While Serre duality uses a line bundle or invertible sheaf as a dualizing sheaf, the general theory (it turns out) cannot be quite so simple. (More precisely, it can, but at the cost of imposing the Gorenstein ring condition.) In a characteristic turn, Grothendieck reformulated general coherent duality as the existence of a right adjoint functor ${\displaystyle f^{!}}$ , called twisted or exceptional inverse image functor, to a higher direct image with compact support functor ${\displaystyle Rf_{!}}$ .

Higher direct images are a sheafified form of sheaf cohomology in this case with proper (compact) support; they are bundled up into a single functor by means of the derived category formulation of homological algebra (introduced with this case in mind). If ${\displaystyle f}$  is proper, then ${\displaystyle Rf_{!}=Rf_{\ast }}$  is a right adjoint to the inverse image functor ${\displaystyle f^{\ast }}$ . The existence theorem for the twisted inverse image is the name given to the proof of the existence for what would be the counit for the comonad of the sought-for adjunction, namely a natural transformation

${\displaystyle Rf_{!}f^{!}\rightarrow id}$ ,

which is denoted by ${\displaystyle Tr_{f}}$  (Hartshorne) or ${\displaystyle \int _{f}}$  (Verdier). It is the aspect of the theory closest to the classical meaning, as the notation suggests, that duality is defined by integration.

To be more precise, ${\displaystyle f^{!}}$  exists as an exact functor from a derived category of quasi-coherent sheaves on ${\displaystyle Y}$ , to the analogous category on ${\displaystyle X}$ , whenever

${\displaystyle f:X\rightarrow Y}$ 

is a proper or quasi projective morphism of noetherian schemes, of finite Krull dimension.^[1] From this the rest of the theory can be derived: dualizing complexes pull back via ${\displaystyle f^{!}}$ , the Grothendieck residue symbol, the dualizing sheaf in the Cohen–Macaulay case.

In order to get a statement in more classical language, but still wider than Serre duality, Hartshorne (Algebraic Geometry) uses the Ext functor of sheaves; this is a kind of stepping stone to the derived category.

The classical statement of Grothendieck duality for a projective or proper morphism ${\displaystyle f:X\rightarrow Y}$  of noetherian schemes of finite dimension, found in Hartshorne (Residues and duality) is the following quasi-isomorphism

${\displaystyle Rf_{\ast }RHom_{X}(F^{\bullet },f^{!}G^{\bullet })\to RHom_{Y}(Rf_{\ast }F^{\bullet },G^{\bullet })}$ 

for ${\displaystyle F^{\bullet }}$  a bounded above complex of ${\displaystyle O_{X}}$ -modules with quasi-coherent cohomology and ${\displaystyle G^{\bullet }}$  a bounded below complex of ${\displaystyle O_{Y}}$ -modules with coherent cohomology. Here the ${\displaystyle Hom}$ 's are sheaves of homomorphisms.

Over the years, several approaches for constructing the ${\displaystyle f^{!}}$  pseudofunctor emerged. One quite recent successful approach is based on the notion of a rigid dualizing complex. This notion was first defined by Van den Bergh in a noncommutative context.^[2] The construction is based on a variant of derived Hochschild cohomology (Shukla cohomology): Let ${\displaystyle k}$  be a commutative ring, and let ${\displaystyle A}$  be a commutative ${\displaystyle k-}$ algebra. There is a functor ${\displaystyle RHom_{A\otimes _{k}^{L}A}(A,M\otimes _{k}^{L}M)}$  which takes a cochain complex ${\displaystyle M}$  to an object ${\displaystyle RHom_{A\otimes _{k}^{L}A}(A,M\otimes _{k}^{L}M)}$  in the derived category over ${\displaystyle A}$ .^[3][4]

Assuming ${\displaystyle A}$  is noetherian, a rigid dualizing complex over ${\displaystyle A}$  relative to ${\displaystyle k}$  is by definition a pair ${\displaystyle (R,\rho )}$  where ${\displaystyle R}$  is a dualizing complex over ${\displaystyle A}$  which has finite flat dimension over ${\displaystyle k}$ , and where ${\displaystyle \rho :R\to RHom_{A\otimes _{k}^{L}A}(A,R\otimes _{k}^{L}R)}$  is an isomorphism in the derived category ${\displaystyle D(A)}$ . If such a rigid dualizing complex exists, then it is unique in a strong sense.^[5]

