tensderived

Tensderived is a term found in discussions of homological algebra and related areas to denote the derived tensor product operation. It is not universally standardized as a separate formal label; in most contexts tensderived is understood as the left derived functor of the tensor product, commonly written as ⊗^L_R.

Definition and scope: For a ring R and chain complexes A and B of R-modules, tensderived A ⊗^L_R B refers to the homotopy-invariant tensor product obtained by replacing A with a projective (or flat) resolution P• and forming P• ⊗_R B. The resulting complex encodes the Tor groups as its homology: H_n(A ⊗^L_R B) ≅ Tor_n^R(A,B). In the language of derived categories, this object is the derived tensor product L⊗^L_R B.

Construction: The standard construction uses projective resolutions, flat resolutions, or, more generally, a cofibrant replacement in a suitable model category of chain complexes. Different equivalent approaches yield isomorphic objects in the derived category, ensuring a well-defined tensderived operation up to isomorphism.

Properties and applications: Tensderived is bifunctorial in both arguments and behaves well with respect to quasi-isomorphisms. It gives rise to spectral sequences, such as those computing Tor from the homology of the factors, and it plays a central role in base-change theorems, deformation theory, and derived algebraic geometry. It also underpins computational tools for invariants that arise from tensoring complexes.

Terminology notes: Some authors use tensderived informally as a shorthand for the derived tensor product, while others reserve tensderived for specific presentations (for example, via particular resolutions). The prevailing notation in most texts is ⊗^L or L⊗.