMPol, RngMPol -> ModMPol
      ColonIdeal(M, N) : ModMPol, ModMPol -> RngMPol
      Annihilator(M) : ModMPol -> RngMPol
      FittingIdeal(M, i) : ModMPol, RngIntElt -> RngMPol
      FittingIdeals(M) : ModMPol, RngIntElt -> RngMPol
      SyzygyModule(M) : ModMPol -> [ ModMPolElt ]
      MinimalSyzygyModule(M) : ModMPol -> [ ModMPolElt ]
      SyzygyModule(Q) : [ RngMPolElt ] -> ModTupRng
      Example PMod_Rank (H109E5)

 
Changing Ring
      ChangeRing(M, S) : RngMPol, Rng -> RngMPol

 
Hilbert Series
      HilbertSeries(M) : ModMPol -> FldFunElt
      HilbertSeries(M, p) : ModMPol, RngIntElt -> RngSerLaurElt
      HilbertDenominator(M) : ModMPol -> RngUPol
      HilbertNumerator(M) : ModMPol -> RngUPolElt, RngIntElt
      HilbertPolynomial(I) : ModMPol -> RngUPolElt, RngIntElt
      Example PMod_Hilbert (H109E6)

 
Free Resolutions

      Constructing Free Resolutions
            FreeResolution(M) : ModMPol -> ModCpx, ModMPolHom
            SetVerbose("Resolution", v) : MonStgElt, RngIntElt ->
            Example PMod_FreeResolution (H109E7)
            Example PMod_FreeResolution1 (H109E8)

      Betti Numbers and Related Invariants
            BettiNumbers(M) : ModMPol -> [ RngIntElt ]
            BettiNumber(M, i, j) : ModMPol, RngIntElt -> RngIntElt
            MaximumBettiDegree(M, i) : ModMPol -> RngIntElt
            BettiTable(M) : ModMPol -> [[ RngIntElt ]], RngIntElt
            Regularity(M) : ModMPol -> RngInt
            HomologicalDimension(M) : ModMPol -> RngInt
            Example PMod_Koszul (H109E9)
            Example PMod_ExteriorAlgebra (H109E10)
            Example PMod_FreeResolutionLocal (H109E11)
            Example PMod_HilbertSeriesBetti (H109E12)
            Example PMod_Regularity (H109E13)
            Example PMod_Genus11Curve (H109E14)

 
The Hom Module and Ext
      Hom(M, N) : ModMPol, ModMPol -> ModMPol, Map
      Hom(C, N) : ModCpx, ModMPol -> ModMPol
      Ext(i, M, N) : RngIntElt, ModMPol, ModMPol -> ModMPol
      Example PMod_Hom (H109E15)

 
Tensor Products and Tor
      TensorProduct(M, N) : ModMPol, ModMPol -> ModMPol, Map
      TensorProduct(C, N) : ModCpx, ModMPol -> ModMPol
      Tor(i, M, N) : RngIntElt, ModMPol, ModMPol -> ModMPol
      Example PMod_Hom (H109E16)

 
Cohomology Of Coherent Sheaves
      CohomologyDimension(M,r,n) : ModMPolGrd, RngIntElt, RngIntElt -> RngIntElt
      Example PMod_cohom (H109E17)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012