MODULES WITH FINITE F-REPRESENTATION TYPE

Finitely generated modules with finite F-representation type over Noetherian (local) rings of prime characteristic p are studied. If a ring R has finite F-representation type or, more generally, if a faithful R-module has finite F-representation type, then tight closure commutes with localizations over R. F-contributors are also defined, and they are used as an effective way of characterizing tight closure. Then it is shown that lime→∞(#( eM,Mi)/(apd)e) always exists under the assumption that (R,m) satisfies the Krull–Schmidt condition and M has finite F-representation type by {M1,M2, . . . , Ms}, in which all the Mi are indecomposable R-modules that belong to distinct isomorphism classes and a = [R/m: (R/m)p].