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Packages » BernsteinSato :: localCohom(...,Strategy=>...)

Description

There are two main strategies, Walther and OaTa. If the user selects Walther, which is the default, then LocStrategy determines the localization strategy for localCohom(...Ideal...) and localCohom(...Ideal, Module...).

• Walther -- the algorithm of U. Walther that uses Cech complex.
• • LocStrategy => null -- used only for localCohom(...Ideal...), localizations are done by straightforward computation of annihilators and b-polynomials as described in [1].
  • LocStrategy => OaTaWa -- localizations are done following Oaku-Takayama-Walther method [2].
  • LocStrategy => Oaku -- localizations are done following Oaku's algorithm.
• OaTa -- restriction from the graph embedding is used, which is due to T. Oaku and N. Takayama [3]. See Drestriction.

For detailed description of the algorithms see

• [1] Walther, Algorithmic computation of local cohomology modules and the local cohomological dimension of algebraic varieties (JPAA (139), 1999.)
• [2] Oaku, Takayama, Walther, A Localization Algorithm for D-modules (J. Symbolic Computation (29), 2000.)
• [3] Oaku, Takayama, Algorithms for D-modules -- restriction, tensor product, localization, and local cohomology groups (JPAA (156), 2001.)

Further information

• Default value: Walther
• Function: localCohom -- local cohomology
• Option key: Strategy -- an optional argument

Caveat

localCohom(...Ideal, Module...) with the default strategy computes presentations for all the terms in the Cech complex regardless of the requested homological degrees. All strategies use the given generators of the ideal; the user is advised to call mingens before calling localCohom.

Functions with optional argument named Strategy:

