Acyclicity of Tate Constructions

We prove that a Tate construction A〈u1, . . . , un | ∂(ui) = zi〉 over a DG algebra A, on cycles z1, . . . , zn in A> 1, is acyclic if and only the map of graded-commutative algebras H0(A)[y1, . . . , yn] → H(A), with yi 7→ cls(zi), is an isomorphism. This is used to establish that if a large homomorphism R → S has an acyclic closure R〈U〉 with sup{i | Ui 6= ∅} = s < ∞, then s is either 1 or an even integer. Let A be a Differential Graded algebra (henceforth abbreviated to DG algebra), and let w1, . . . , wn be classes in H(A). Choose cycles z1, . . . , zn in A with cls(zi) = wi for each i, and consider the DG algebra A〈U〉 = A〈u1, . . . , un | ∂(ui) = zi〉; the Tate construction on A over the cycles w1, . . . wn, cf. [1, (6.1)]. Our main result, Theorem 2.3, asserts when |wi| ≥ 1 for each i, the Tate construction A〈U〉 is acyclic if and only if the canonical map of graded-commutative algebras: H0(A)[y1, . . . , yn] → H(A) where yi 7→ wi , is an isomorphism; here H0(A)[y1, . . . , yn] denotes the free graded-commutative algebra over H0(A) on the graded set {y1, . . . , yn}, cf. 1.1. Our theorem generalizes a result of Blanco, Majadas, and Rodicio [4, Theorem 1] who consider the case where A is the Koszul complex on a set of elements in a ring and each class wi has degree one. The crux of the matter is the interplay between regularity, in the sense of Tate [10], of a sequence of elements in a graded-commutative algebra and quasiregularity, a concept introduced here by extrapolating from commutative rings, which are viewed as graded-commutative algebras concentrated in degree 0. In this context, we prove that, as in the classical case, a regular sequence is quasiregular, while the converse holds under suitable separatedness assumptions. This is the content of Section 1. Section 2 combines the tools developed in Section 1 with those of Tate, to arrive at the following: Let A〈U〉 be the Tate construction over A on a classes w = w1, . . . , wn in H(A); we make no assumptions on the degrees of the elements. Then, under separatedness assumptions, the following are equivalent. (a) the sequence w1, . . . , wn is regular; (b) the sequence w1, . . . , wn is quasi-regular; (c) the canonical map H(A) → H(A〈U〉) is surjective; (d) the canonical map H(A) (w)H(A) → H(A〈U〉) is an isomorphism; Date: May 30, 2000. 1991 Mathematics Subject Classification. 13C99, 13D99.