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# extensionAlgebra -- extend a graded algebra by a graded module

- 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

## Ways to use extensionAlgebra:

- `extensionAlgebra(RingMap,RingElement)`

## For the programmer

The object extensionAlgebra is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Schubert2.m2:1857:0.