PHANTOM HOMOLOGY 9 In a later paper [HH5] we will return to the problem of the existence of test elements and in particular show that any reduced finitely generated algebra over an excellent local ring has a completely stable test element. Finally we recall one more definition from [HH4]: (1.4) DEFINITION. A Noetherian ring of characteristic p is called weakly F-regular if every ideal is tightly closed. If every localization of R at a multiplicative system is weakly F-regular, we say that R is F-regular. We recall from [HH4] that a weakly F-regular ring is normal and that if it is a homomorphic image of a C-M ring then it is itself C-M. In §8 of [HH4] we also show that every J?-submodule of every finitely generated .R-module is tightly closed iff R is weakly F-regular. In general, we do not know whether tight closure commutes with localization, nor even whether weakly F-regular implies F-regular. A C K N O W L E D G M E N T We would like to thank the University of Stockholm, where parts of this research were done, for its hospitality. 2. Minheight and the weak Cohen-Macaulay property, Our objective in this section and the next is to develop parallels for the results of §9 and §11 of [HH5] for complexes of modules which, while finitely generated, are no longer necessarily free, and to allow our base rings to be mixed. We give results, for example, which may be considered a generalization of the Buchsbaum-Eisenbud criterion for acyclic- ity to complexes of not necessarily free modules (see (3.11) and (3.13)). However, whenever we develop an "ordinary" (as opposed to phantom) acyclicity criterion, we develop a ver- sion "with denominators". We then use the acyclicity criteria with denominators to prove results, like (3.22) below, that are related to our earlier one on phantom acyclicity for free complexes, although these new results (e.g., (3.18)) are not always expressible directly in terms phantom acyclicity.
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