# Six operations

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Formalism in homological algebra

In [mathematics](/source/Mathematics), **Grothendieck's six operations**, named after [Alexander Grothendieck](/source/Alexander_Grothendieck), is a formalism in [homological algebra](/source/Homological_algebra), also known as the **six-functor formalism**.[1] It originally sprang from the relations in [étale cohomology](/source/%C3%89tale_cohomology) that arise from a [morphism](/source/Morphism_of_schemes) of [schemes](/source/Scheme_(mathematics)) *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](/source/D-module) on [algebraic varieties](/source/Algebraic_varieties), [sheaves](/source/Sheaf_(mathematics)) on [locally compact topological spaces](/source/Locally_compact_topological_space), and [motives](/source/Motive_(mathematics)).

## The operations

The operations are six [functors](/source/Functor). Usually these are functors between [derived categories](/source/Derived_categories) and so are actually left and right [derived functors](/source/Derived_functor).

- the [direct image](/source/Direct_image_functor) f ∗ {\displaystyle f_{*}}

- the [inverse image](/source/Inverse_image_functor) f ∗ {\displaystyle f^{*}}

- the [proper (or extraordinary) direct image](/source/Direct_image_with_compact_support) f ! {\displaystyle f_{!}}

- the [proper (or extraordinary) inverse image](/source/Exceptional_inverse_image_functor) f ! {\displaystyle f^{!}}

- internal [tensor product](/source/Tensor_product)

- [internal Hom](/source/Internal_Hom)

The functors f ∗ {\displaystyle f^{*}} and f ∗ {\displaystyle f_{*}} form an [adjoint functor](/source/Adjoint_functor) pair, as do f ! {\displaystyle f_{!}} and f ! {\displaystyle f^{!}} .[2] Similarly, internal tensor product is left adjoint to internal Hom.

## Six operations in étale cohomology

Let *f* : *X* → *Y* be a morphism of schemes. The morphism *f* induces several functors. Specifically, it gives [adjoint functors](/source/Adjoint_functors) f ∗ {\displaystyle f^{*}} and f ∗ {\displaystyle f_{*}} between the categories of sheaves on *X* and *Y*, and it gives the functor f ! {\displaystyle f_{!}} of direct image with proper support. In the [derived category](/source/Derived_category), *Rf*! admits a right adjoint f ! {\displaystyle f^{!}} . Finally, when working with abelian sheaves, there is a tensor product functor ⊗ and an internal Hom functor, and these are adjoint. The six operations are the corresponding functors on the derived category: *Lf**, *Rf**, *Rf*!, *f*!, ⊗*L*, and RHom.

Suppose that we restrict ourselves to a category of ℓ {\displaystyle \ell } -adic [torsion sheaves](/source/Torsion_sheaf), where ℓ {\displaystyle \ell } is coprime to the characteristic of *X* and of *Y*. In [SGA](/source/S%C3%A9minaire_de_G%C3%A9om%C3%A9trie_Alg%C3%A9brique_du_Bois_Marie) 4 III, Grothendieck and [Artin](/source/Michael_Artin) proved that if *f* is [smooth](/source/Smooth_morphism) of [relative dimension](/source/Relative_dimension_(scheme_theory)) *d*, then L f ∗ {\displaystyle Lf^{*}} is isomorphic to *f*!(−*d*)[−2*d*], where (−*d*) denotes the *d*th inverse [Tate twist](/source/Tate_twist) and [−2*d*] denotes a shift in degree by −2*d*. Furthermore, suppose that *f* is [separated](/source/Separated_morphism) and [of finite type](/source/Morphism_of_finite_type). If *g* : *Y*′ → *Y* is another morphism of schemes, if *X*′ denotes the base change of *X* by *g*, and if *f*′ and *g*′ denote the base changes of *f* and *g* by *g* and *f*, respectively, then there exist natural isomorphisms:

- L g ∗ ∘ R f ! → R f ! ′ ∘ L g ′ ∗ , {\displaystyle Lg^{*}\circ Rf_{!}\to Rf'_{!}\circ Lg'^{*},}

- R g ∗ ′ ∘ f ′ ! → f ! ∘ R g ∗ . {\displaystyle Rg'_{*}\circ f'^{!}\to f^{!}\circ Rg_{*}.}

Again assuming that *f* is separated and of finite type, for any objects *M* in the derived category of *X* and *N* in the derived category of *Y*, there exist natural isomorphisms:

- ( R f ! M ) ⊗ Y N → R f ! ( M ⊗ X L f ∗ N ) , {\displaystyle (Rf_{!}M)\otimes _{Y}N\to Rf_{!}(M\otimes _{X}Lf^{*}N),}

- RHom Y ⁡ ( R f ! M , N ) → R f ∗ RHom X ⁡ ( M , f ! N ) , {\displaystyle \operatorname {RHom} _{Y}(Rf_{!}M,N)\to Rf_{*}\operatorname {RHom} _{X}(M,f^{!}N),}

- f ! RHom Y ⁡ ( M , N ) → RHom X ⁡ ( L f ∗ M , f ! N ) . {\displaystyle f^{!}\operatorname {RHom} _{Y}(M,N)\to \operatorname {RHom} _{X}(Lf^{*}M,f^{!}N).}

