duality

Contents

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).

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

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

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

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

• H. Fausk, P. Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (TAC, pdf)