Six operations

In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra. It originally sprang from the relations in étale cohomology that arise from a morphism of schemes f : X → Y. The basic insight was that many of the elementary facts relating cohomology on X and Y were formal consequences of a small number of axioms. These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold. The six operations formalism has since been shown to apply to contexts such as D-modules on algebraic varieties, sheaves on locally compact topological spaces, and motives.

The operations are six functors. Usually these are functors between derived categories and so are actually left and right derived functors.

The functors ${\displaystyle f^{*}}$ and ${\displaystyle f_{*}}$ form an adjoint functor pair, as do ${\displaystyle f_{!}}$ and ${\displaystyle f^{!}}$. Similarly, internal tensor product is left adjoint to internal Hom.

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