extensionAlgebra -- extend a graded algebra by a graded module

Synopsis

• Usage:

  extensionAlgebra(f,c)

  extensionAlgebra(f,c,Codimension => r)

  extensionAlgebra(f,c,CoefficientRing => S)

  extensionAlgebra(f,c,Codimension => r, CoefficientRing => S)

• Inputs:
  • f, from A to B
  • c, a homogeneous element of B
• Optional inputs:
  • Codimension => an integer, default value null, the desired degree of the inclusion map from B to E; only required if c = 0, and otherwise must match degree of c
  • CoefficientRing => a ring, default value null, a coefficient ring of A, which will be used as the coefficient ring of the output
• Outputs:
  • a ring, the algebra obtained by making the direct sum of A and B[-r] into an algebra with the multiplication rule (a+b)(a'+b') = aa' + ab' + a'b + cbb'.
• Consequences:

  • An algebra E is created and promotion methods from A and B to E and vice-versa are installed. The natural pullback map from E to B given by sending a+b to f(a) + cb is stored in E.PullBack, see PullBack.

Description

i1 : A = QQ[x]

o1 = A

o1 : PolynomialRing

i2 : B = QQ[y]

o2 = B

o2 : PolynomialRing

i3 : c = 2_B

o3 = 2

o3 : B

i4 : f = map(B,A,gens B)

o4 = map (B, A, {y})

o4 : RingMap B <-- A

i5 : extensionAlgebra(f,c)

     QQ[E , x]
         0
o5 = ---------
       2
      E  - 2E
       0     0

o5 : QuotientRing

i6 : oo.PullBack

             QQ[E , x]
                 0
o6 = map (B, ---------, {2, y})
               2
              E  - 2E
               0     0

                   QQ[E , x]
                       0
o6 : RingMap B <-- ---------
                     2
                    E  - 2E
                     0     0