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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→∞(#(M,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].