  In this paper we study various equivalent conditions for tight closure to
commute with localization. If N is a submodule of a finitely generated module M
over a Noetherian commutative ring of characteristic p, then a test exponent
for c,N,M is defined to be a power q' of p such that u is in the tight closure
of N in M whenever cu^q is in the qth Frobenius power of N for some q \ge q'.
We prove that that a test exponent for a locally stable test element c and for
N,M as above exists if and only if the tight closure of N in M commutes with
localization. Other equivalent conditions are given for tight closure to
commute with localization.