If *i* is a closed immersion of *Z* into *S* with complementary open immersion *j*, then there is a distinguished triangle in the derived category:

- R j ! j ! → 1 → R i ∗ i ∗ → R j ! j ! [ 1 ] , {\displaystyle Rj_{!}j^{!}\to 1\to Ri_{*}i^{*}\to Rj_{!}j^{!}[1],}

where the first two maps are the counit and unit, respectively, of the adjunctions. If *Z* and *S* are [regular](/source/Regular_scheme), then there is an isomorphism:

- 1 Z ( − c ) [ − 2 c ] → i ! 1 S , {\displaystyle 1_{Z}(-c)[-2c]\to i^{!}1_{S},}

where 1*Z* and 1*S* are the units of the tensor product operations (which vary depending on which category of ℓ {\displaystyle \ell } -adic torsion sheaves is under consideration).

If *S* is regular and *g* : *X* → *S*, and if *K* is an invertible object in the derived category on *S* with respect to ⊗*L*, then define *D**X* to be the functor RHom(—, *g*!*K*). Then, for objects *M* and *M*′ in the derived category on *X*, the canonical maps:

- M → D X ( D X ( M ) ) , {\displaystyle M\to D_{X}(D_{X}(M)),}

- D X ( M ⊗ D X ( M ′ ) ) → RHom ⁡ ( M , M ′ ) , {\displaystyle D_{X}(M\otimes D_{X}(M'))\to \operatorname {RHom} (M,M'),}

are isomorphisms. Finally, if *f* : *X* → *Y* is a morphism of *S*-schemes, and if *M* and *N* are objects in the derived categories of *X* and *Y*, then there are natural isomorphisms:

- D X ( f ∗ N ) ≅ f ! ( D Y ( N ) ) , {\displaystyle D_{X}(f^{*}N)\cong f^{!}(D_{Y}(N)),}

- D X ( f ! N ) ≅ f ∗ ( D Y ( N ) ) , {\displaystyle D_{X}(f^{!}N)\cong f^{*}(D_{Y}(N)),}

- D Y ( f ! M ) ≅ f ∗ ( D X ( M ) ) , {\displaystyle D_{Y}(f_{!}M)\cong f_{*}(D_{X}(M)),}

- D Y ( f ∗ M ) ≅ f ! ( D X ( M ) ) . {\displaystyle D_{Y}(f_{*}M)\cong f_{!}(D_{X}(M)).}

## See also

- [Coherent duality](/source/Coherent_duality)

- [Grothendieck local duality](/source/Grothendieck_local_duality)

- [Image functors for sheaves](/source/Image_functors_for_sheaves)

- [Verdier duality](/source/Verdier_duality)

- [Change of rings](/source/Change_of_rings)

## References

1. **[^](#cite_ref-gallauer_1-0)** Gallauer, Martin (2021). ["An introduction to six-functor formalism"](https://guests.mpim-bonn.mpg.de/gallauer/docs/m6ff.pdf) (PDF).

1. **[^](#cite_ref-Fausk2003_2-0)** Fausk, H.; P. Hu; J. P. May (2003). ["Isomorphisms between left and right adjoints"](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf) (PDF). *Theory Appl. Categ.*: 107–131. [arXiv](/source/ArXiv_(identifier)):[math/0206079](https://arxiv.org/abs/math/0206079). [Bibcode](/source/Bibcode_(identifier)):[2002math......6079F](https://ui.adsabs.harvard.edu/abs/2002math......6079F). Retrieved 6 June 2013.

- [Laszlo, Yves](/source/Yves_Laszlo); Olsson, Martin (2005). "The six operations for sheaves on Artin stacks I: Finite coefficients". [arXiv](/source/ArXiv_(identifier)):[math/0512097](https://arxiv.org/abs/math/0512097).

- Ayoub, Joseph. [*Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique*](https://faculty.math.illinois.edu/K-theory/0761/THESE.pdf) (PDF) (Thesis).

- Cisinski, Denis-Charles; Déglise, Frédéric (2019). *Triangulated categories of mixed motives*. Springer Monographs in Mathematics. [arXiv](/source/ArXiv_(identifier)):[0912.2110](https://arxiv.org/abs/0912.2110). [doi](/source/Doi_(identifier)):[10.1007/978-3-030-33242-6](https://doi.org/10.1007%2F978-3-030-33242-6). [ISBN](/source/ISBN_(identifier)) [978-3-030-33241-9](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-33241-9). [S2CID](/source/S2CID_(identifier)) [115163824](https://api.semanticscholar.org/CorpusID:115163824).

- Mebkhout, Zoghman (1989). *Le formalisme des six opérations de Grothendieck pour les DX-modules cohérents*. Travaux en Cours. Vol. 35. Paris: Hermann. [ISBN](/source/ISBN_(identifier)) [2-7056-6049-6](https://en.wikipedia.org/wiki/Special:BookSources/2-7056-6049-6).

## External links

- [six operations](https://ncatlab.org/nlab/show/six+operations) at the [*n*Lab](/source/NLab)

- [What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?](https://mathoverflow.net/q/170319)

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