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?<\/p>\n
In a paper<\/a> 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<\/a>) together with a given augmentation \u03b5 : A —> R\/I such that<\/p>\n 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.<\/p>\n 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<\/a> by Gulliksen and Levin it is checked that the dga which Tate gets is endowed with divided powers.<\/p>\n 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<\/a>.<\/p>\n Namely, let B be a dga with divided powes. It turns out that the only obstruction to defining \u03b3_n on H_*(B) is that it may happen that y \u2208 B of even degree is a coboundary but \u03b3_n(y) isn’t.\u00a0 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 \u03b3_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’\u00a0 \u2208 A, then y’ = d(x’) is a coboundary in A, hence \u03b3_n(y’) is a cocycle in A by the compatibility of divided powers with d, hence \u03b3_n(y’) = d(x”) as H_i(A) = 0, hence \u03b3_n(y) = d(φ(x”)).<\/p>\n 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!<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading \n