ModulFunktor

ModulFunktor is a term used in category theory and algebra to denote a functor between categories of modules, typically between Mod-R and Mod-S for rings R and S. Such a functor F assigns to each left R-module M an S-module F(M) and to each R-linear map f: M → N a corresponding S-linear map F(f): F(M) → F(N), preserving composition and identities. The construction expresses how module structures and homomorphisms behave under changes of scalars or other module-construction processes.

Canonical examples include extension of scalars along a ring homomorphism φ: R → S, yielding a functor φ_! : Mod-R → Mod-S with φ_!(M) ≅ S ⊗_R M; restriction of scalars along φ, φ^* : Mod-S → Mod-R; and the tensor product with a fixed bimodule N, giving a functor - ⊗_R N : Mod-R → Mod-S. The Hom functor, Hom_R(-, N), is a contravariant ModulFunktor Mod-R → Ab, and when N carries an S-module structure, it can target Mod-S.

Important properties: A ModulFunktor can be additive, preserving zero objects and finite direct sums. It may be exact, left-exact, or right-exact, depending on how it preserves exact sequences. Many ModulFunktoren admit adjoints; for example, the extension-of-scalars functor is left adjoint to restriction of scalars under suitable hypotheses.

Applications: ModulFunktoren are central in representation theory, algebraic geometry, and homological algebra, where they transfer module structures across rings, analyze change of rings, and define derived functors such as Tor and Ext. They also organize constructions like pushforwards and pullbacks of sheaves in a ringed space setting.

Notes: The term is more common in German-language texts; in English literature such functors are usually called module functors or described by the specific construction (tensor, Hom, extension of scalars, etc.).