Modules over Multivariate Rings
- Introduction
- Module Basics: Embedded and Reduced Modules
- Monomial Orders
- Basic Creation and Access
- Creation of Ambient Embedded Modules
- Creation of Reduced Modules
- Localization
- Basic Invariants
Ambient(M): ModMPol → ModMPolGeneric(M): ModMPol → ModMPolIsAmbient(M): ModMPol → BoolEltIsEmbedded(M): ModMPol → BoolEltIsReduced(M): ModMPol → BoolEltIsRoot(M): ModMPol → BoolEltCoefficientRing(M): ModMPol → ModMPolBaseRing(M): ModMPol → RngDegree(M): ModMPol → RngIntEltColumnWeights(M): ModMPol → [ RngIntElt ]Grading(M): ModMPol → [ RngIntElt ]RelationModule(M): ModMPol → [ ModMPol ]Relations(M): ModMPol → [ ModMPol ]RelationMatrix(M): ModMPol → ModMatRngEltPresentation(M): ModMPol → [ ModMPol ]IsGraded(M): ModMPol → BoolEltIsHomogeneous(M): ModMPol → BoolElt
- Creation of Module Elements
- Element Operations
- Access
- Arithmetic
f + g: ModMPolElt, ModMPolElt → ModMPolEltf - g: ModMPolElt, ModMPolElt → ModMPolElt- f: ModMPolElt → ModMPolEltr * f: RngElt, ModMPolElt → ModMPolEltf * r: ModMPolElt, RngElt → ModMPolEltf div s: ModMPolElt, RngMPolElt → ModMPolEltSPolynomial(f, g): ModMPolElt, ModMPolElt → ModMPolEltNormalize(f): ModMPolElt → ModMPolEltNormalForm(f, S): ModMPolElt, ModMPol → ModMPolEltCoordinates(M, f): ModMPolElt, ModMPol → [ RngMPolElt ]
- Accessing the Underlying Representation
Coefficients(f): ModMPolElt → [RngElt]Monomials(f): ModMPolElt → [ModMPolElt]Terms(f): ModMPolElt → [ModMPolElt]LeadingCoefficient(f): ModMPolElt → RngEltLeadingMonomial(f): ModMPolElt → ModMPolEltLeadingTerm(f): ModMPolElt → ModMPolEltCoefficientsAndMonomials(f): ModMPolElt → [RngElt], [ModMPolElt]Column(f): ModMPolElt → RngIntEltDegree(f): ModMPolElt → RngIntEltWeightedDegree(f): ModMPolElt → RngIntEltIsHomogeneous(f): ModMPolElt → BoolElt
- Predicates
- The Homomorphism Type
Homomorphism(M, N, A): ModMPol, ModMPol, Mtrx → ModMPolHomDomain(f): ModMPolHom → ModMPolCodomain(f): ModMPolHom → ModMPolPresentationMatrix(f): ModMPolHom → ModMatRngEltMatrix(f): ModMPolHom → ModMatRngEltAmbientMatrix(f): ModMPolHom → ModMatRngEltMatrix(f): ModMPolHom → ModMatRngEltf(v): ModMPolHom, RngMPolElt → RngMPolEltv * f: ModMPolHom, RngMPolElt → RngMPolEltf[i]: ModMPolHom, RngIntElt → RngMPolEltImage(f): ModMPolHom → ModMPolKernel(f): ModMPolHom → ModMPolCokernel(f): ModMPolHom → ModMPolIsZero(f): ModMPolHom → BoolEltIsInjective(f): ModMPolHom → BoolEltIsSurjective(f): ModMPolHom → BoolEltIsBijective(f): ModMPolHom → BoolEltIsGraded(f): ModMPolHom → BoolEltIsHomogeneous(f): ModMPolHom → BoolEltDegree(f): ModMPolHom → RngIntElt- Example: Inclusion Homomorphism of Submodules
- Submodules and Quotient Modules
- Basic Module Constructions
M + N: ModMPol, ModMPol → ModMPolM meet N: ModMPol, ModMPol → ModMPolf * M: ModMPol, RngElt → ModMPolM * f: ModMPol, RngElt → ModMPolI * M: RngMPol, ModMPol → ModMPolM * I: ModMPol, RngMPol → ModMPolM / N: ModMPol, ModMPol → ModMPolDirectSum(M, N): ModMPol, ModMPol → ModMPol, [ModMPolHom], [ModMPolHom]DirectSum(S): [ModMPol] → ModMPol, [ModMPolHom], [ModMPolHom]DirectSum(S): [* ModMPol *] → ModMPol, [ModMPolHom], [ModMPolHom]Twist(M, d): ModMPol, RngIntElt → [ ModMPolElt ], ModMPolHom
- Predicates
- Module Operations
MinimalBasis(M): ModMPol → [ ModMPolElt ]MinimalBasis(S): [ ModMPolElt ] → [ ModMPolElt ]MinimalBasis(S): { ModMPolElt } → [ ModMPolElt ]Rank(M): ModMPol → RngIntEltColonModule(M, J): ModMPol, RngMPol → ModMPolColonIdeal(M, N): ModMPol, ModMPol → RngMPolAnnihilator(M): ModMPol → RngMPolFittingIdeal(M, i): ModMPol, RngIntElt → RngMPolFittingIdeals(M): ModMPol → RngMPolSyzygyModule(M): ModMPol → [ ModMPolElt ]MinimalSyzygyModule(M): ModMPol → [ ModMPolElt ]SyzygyModule(Q): [ RngMPolElt ] → ModTupRng- Example: Rank of a Module and Its Localization
- Changing Ring
- Hilbert Series
- Free Resolutions
- Constructing Free Resolutions
- Betti Numbers and Related Invariants
BettiNumbers(M): ModMPol → [ RngIntElt ]BettiNumber(M, i, j): ModMPol, RngIntElt, RngIntElt → RngIntEltMaximumBettiDegree(M, i): ModMPol, RngIntElt → RngIntEltBettiTable(M): ModMPol → [[ RngIntElt ]], RngIntEltRegularity(M): ModMPol → RngIntHomologicalDimension(M): ModMPol → RngInt- Example: Koszul Complex
- Example: Resolution over an Exterior Algebra
- Example: Betti Numbers of a Localized Module
- Example: Hilbert Series from Betti Numbers
- Example: Bounding the Regularity of a Module
- Example: Random Space Curve of Genus 11
- The Hom Module and Ext
- Tensor Products and Tor
- Cohomology Of Coherent Sheaves
- Bibliography