Computing Tor

Let R be a Noetherian ring and let I, J be ideals of R. Then Tor^R_*(R/I, R/J) is a differential graded algebra (with zero differential). How does one get this algebra structure?

In a paper published in 1957, John Tate came up with the following strategy: Try to find a strictly commutative differential graded R-algebra A endowed with divided powers (as in this post) together with a given augmentation ε : A —> R/I such that

  1. H_i(A) = 0 for i > 0 and H_0(A) = R/I, and
  2. A is obtained from R by successively adjoining divided power variables.

The first condition means that A is quasi-isomorphic to R/I as a dga (with divided powers) and the second implies that A is a free resolution of R/I as an R-module. Hence we see that Tor_*(R/I, R/J) is the homology of A \otimes_R R/J = A/JA which is a dga with divided powers.

Tate shows that you can construct such dga resolutions of R/I by successively adjoining variables to kill cycles; starting with the Koszul complex for a set of generators of I. In the book by Gulliksen and Levin it is checked that the dga which Tate gets is endowed with divided powers.

I’d just like to make here the observation that this also determines divided powers on Tor^R_*(R/I, R/J). This despite the problem that in general the homology of a dga with divided powers isn’t endowed with divided powers as I mentioned here.

Namely, let B be a dga with divided powes. It turns out that the only obstruction to defining γ_n on H_*(B) is that it may happen that y ∈ B of even degree is a coboundary but γ_n(y) isn’t.  For example if B is the divided power algebra over F_2 on x in degree 1 and y in degree 2 and d(x) = y, then γ_2(y) isn’t a coboundary! But, if there exists a surjection φ : A —> B of dgas with divided powers where A is such that H_i(A) = 0 for i > 0, then this disaster doesn’t happen. The reason is that writing y = d(x) and x = φ(x’) for some x’  ∈ A, then y’ = d(x’) is a coboundary in A, hence γ_n(y’) is a cocycle in A by the compatibility of divided powers with d, hence γ_n(y’) = d(x”) as H_i(A) = 0, hence γ_n(y) = d(φ(x”)).

And of course, in the situation of Tate’s construction above, we have that A/JA is the quotient of a dga acyclic in positive degrees!