Finite generation of Hochschild homology algebras

André–Quillen homology of algebra retracts

“…The proof draws on results obtained above and on earlier results from [13]. One of them is for a homomorphism ϕ : (R, m, k) → (S, n, k) of local rings that is large in the sense of Levin [23], meaning that the map Tor Thus, [13, (3.2)] shows that if Tor R • (S, S) is finitely generated, then AQ-dim S R is finite, and [21, (3.1)] proves that in this case AQ-dim S R is equal to 1 or is even.…”

“…If S is a flat algebra over some ring A, then Tor S⊗AS • (S, S) is isomorphic to the Hochschild homology algebra HH • (S|A) of S over A. Our main result in [13] shows that if the ring R = S ⊗ A S is noetherian, and HH • (S|A) is finitely generated as an algebra over S, then S is regular over A. On the other hand, by the Hochschild-Kostant-Rosenberg Theorem [20], as generalized by André [3], if S is regular over A then HH • (S|A) ∼ = S D 1 .…”

“…We use Theorem I together with our results in [13] in a situation that does not a priori involve André-Quillen homology-the classical homology of an algebra retract S → R → S. In that case Tor R 0 (S, S) = S and Tor R • (S, S) is a gradedcommutative algebra with divided powers, but precise information on its structure is available in two instances only: when S is a field, cf. [22], [19], or when R → S is locally complete intersection.…”

“…To obtain converses, we assume that D n (S |R; S) is free for n ≤ 2, and D 3 (S |R; S) = 0. From (13) and (10) we get…”

André-Quillen homology of commutative algebras

“…André-Quillen homology does not appear in the statement of Theorem 9.9. This situation is typical: André-Quillen theory provides streamlined proofs of many results concerning Hochschild homology, and is sometimes indispensable, see [9]. There is a mathematical reason for this, see [22, (8.1)].…”