Cotangentkomplekset

Cotangentkomplekset, known in English as the cotangent complex, is a fundamental object in algebraic geometry and related fields. It is an object in the derived category of sheaves (or modules) on a scheme or ringed space that generalizes the classical sheaf of differentials and encodes higher infinitesimal information about a morphism of spaces.

For a morphism f: X → S of schemes, the cotangent complex L_{X/S} is defined in the derived category of O_X-modules. Its zeroth cohomology H^0(L_{X/S}) is the classical sheaf of differentials Ω^1_{X/S}, while the higher and lower cohomology groups capture higher-order and obstruction data. In general, L_{X/S} may have nontrivial components in negative degrees, reflecting the derived structure needed to describe deformations accurately. When the morphism is smooth, L_{X/S} is quasi-isomorphic to Ω^1_{X/S} concentrated in degree 0; for étale morphisms, it is quasi-isomorphic to zero.

A key structural feature of the cotangent complex is the transitivity triangle. If X → Y → S are morphisms, there is a distinguished triangle in the derived category relating L_{X/S}, L_{Y/S}, and L_{X/Y}, expressing how infinitesimal behavior composes along a chain of morphisms:

L_{X/S} → L_{X/Y} → f^*L_{Y/S}[1].

Applications of the cotangent complex are central to deformation theory: Ext groups of L_{X/S} with coefficients in a module govern deformations and obstructions of X over S and of morphisms. It also serves as a foundation for derived algebraic geometry, where it plays a universal role in describing infinitesimal and derived structures.

See also deformations, Illusie, and derived categories for related background.