1. Introduction and background. Throughout this paper all rings are commutative, with identity, and Noetherian, unless otherwise specified. In [HH4] we introduced the notion of tight closure for ideals and then submodules, both for Noetherian rings of characteristic p and for finitely generated algebras over a field of characteristic zero (where the definition depends on reduction to characteristic p). See Definition 1.1 below, and [HH1-4]. The tight closure operation assigns to each ideal / of the ring R (respectively, to each submodule N of a finitely generated i2-module M) an ideal I* D I (respectively, a submodule N%j = iV* ~D N) in such a way that the usual properties of a closure operation are satisfied, e.g. (N*)* = N*. The case of ideals I C R is a special case of the theory for submodules of modules, but is especially important and gets special emphasis.) The idea of tight closure leads to the notion of phantom (co)homology for a complex Gm: at the ith spot, the complex has phantom homology if and only if the cycles are in the tight closure of the boundaries within Gt. In this paper we study phantom homology of complexes in characteristic p, and use the results to obtain greatly improved versions of a multitude of homological theorems in local algebra of the kind related to the existence of big Cohen-Macaulay modules. We also obtain many strong results which are entirely new. The methods used here do more than provide new theorems and strengthened versions of old theorems: they give a new insight into why theorems of this kind are true, and how to discover more of them. In a subsequent paper, [HH10], we shall develop the corresponding theory in characteristic zero. The question of whether there is an analogous theory in mixed characteristic remains open. A key point about phantom homology is that it is killed by maps to regular rings under mild hypotheses. This produces very strong vanishing theorems which appear to be unattainable by other means. One obtains, for example, a new proof that direct summands of regular rings are Cohen-Macaulay (cf. [HRl], [Ke], and [B]): the result may be deduced from a very strong vanishing theorem for maps of Tor: a non-technical version of that result 1 Both authors were partially supported by the NSF. 2 Received by the editor July 24, 1991 and in revised form September 27, 1991. 1
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