Phantom depth and stable phantom exactness

Abstract

Phantom depth, phantom nonzerodivisors, and phantom exact sequences are analogues of the non-"phantom" notions which have been useful in tackling the (very difficult) localization problem in tight closure theory. In the present paper, these notions are developed further and partially reworked. For instance, although no analogue of a long exact sequence arises from a short stably phantom exact sequence of complexes, we provide a method for recovering the kind of information obtainable from such a long sequence. Also, we give alternate characterizations of the notion of phantom depth, including one based on Koszul homology which we use to show that with very mild conditions on a finitely generated module $M$, any two maximal phantom $M$-regular sequences in an ideal $I$ have the same length. In order to do so, we prove a "Nakayama lemma for tight closure" which is of independent interest. We strengthen the connection of phantom depth with minheight, we explore several analogues of "associated prime" in tight closure theory, and we discuss a connection with the problem of when tight closure commutes with localization.