| In studying closure operators on ideals or submodules, tools developed in studying one closure often translate into useful tools for other closures. Our main focus is tight closure theory. First we introduce Frobenius, tight, plus, and integral closures. Then we enter a lengthy exploration of tight closure analogues of depth, associated primes, regular sequences, exact sequences, and long exact sequences associated to short exact sequences of complexes. We show that the analogues with the classical notions are stronger than previously thought, and in some cases we give new analogues. Next, we give tight closure analogues of the integral closure notions of reductions and analytic spread, using the tool of “special tight closure”. Then we analogize this tool by giving axioms for what the “special part” of a closure should satisfy, we give examples of such special parts for Frobenius and integral closures, and we give several applications, including general circumstances under which a closure yields an analogue of analytic spread. |
