# resLengthThreeTorAlg -- the Tor algebra presented as a graded-commutative ring

## Synopsis

• Usage:
resLengthThreeTorAlg F, resLengthThreeTorAlg(F,L)
• Inputs:
• F, , a length three free resolution of a cyclic module
• L, a list, of three symbols
• Outputs:
• , the Tor algebra presented as a quotient of a graded-commutative free algebra over the residue field of the ambient ring

## Description

For a free resolution F over a ring Q, the function returns the algebra $Tor^Q(R,k)$ as a quotient of a graded-commutative free algebra over the residue field of 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 = resLengthThreeTorAlg res ideal (x^2,y^2,z^2) o2 = A o2 : QuotientRing i3 : describe A QQ[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 : A = resLengthThreeTorAlg ( res ideal (x^2,x*y,y^2,z^2), {a,b,c} ) o10 = A o10 : QuotientRing i11 : describe A QQ[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 , a a , a a , 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 3 1 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 = resLengthThreeTorAlg ( res ideal (x^2,x*y,y^2,z^2), {a,b,c} ) o14 = A o14 : QuotientRing i15 : describe A QQ[a ..a , b ..b , c ..c ] 1 4 1 5 1 2 o15 = --------------------------------------------------------------------------------------------------------------------------- 2 2 (a a - b , a a - b , a a - b , a a , a a , a a , a b - c , a b , a b , a b , a b - c , a b , a b , a b , b , b b , b ) 3 4 5 2 4 4 1 4 3 2 3 1 3 1 2 4 2 2 3 2 2 2 1 2 4 1 1 3 1 2 1 1 1 2 1 2 1

## Caveat

For the function to return an algebra over the residue field of the ambient ring Q that ring must a homogeneous quotient of a polynomial algebra over a field.

## Ways to use resLengthThreeTorAlg :

• "resLengthThreeTorAlg(ChainComplex)"
• "resLengthThreeTorAlg(ChainComplex,List)"

## For the programmer

The object resLengthThreeTorAlg is .