PHANTOM HOMOLOGY

This paper deals almost exclusively with Noetherian commutative rings with identity. In the discussion of tight closure, phantom homology, a nd phantom acyclicity, positive prime characteristic p is assumed. The minheight, mnht(I)R, of a finitely generated module M on an ideal I o f a Noetherian ring R is defined to be +infinity if IM = M and otherwi se to be the infimum of the integers height I(R/Q) (in R/Q) as Q runs through the minimal primes of M. When M = R, mnht(I)R is also referred to as the minheight of I. This terminology is parallel to the termino logy for ''depth''. M is called weakly Cohen-Macaulay if for every ide al I of R and every prime ideal P of R, mnht(IP) M(P) = depth(IP) M(P) . These notions are studied and the prerequisite definitions and resul ts from earlier work on tight closure are reviewed. A characterization is then obtained for when a finite free complex over a Noetherian rin g of characteristic p is stably phantom acyclic: a complex is said to have phantom homology at a certain spot if the cycles are in the tight closure of the boundaries in the ambient module of chains, to be phan tom acyclic if all the higher homology is phantom, and to be stably ph antom acyclic if this remains true as one applies the iterates of the Frobenius endomorphism. Ordinary acyclicity for a finite free complex is characterized by certain standard conditions on the ranks of the ma ps and on the depths of the largest non-vanishing ideals of minors of the matrices of the maps. The phantom acyclicity criterion asserts tha t, when R is of finite Krull dimension and a homomorphic image of a Go renstein ring, the same conditions, but with rank calculated modulo ni lpotents and with ''minheight'' replacing ''depth'', characterize stab le phantom acyclicity. The use of minheight is made necessary by the f act that there are no equidimensionality conditions on the ring. The a rgument depends on a thorough study of acyclicity criteria with demoni nators for finite complexes. Results of this sort are obtained even wh en the complex is not necessarily projective. The phrase ''with denomi nators'' may be explained as follows: in certain instances one knows t hat a complex becomes acyclic after localizing at an element a of the ring, which implies that some power of a kills the homology at each sp ot. But one often wants to know a bound for this power, and this is th e goal of the acyclicity criteria ''with denominators''. It turns out that there is a fixed power J of the ideal that defines the locus of p rimes P in Spec(R) such that R(p) is not weakly Cohen-Macaulay such th at J kills all homology of any finite free stably phantom acyclic comp lex. This yields a much richer supply of ''test elements'' than is ava ilable ordinarily for tight closure tests. These results are applied t o give new insight into the local homological conjectures, which appea r in a greatly improved and strengthened form, and seem more natural f rom the perspective of tight closure theory. New results axe obtained, including some very powerful vanishing theorems for maps of Tor. Thes e theorems, even when greatly specialized, suffice to prove that direc t summands of regular rings are Cohen-Macaulay. A generalization of th e new intersection theorem called the phantom intersection theorem is developed, and a new proof is given for a characteristic p result of P aul Roberts that is crucial in his proof of the new intersection theor em in mixed characteristic. The notion of the regular closure of an id eal or submodule is also developed: when tight closure is defined, the tight closure is contained in the regular closure (usually strictly), and this permits parallels of the characteristic p results of this pa per to be stated in characteristic zero. In the equal characteristic c ase these parallels can frequently be proved by reduction to character istic p: this program will be carried through in a later paper. In mix ed characteristic many of the parallels axe open questions.