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
|
The object resLengthThreeAlg is a method function.