  
  [1X3 [33X[0;0YQuick Start[133X[101X
  
  
  [1X3.1 [33X[0;0YLocalization of ℤ[133X[101X
  
  [33X[0;0YThe following example is taken from Section 2 of [BR06].[133X
  [33X[0;0YThe  computation  takes place over the local ring [22XR=ℤ_⟨ 2⟩[122X (i.e. ℤ localized
  at the maximal ideal generated by [22X2[122X).[133X
  
  [33X[0;0YHere we compute the (infinite) long exact homology sequence of the covariant
  functor [22XHom(Hom(-,R/2^7R),R/2^4R)[122X (and its left derived functors) applied to
  the short exact sequence[133X
  [33X[0;0Y[22X0 -> M_=R/2^2R --alpha_1--> M=R/2^5R --alpha_2--> _M=R/2^3R -> 0[122X.[133X
  
  [33X[0;0YWe   want   to   lead   your   attention  to  the  commands  [9XLocalizeAt[109X  and
  [9XHomalgLocalMatrix[109X.  The first one creates a localized ring from a global one
  and  generators of a maximal ideal and the second one creates a local matrix
  from  a  global  matrix.  The  other  commands used here are well known from
  [5Xhomalg[105X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X  [128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "LocalizeRingForHomalg" );;[127X[104X
    [4X[25Xgap>[125X [27Xzz := HomalgRingOfIntegers(  );[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27XR := LocalizeAt( zz , [ 2 ] );[127X[104X
    [4X[28XZ_< 2 >[128X[104X
    [4X[25Xgap>[125X [27XDisplay( R );[127X[104X
    [4X[28X<A local ring>[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "Modules" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XM := LeftPresentation( HomalgMatrix( [ 2^5 ], R ) );[127X[104X
    [4X[28X<A cyclic left module presented by 1 relation for a cyclic generator>[128X[104X
    [4X[25Xgap>[125X [27X_M := LeftPresentation( HomalgMatrix( [ 2^3 ], R ) );[127X[104X
    [4X[28X<A cyclic left module presented by 1 relation for a cyclic generator>[128X[104X
    [4X[25Xgap>[125X [27Xalpha2 := HomalgMap( HomalgMatrix( [ 1 ], R ), M, _M );[127X[104X
    [4X[28X<A "homomorphism" of left modules>[128X[104X
    [4X[25Xgap>[125X [27XM_ := Kernel( alpha2 );[127X[104X
    [4X[28X<A cyclic left module presented by yet unknown relations for a cyclic generato\[128X[104X
    [4X[28Xr>[128X[104X
    [4X[25Xgap>[125X [27Xalpha1 := KernelEmb( alpha2 );[127X[104X
    [4X[28X<A monomorphism of left modules>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( M_ );[127X[104X
    [4X[28XZ_< 2 >/< -4/1 >[128X[104X
    [4X[25Xgap>[125X [27XDisplay( alpha1 );[127X[104X
    [4X[28X[ [  8 ] ][128X[104X
    [4X[28X/ 1[128X[104X
    [4X[28X[128X[104X
    [4X[28Xthe map is currently represented by the above 1 x 1 matrix[128X[104X
    [4X[25Xgap>[125X [27XByASmallerPresentation( M_ );[127X[104X
    [4X[28X<A cyclic left module presented by 1 relation for a cyclic generator>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( M_ );[127X[104X
    [4X[28XZ_< 2 >/< 4/1 >[128X[104X
  [4X[32X[104X
  
