torAlgebra(Ring,Ring) -- Computes Tor_R(S,k) up to a specified generating and relating degree.

Synopsis

Usage:
TorRS = torAlgebra(R,S,GenDegreeLimit=>m,RelDegreeLimit=>n)
Function: torAlgebra
Inputs:
- R, a ring
- S, a ring
Optional inputs:
- GenDegreeLimit => ..., -- Option to specify the maximum degree to look for generators
- RelDegreeLimit => ..., -- Option to specify the maximum degree to look for relations
Outputs:
- TorRS, a ring

Description

i1 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4}

o1 = R

o1 : QuotientRing

i2 : M = coker matrix {{a^3*b^3*c^3*d^3}};

i3 : S = R/ideal{a^3*b^3*c^3*d^3}

o3 = S

o3 : QuotientRing

i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
Computing generators in degree 1 :      -- used 0.0195185 seconds
Computing generators in degree 2 :      -- used 0.0344045 seconds
Computing generators in degree 3 :      -- used 0.0620442 seconds
Computing generators in degree 4 :      -- used 0.0581557 seconds
Finding easy relations           :      -- used 1.22445 seconds
Computing relations in degree 1  :      -- used 0.0700999 seconds
Computing relations in degree 2  :      -- used 0.0728236 seconds
Computing relations in degree 3  :      -- used 0.238993 seconds
Computing relations in degree 4  :      -- used 0.248717 seconds
Computing relations in degree 5  :      -- used 0.690785 seconds
Computing relations in degree 6  :      -- used 0.891095 seconds
Computing relations in degree 7  :      -- used 1.21839 seconds
Computing relations in degree 8  :      -- used 1.673 seconds

o4 = HB

o4 : QuotientRing

i5 : numgens HB

o5 = 35

i6 : apply(5,i -> #(flatten entries getBasis(i,HB)))

o6 = {1, 1, 4, 10, 20}

o6 : List

i7 : Mres = res(M, LengthLimit=>8)

      1      1      4      10      20      35      56      84      120
o7 = R  <-- R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R   <-- R
                                                                   
     0      1      2      3       4       5       6       7       8

o7 : ChainComplex

Note that in this example, Tor_*^R(S,k) has trivial multiplication, since the map from R to S is a Golod homomorphism by a theorem of Levin and Avramov.