• addHook(...,Strategy=>...) -- see addHook -- add a hook function to an object for later processing
• adjoint(...,Strategy=>...) (missing documentation)
• annihilator(...,Strategy=>...) -- see annihilator -- the annihilator ideal
• associatedPrimes(...,Strategy=>...) -- see associatedPrimes -- find associated primes
• basis(...,Strategy=>...) -- see basis -- basis or generating set of all or part of a ring, ideal or module
• bFunction(...,Strategy=>...) -- specify strategy for computing b-function
• canonicalBundle(...,Strategy=>...) (missing documentation)
• mingens(...,Strategy=>...) -- see Complement -- a Strategy option value
• trim(...,Strategy=>...) -- see Complement -- a Strategy option value
• compose(Module,Module,Module,Strategy=>...) -- see compose -- composition as a pairing on Hom-modules
• cotangentSheaf(...,Strategy=>...) (missing documentation)
• deRham(...,Strategy=>...)
• deRhamAll(...,Strategy=>...)
• determinant(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic algorithms
• DExt(...,Strategy=>...)
• DHom(...,Strategy=>...)
• Dintegration(...,Strategy=>...)
• DintegrationAll(...,Strategy=>...) -- see DintegrationIdeal(...,Strategy=>...)
• DintegrationClasses(...,Strategy=>...) -- see DintegrationIdeal(...,Strategy=>...)
• DintegrationComplex(...,Strategy=>...) -- see DintegrationIdeal(...,Strategy=>...)
• DintegrationIdeal(...,Strategy=>...)
• Drestriction(...,Strategy=>...) -- see DintegrationIdeal(...,Strategy=>...)
• DrestrictionAll(...,Strategy=>...) -- see DintegrationIdeal(...,Strategy=>...)
• DrestrictionClasses(...,Strategy=>...) -- see DintegrationIdeal(...,Strategy=>...)
• DrestrictionComplex(...,Strategy=>...) -- see DintegrationIdeal(...,Strategy=>...)
• DrestrictionIdeal(...,Strategy=>...) -- see DintegrationIdeal(...,Strategy=>...)
• Dlocalize(...,Strategy=>...) -- strategy for computing a localization of a D-module
• DlocalizeAll(...,Strategy=>...) -- see Dlocalize(...,Strategy=>...) -- strategy for computing a localization of a D-module
• DlocalizeMap(...,Strategy=>...) -- see Dlocalize(...,Strategy=>...) -- strategy for computing a localization of a D-module
• Dresolution(...,Strategy=>...) -- strategy for computing a resolution of a D-module
• dual(ChainComplex,Strategy=>...) (missing documentation)
• dual(Matrix,Strategy=>...) (missing documentation)
• dual(MonomialIdeal,List,Strategy=>...) -- see dual(MonomialIdeal,Strategy=>...)
• dual(MonomialIdeal,RingElement,Strategy=>...) -- see dual(MonomialIdeal,Strategy=>...)
• dual(MonomialIdeal,Strategy=>...)
• dual(SheafMap,Strategy=>...) (missing documentation)
• End(...,Strategy=>...) -- see End -- module of endomorphisms
• exteriorPower(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic algorithms
• gb(...,Strategy=>...) -- see gb -- compute a Gröbner basis
• gbw(...,Strategy=>...) -- see gbw -- compute a Gröbner basis with respect to a weight vector
• gcdLLL(...,Strategy=>...) (missing documentation)
• generalB(...,Strategy=>...) -- specify strategy for computing generalized Bernstein-Sato polynomial
• GF(...,Strategy=>...) -- see GF -- make a finite field
• globalBFunction(...,Strategy=>...) -- specify strategy for computing global b-function
• groebnerBasis(...,Strategy=>...) -- see groebnerBasis -- Gröbner basis, as a matrix
• hermite(...,Strategy=>...) (missing documentation)
• Hom(...,Strategy=>...) -- see Hom -- module of homomorphisms
• homomorphism'(...,Strategy=>...) -- see homomorphism' -- get the element of Hom from a homomorphism
• hooks(...,Strategy=>...) -- see hooks -- list hooks attached to a key
• idealizer(...,Strategy=>...) -- see idealizer -- compute Hom(I,I) as a quotient ring
• integralClosure(...,Strategy=>...) -- control the algorithm used
• intersect(Ideal,Ideal,Strategy=>...) -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
• intersect(Module,Module,Strategy=>...) -- see intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
• intersectInP(...,Strategy=>...) -- see intersectInP(...,BasisElementLimit=>...) -- Option for intersectInP
• intersectionCohomology(...,Strategy=>...) -- see intersectionCohomology -- intersection cohomology of an irreducible affine variety
• isInMultiplierIdeal(...,Strategy=>...) -- see isInMultiplierIdeal(RingElement,Ideal,QQ) -- multiplier ideal membership test
• isPrimary(...,Strategy=>...) -- see isPrimary -- determine whether a submodule is primary
• isPrime(Ideal,Strategy=>...) -- see isPrime(Ideal) -- whether an ideal is prime
• lct(...,Strategy=>...) -- specify strategy for computing lct
• LLL(...,Strategy=>...) -- choose among different algorithms
• localCohom(...,Strategy=>...) -- specify strategy for local cohomology
• localize(...,Strategy=>...) -- see localize -- localize an ideal at a prime ideal
• match(...,Strategy=>...) -- see match -- regular expression matching
• decompose(Ideal,Strategy=>...) -- see minimalPrimes -- minimal primes of an ideal
• minimalPrimes(...,Strategy=>...) -- see minimalPrimes -- minimal primes of an ideal
• minors(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic algorithms
• jumpingCoefficients(...,Strategy=>...) -- see multiplierIdeal(Ideal,QQ) -- multiplier ideal
• multiplierIdeal(...,Strategy=>...) -- see multiplierIdeal(Ideal,QQ) -- multiplier ideal
• parallelApply(...,Strategy=>...) -- see parallelApply -- apply a function to each element in parallel
• polynomialExt(...,Strategy=>...)
• primaryComponent(...,Strategy=>...) -- see primaryComponent -- find a primary component corresponding to an associated prime
• pushForward(...,Strategy=>...) (missing documentation)
• quotient(...,Strategy=>...)
• radical(...,Strategy=>...) -- see radical -- the radical of an ideal
• radicalContainment(...,Strategy=>...) -- see radicalContainment -- whether an element is contained in the radical of an ideal
• rationalFunctionExt(...,Strategy=>...)
• analyticSpread(...,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• associatedGradedRing(...,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• distinguished(...,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• isLinearType(...,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• isReduction(...,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• minimalReduction(...,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• multiplicity(...,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• normalCone(Ideal,RingElement,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• normalCone(Ideal,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• reesAlgebra(...,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• specialFiber(...,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• specialFiberIdeal(...,Strategy=>...) -- see reesIdeal(...,Strategy=>...) -- Choose a strategy for the saturation step
• regSeqInIdeal(...,Strategy=>...) -- see regSeqInIdeal -- a regular sequence contained in an ideal
• resolution(...,Strategy=>...)
• saturate(...,Strategy=>...)
• sheafHom(...,Strategy=>...) (missing documentation)
• primaryDecomposition(...,Strategy=>...) -- see strategies for computing primary decomposition
• syz(...,Strategy=>...) -- see syz(Matrix) -- compute the syzygy matrix
• tangentCone(...,Strategy=>...) -- see tangentCone(Ideal)
• tangentSheaf(...,Strategy=>...) (missing documentation)