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# reesAlgebra -- Compute the defining ideal of the Rees Algebra

- Usage:
    `A = reesAlgebra M`
    `A = reesAlgebra(M,f)`
- Inputs:
  - M, a module, or an ideal of a quotient polynomial ring $R$
  - f, a ring element, any non-zerodivisor in ideal or the first Fitting ideal of the module. Optional
- Optional inputs:
  - BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
  - DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step. Defaults to infinity
  - Jacobian (missing documentation) => ..., default value false,
  - MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
  - PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
  - Strategy => ..., default value null, Choose a strategy for the saturation step
  - Variable => ..., default value "w", Choose name for variables in the created ring
- Outputs:
  - A, a ring, defining the Rees algebra of M

## Description

If $M$ is an ideal or module over a ring $R$, and $F\to M$ is a surjection from a free module, then reesAlgebra(M) returns the ring $Sym(F)/J$, where $J = reesIdeal(M)$.

In the following example, we find the Rees Algebra of a monomial curve singularity. We also demonstrate the use of reesIdeal, symmetricKernel, isLinearType, normalCone(Ideal,RingElement), associatedGradedRing, specialFiberIdeal.

    i1 : S = QQ[x_0..x_3]

    o1 = S

    o1 : PolynomialRing

    i2 : i = monomialCurveIdeal(S,{3,7,8})

                   2    2       3      3   3      2 2   5      4   5    3
    o2 = ideal (x x  - x x , x x  - x x , x x  - x x , x  - x x , x  - x x x )
                 0 2    1 3   1 2    0 3   1 2    0 3   2    1 3   1    0 2 3

    o2 : Ideal of S

    i3 : I = reesIdeal i;

    o3 : Ideal of S[w ..w ]
                     0   4

    i4 : reesIdeal(i, Variable=>v)

                                       3      2              2      2      2
    o4 = ideal (x x v  - x v  + x v , x v  + x v  - x v , x x v  - x v  + x v ,
                 1 2 0    0 1    3 2   3 0    2 1    1 3   0 3 0    1 1    2 2
         ------------------------------------------------------------------------
          2        2            3                     3                       2 2
         x x v  - x v  + x v , x v  + x x v  - x v , x v  - x x v  + x v , x x v
          0 3 0    1 2    2 4   2 0    1 3 1    0 3   1 0    0 2 2    3 4   2 3 0
         ------------------------------------------------------------------------
              2                2      2             2      2 2  2
         - x v  + v v , x x x v  - x v  + v v , (x x x  + x x )v  - x x v v  +
            1 1    2 3   0 1 3 0    2 2    1 4    0 2 3    1 3  0    1 2 1 2
         ------------------------------------------------------------------------
         v v )
          3 4

    o4 : Ideal of S[v ..v ]
                     0   4

    i5 : I=reesIdeal(i,i_0);

    o5 : Ideal of S[w ..w ]
                     0   4

    i6 : (J=symmetricKernel gens i);

    o6 : Ideal of S[w ..w ]
                     0   4

    i7 : isLinearType(i,i_0)

    o7 = false

    i8 : isLinearType i

    o8 = false

    i9 : reesAlgebra (i,i_0)

                                                                                                               S[w ..w ]
                                                                                                                  0   4
    o9 = ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                 3      2              2      2      2     2        2            3                     3                       2 2      2                2      2             2      2 2  2
         (x x w  - x w  + x w , x w  + x w  - x w , x x w  - x w  + x w , x x w  - x w  + x w , x w  + x x w  - x w , x w  - x x w  + x w , x x w  - x w  + w w , x x x w  - x w  + w w , (x x x  + x x )w  - x x w w  + w w )
           1 2 0    0 1    3 2   3 0    2 1    1 3   0 3 0    1 1    2 2   0 3 0    1 2    2 4   2 0    1 3 1    0 3   1 0    0 2 2    3 4   2 3 0    1 1    2 3   0 1 3 0    2 2    1 4    0 2 3    1 3  0    1 2 1 2    3 4

    o9 : QuotientRing

    i10 : trim ideal normalCone (i, i_0)

                    2    2       3      3   3      2 2   5      4   5    3
    o10 = ideal (x x  - x x , x x  - x x , x x  - x x , x  - x x , x  - x x x )
                  0 2    1 3   1 2    0 3   1 2    0 3   2    1 3   1    0 2 3

                                                                                                                         S[w ..w ]
                                                                                                                            0   4
    o10 : Ideal of ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                           3      2              2      2      2     2        2            3                     3                       2 2      2                2      2             2      2 2  2
                   (x x w  - x w  + x w , x w  + x w  - x w , x x w  - x w  + x w , x x w  - x w  + x w , x w  + x x w  - x w , x w  - x x w  + x w , x x w  - x w  + w w , x x x w  - x w  + w w , (x x x  + x x )w  - x x w w  + w w )
                     1 2 0    0 1    3 2   3 0    2 1    1 3   0 3 0    1 1    2 2   0 3 0    1 2    2 4   2 0    1 3 1    0 3   1 0    0 2 2    3 4   2 3 0    1 1    2 3   0 1 3 0    2 2    1 4    0 2 3    1 3  0    1 2 1 2    3 4

    i11 : trim ideal associatedGradedRing (i,i_0)

                    2    2       3      3   3      2 2   5      4   5    3
    o11 = ideal (x x  - x x , x x  - x x , x x  - x x , x  - x x , x  - x x x )
                  0 2    1 3   1 2    0 3   1 2    0 3   2    1 3   1    0 2 3

                                                                                                                         S[w ..w ]
                                                                                                                            0   4
    o11 : Ideal of ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                           3      2              2      2      2     2        2            3                     3                       2 2      2                2      2             2      2 2  2
                   (x x w  - x w  + x w , x w  + x w  - x w , x x w  - x w  + x w , x x w  - x w  + x w , x w  + x x w  - x w , x w  - x x w  + x w , x x w  - x w  + w w , x x x w  - x w  + w w , (x x x  + x x )w  - x x w w  + w w )
                     1 2 0    0 1    3 2   3 0    2 1    1 3   0 3 0    1 1    2 2   0 3 0    1 2    2 4   2 0    1 3 1    0 3   1 0    0 2 2    3 4   2 3 0    1 1    2 3   0 1 3 0    2 2    1 4    0 2 3    1 3  0    1 2 1 2    3 4

    i12 : trim specialFiberIdeal (i,i_0)

    o12 = ideal (w w , w w , w w )
                  3 4   1 4   2 3

    o12 : Ideal of QQ[w ..w ]
                       0   4

## See also

- reesIdeal -- Compute the defining ideal of the Rees Algebra
- symmetricKernel -- Compute the Rees ring of the image of a matrix

## Ways to use reesAlgebra:

- `reesAlgebra(Ideal)`
- `reesAlgebra(Ideal,RingElement)`
- `reesAlgebra(Module)`
- `reesAlgebra(Module,RingElement)`

## For the programmer

The object reesAlgebra is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/ReesAlgebra.m2:1160:0.