Module Basics: Embedded and Reduced Modules
Term Over Position (Weighted): TOPW
Position Over Term (Permutation): POT-PERM
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
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)
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
Wed Apr 24 15:09:57 EST 2013