nLab Grothendieck context

Idea

A Grothendieck context is a pair of two symmetric monoidal categories $(\mathcal{X}, \otimes_X, 1_{X})$, $(\mathcal{Y}, \otimes_Y, 1_Y)$ which are connected by an adjoint triple of functors such that the leftmost one is a closed monoidal functor.

This is the variant/special case of the yoga of six operations with two adjoint pairs $(f_! \dashv f^!)$ and $(f^\ast \dashv f_\ast)$ for $f_! \simeq f_\ast$.

$$f^\ast \dashv (f_\ast = f_!) \dashv f^! \;\colon\; \mathcal{X} \; \array{ \overset{f^\ast}{\longleftarrow} \\ \overset{f_\ast = f_! }{\longrightarrow} \\ \overset{f^!}{\longleftarrow} } \; \mathcal{Y} \,.$$

(The other specialization of the six operations where $f^\ast \simeq f^!$ is called the Wirthmüller context).

The existence of the (derived) right adjoint $f^!$ to $f_\ast$ is what is called Grothendieck duality.

Examples

Quasicoherent sheaves on schemes

A homomorphism of schemes $f \;\colon\; X \longrightarrow Y$ induces an inverse image $\dashv$ direct image adjunction on the derived categories $QCoh(-)$ of quasicoherent sheaves

$$(f^\ast \dashv f_\ast) \;\colon\; QCoh(X) \underoverset \overset{f^\ast}{\longleftarrow} \overset{f_\ast}{\longrightarrow} {\bot} QCoh(Y) \,.$$

(all derived functors) If $f$ is a proper morphism of schemes then under mild further conditions there is a further right adjoint $f^!$

$$(f^\ast \dashv f_\ast \dashv f^!) \;\colon\; QCoh(X) \; \array{ \overset{f^\ast}{\longleftarrow} \\ \overset{f_\ast}{\longrightarrow} \\ \overset{f^!}{\longleftarrow} } \; QCoh(Y) \,.$$

This is originally due to Grothendieck, whence the name. Refined accounts are in (Deligne 66, Verdier 68, Neeman 96).

Quasicoherent sheaves in $E_\infty$-geometry

Generalization of the pull-push adjoint triple to E-∞ geometry is in (LurieQC, prop. 2.5.12) and the projection formula for this is in (LurieProp, remark 1.3.14).

Related concepts

• Wirthmüller context

• Verdier-Grothendieck context

References

The original construction for quasicoherent sheaves on schemes is due to Alexander Grothendieck, whence the name “Grothendieck context”.

Further stream-lined accounts then appeared in

• Pierre Deligne, Cohomology à support propre en construction du foncteur $f^!$, Appendix to: Residues and Duality, Lecture Notes in Math., vol. 20, Springer-Verlag, Heidelberg, 1966, pp. 404{421. MR 36:5145

• Jean-Louis Verdier, Base change for twisted inverse images of coherent sheaves, Collection: Algebraic Geometry (Internat. Colloq.), Tata Inst. Fund. Res., Bombay, 1968, pp. 393-408. MR 43:227

Further refinement and highlighting of the close relation to the categorical Brown representability theorem is in

• Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205-236 (web)

Discussion of integral transforms in Grothendieck contexts is in

• Alexander Polishchuk, Kernel algebras and generalized Fourier-Mukai transforms (arXiv:0810.1542)

Generalization of the pull-push adjoint triple to E-∞ geometry is in

• Jacob Lurie, section 2.5 of Quasi-Coherent Sheaves and Tannaka Duality Theorems

and the projection formula for this triple appears as remark 1.3.14 of

• Jacob Lurie, Proper Morphisms, Completions, and the Grothendieck Existence Theorem

A clear discussion of axioms of six operations, their specialization to Grothendieck context and Wirthmüller context and their consequences is in

• Halvard Fausk, Po Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories, 11 4 (2003) 107-131 [tac:11-04, pdf

• Paul Balmer, Ivo Dell'Ambrogio, Beren Sanders, Grothendieck-Neeman duality and the Wirthmüller isomorphism, arXiv:1501.01999.