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ResLengthThree :: resLengthThreeAlg

resLengthThreeAlg -- the minimal free resolution presented as a graded-commutative ring

Synopsis

Description

For a free resolution F over a ring Q, the function returns the resolution F as a quotient of a graded-commutative free algebra over Q. The basis vectors in degrees 1, 2, and 3 are named with the symbols from the list L. The default symbols are e, f, and g.

i1 : Q = QQ[x,y,z];
i2 : A = resLengthThreeAlg res ideal (x^2,y^2,z^2)

o2 = A

o2 : QuotientRing
i3 : describe A

                     Q[e ..f , g ]
                        1   3   1
o3 = --------------------------------------------
     (e e  - f , e e  - f , e e  - f , e f  - g )
       2 3    3   1 3    2   1 2    1   1 3    1
i4 : e_1*e_2

o4 = f
      1

o4 : A
i5 : e_1*f_2

o5 = 0

o5 : A
i6 : e_1*f_3

o6 = g
      1

o6 : A
i7 : f_1*f_2

o7 = 0

o7 : A

The ambient ring Q does not need to be a polynomial algebra.

i8 : P = QQ[u,v,x,y,z];
i9 : Q = P/ideal(u^2,u*v);
i10 : F = resLengthThreeAlg ( res ideal (x^2,x*y,y^2,z^2), {a,b,c} )

o10 = F

o10 : QuotientRing
i11 : describe F

                                                                     Q[a ..a , b ..b , c ..c ]
                                                                        1   4   1   5   1   2
o11 = -------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                 2         2
      (a a  - b , a a  - b , a a  - b , a a  - y*b , a a  - x*b  - y*b , a a  - x*b , a b  + c , a b , a b , a b , a b  + c , a b , a b , a b , b , b b , b )
        3 4    3   2 4    4   1 4    5   2 3      1   1 3      1      2   1 2      2   4 2    2   3 2   2 2   1 2   4 1    1   3 1   2 1   1 1   2   1 2   1

i12 : P = QQ[u,v];
i13 : Q = (P/ideal(u^2,u*v))[x,y,z];
i14 : A = resLengthThreeAlg res ideal (x^2,x*y,y^2,z^2)

o14 = A

o14 : QuotientRing
i15 : describe A

                                                                     Q[e ..e , f ..f , g ..g ]
                                                                        1   4   1   5   1   2
o15 = -------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                 2         2
      (e e  - f , e e  - f , e e  - f , e e  - y*f , e e  - y*f  - x*f , e e  - x*f , e f  - g , e f , e f , e f , e f  - g , e f , e f , e f , f , f f , f )
        3 4    5   2 4    4   1 4    3   2 3      2   1 3      1      2   1 2      1   4 2    2   3 2   2 2   1 2   4 1    1   3 1   2 1   1 1   2   1 2   1

i16 : P = ZZ[x,y,z];
i17 : Q = P/ideal(4_P);
i18 : A = resLengthThreeAlg res ideal (x^2,y^2,z^2)

o18 = A

o18 : QuotientRing
i19 : describe A

                                                                                    Q[e ..f , g ]
                                                                                       1   3   1
o19 = ------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                2               2         2
      (e e  - f , e e  - f , e e  - f , e f , e f  + g , e f , e f , e f , e f  - g , e f  - g , e f , e f , e g , e g , e g , f , f f , f f , f , f f , f , f g , f g , f g )
        2 3    2   1 3    3   1 2    1   3 3   2 3    1   1 3   3 2   2 2   1 2    1   3 1    1   2 1   1 1   3 1   2 1   1 1   3   2 3   1 3   2   1 2   1   3 1   2 1   1 1

Caveat

The ambient ring Q must be homogeneous.

Ways to use resLengthThreeAlg :

For the programmer

The object resLengthThreeAlg is a method function.