One Hat Cyber Team

View File Name : Dlocalize.m2

document {
     Key => {[Dlocalize,Strategy],[DlocalizeAll,Strategy],[DlocalizeMap,Strategy],
	 Oaku,OTW,OTWcyclic},
     Headline=>"strategy for computing a localization of a D-module",
     UL{
	  {BOLD "Oaku", " -- use Oaku's algorithm"},
	  {BOLD "OTW", " -- use Oaku-Takayama-Walther's algorithm"},
	  {BOLD "OTWcyclic", " -- use Oaku-Takayama-Walther's algorithm for a cyclic module"}
	  }
     }

document {
     Key => {Dlocalize, (Dlocalize,Ideal,RingElement), (Dlocalize,Module,RingElement)},
     Headline => "localization of a D-module",

     Usage => "Dlocalize(M,f), Dlocalize(I,f)",
     Inputs => {
	  "M" => Module => {"over the Weyl algebra ", EM "D"},
	  "I" => Ideal => {"which represents the module ", EM "M = D/I"},
	  "f" => RingElement => "a polynomial",
	  },
     Outputs => {
	  Module => {"the localized module ", TEX "M_f = M[f^{-1}]", " as a D-module"}
	  },
     "One of the nice things about D-modules is that if a finitely
     generated D-module is specializable along ", EM "f", ", then it's localization 
     with respect to ", EM "f", " is also finitely generated.  For instance,
     this is true for all holonomic D-modules.",
     
     PARA{},
     "There are two different algorithms for localization implemented.  
     The first appears in the
     paper 'A localization algorithm for D-modules' by Oaku-Takayama-Walther
     (1999).  The second is due to Oaku and appears in the paper
     'Algorithmic computation of local cohomology modules and the
     cohomological dimension of algebraic varieties' by Walther(1999)",

     EXAMPLE lines ///
	W = QQ[x,y,Dx,Dy, WeylAlgebra => {x=>Dx,y=>Dy}]
     	M = W^1/(ideal(x*Dx+1, Dy))
     	f = x^2-y^3
     	Mf = Dlocalize(M, f)
	///,
     SeeAlso => {"DlocalizeAll", "DlocalizeMap", "AnnFs", "Dintegration"}
     }

document {
     Key => {DlocalizeMap, (DlocalizeMap,Ideal,RingElement), (DlocalizeMap,Module,RingElement)},
     Headline => "localization map from a D-module to its localization",
     Usage => "DlocalizeMap(M,f), DlocalizeMap(I,f)",
     Inputs => {
	  "M" => Module => {"over the Weyl algebra ", EM "D"},
	  "I" => Ideal => {"which represents the module ", EM "M = D/I"},
	  "f" => RingElement => "a polynomial",
	  },
     Outputs => {
	  Matrix => {"which represents the natural map from ", TEX "M", " to ", TEX "M_f = M[f^{-1}]"}
	  },
     "A supplementary function for ", TO "Dlocalize", 
     " that computes the localization map.",
     EXAMPLE lines ///
	W = QQ[x,y,Dx,Dy, WeylAlgebra => {x=>Dx,y=>Dy}]
     	M = W^1/(ideal(x*Dx+1, Dy))
     	f = x^2-y^3
     	DlocalizeMap(M, f)
	///,
     SeeAlso => {"Dlocalize", "AnnFs", "Dintegration"}
     }

doc ///
    Key
    	DlocalizeAll
	(DlocalizeAll,Ideal,RingElement)
	(DlocalizeAll,Module,RingElement)
	LocModule
	GeneratorPower
	LocMap
	annFS
	IntegrateBfunction
	Bfunction
    Headline
    	localization of a D-module (extended version)
    Usage
    	DlocalizeAll(M,f)
	DlocalizeAll(I,f)
    Inputs
    	M:Module
	    over the Weyl algebra $D$
	I:Ideal
	    which represents the module $M=D/I$
	f:RingElement
	    a polynomial
    Outputs
    	:HashTable
	    which contains the localized module $M_f = M[f^{-1}]$ and some additional information
    Description
    	Text
	    An extension of @TO Dlocalize@ that in addition computes the localization map
	    the b-function, and the power $s$ of the generator $f^s$.
	    
	    The keys of the output HashTable depend on which strategy is used. Common to each strategy
	    are the keys @{TT "LocMap"}@ and @{TT "LocModule"}@, which have the localization map
	    and the localized module, respectively; and @{TT "GeneratorPower"}@, which is an integer
	    $s$ such that (the images of) the generators of $M$ are $f^{-s}$ times the generators of $M_f$.
	Example
	    W = makeWeylAlgebra(QQ[x,y])
	    M = W^1/ideal(x*dx + 1, dy)
	    f = x^2 - y^3
	    Mfall = DlocalizeAll(M, f)
	    gens image Mfall.LocMap == f^(-Mfall.GeneratorPower) * gens Mfall.LocModule
	Text
	    
    SeeAlso
        Dlocalize
	AnnFs
	Dintegration
///