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

- Function: torAlgebra
- Usage:
    `TorRS = torAlgebra(R,S,GenDegreeLimit=>m,RelDegreeLimit=>n)`
- Inputs:
  - R, a ring,
  - S, a ring,
- Optional inputs:
  - GenDegreeLimit => ..., default value infinity, Option to specify the maximum degree to look for generators
  - RelDegreeLimit => ..., default value infinity, 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)
     -- used 0.495956s (cpu); 0.450797s (thread); 0s (gc)
    Finding easy relations           :
    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 = freeResolution(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 : Complex

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.

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/DGAlgebras.m2:2265:0.