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 $\lim_{e \to \infty} ({\#(\e M,M_i)}/ {(ap^d)^e})$ always exists under the assumption that $(R,\m)$ satisfies the Krull–Schmidt condition and $M$ has finite $F$-representation type by $\{ M_1, M_2, \dots, M_s \}$, in which all the $M_i$ are indecomposable $R$-modules that belong to distinct isomorphism classes and $a=[R/\m:(R/\m)^p]$.

If (V, 0) is an isolated complete intersection singularity and X a holomorphic vector field tangent to V, one can define an index of X, the so-called GSV index, which generalizes the Poincaré–Hopf index. We prove that the GSV index coincides with the dimension of a certain explicitly constructed vector space, if X is deformable in a certain sense and V is a curve. We also give a sufficient algebraic criterion for X to be deformable in this way. If one considers the real analytic case one can also define an index of X which is called the real GSV index. Under the condition that X has the deformation property, we prove a signature formula for the index generalizing the Eisenbud–Levine Theorem.