Assuming ${\displaystyle A}$  is a localization of a finite type ${\displaystyle k}$ -algebra, existence of a rigid dualizing complex over ${\displaystyle A}$  relative to ${\displaystyle k}$  was first proved by Yekutieli and Zhang^[6] assuming ${\displaystyle k}$  is a regular noetherian ring of finite Krull dimension, and by Avramov, Iyengar and Lipman^[7] assuming ${\displaystyle k}$  is a Gorenstein ring of finite Krull dimension and ${\displaystyle A}$  is of finite flat dimension over ${\displaystyle k}$ .

If ${\displaystyle X}$  is a scheme of finite type over ${\displaystyle k}$ , one can glue the rigid dualizing complexes that its affine pieces have,^[8] and obtain a rigid dualizing complex ${\displaystyle R_{X}}$ . Once one establishes a global existence of a rigid dualizing complex, given a map ${\displaystyle f:X\to Y}$  of schemes over ${\displaystyle k}$ , one can define ${\displaystyle f^{!}:=D_{X}\circ Lf^{*}\circ D_{Y}}$ , where for a scheme ${\displaystyle X}$ , we set ${\displaystyle D_{X}:=RHom_{{\mathcal {O}}_{X}}(-,R_{X})}$ .

Dualizing Complex for a Projective Variety edit

The dualizing complex for a projective variety ${\displaystyle X\subset \mathbb {P} ^{n}}$  is given by the complex

${\displaystyle \omega _{X}^{\bullet }=\mathrm {RHom} _{\mathbb {P} ^{n}}({\mathcal {O}}_{X},\omega _{\mathbb {P} ^{n}}[+n])}$ 

^[9]

Plane Intersecting a Line edit

Consider the projective variety

${\displaystyle X={\text{Proj}}\left({\frac {\mathbb {C} [x,y,z,w]}{(x)(y,z)}}\right)={\text{Proj}}\left({\frac {\mathbb {C} [x,y,z,w]}{(xy,xz)}}\right)}$ 

We can compute ${\displaystyle \mathrm {RHom} _{\mathbb {P} ^{3}}({\mathcal {O}}_{X},\omega _{\mathbb {P} ^{3}}[+3])}$  using a resolution ${\displaystyle {\mathcal {L}}^{\bullet }\to {\mathcal {O}}_{X}}$  by locally free sheaves. This is given by the complex

${\displaystyle 0\to {\mathcal {O}}(-3){\xrightarrow {\begin{bmatrix}z\\-y\end{bmatrix}}}{\mathcal {O}}(-2)\oplus {\mathcal {O}}(-2){\xrightarrow {\begin{bmatrix}xy&xz\end{bmatrix}}}{\mathcal {O}}\to {\mathcal {O}}_{X}\to 0}$ 

Since ${\displaystyle \omega _{\mathbb {P} ^{3}}\cong {\mathcal {O}}(-4)}$  we have that

${\displaystyle \omega _{X}^{\bullet }=\mathrm {RHom} _{\mathbb {P} ^{3}}({\mathcal {L}}^{\bullet },{\mathcal {O}}(-4)[+3])=\mathrm {RHom} _{\mathbb {P} ^{3}}({\mathcal {L}}^{\bullet }\otimes {\mathcal {O}}(4)[-3],{\mathcal {O}})}$ 

This is the complex

${\displaystyle [{\mathcal {O}}(-4){\xrightarrow {\begin{bmatrix}xy\\xz\end{bmatrix}}}{\mathcal {O}}(-2)\oplus {\mathcal {O}}(-2){\xrightarrow {\begin{bmatrix}z&-y\end{bmatrix}}}{\mathcal {O}}(-1)][-3]}$ 

• Verdier duality

• Greenlees, J. P. C.; May, J. Peter (1992), "Derived functors of I-adic completion and local homology", Journal of Algebra, 149 (2): 438–453, doi:10.1016/0021-8693(92)90026-I, ISSN 0021-8693, MR 1172439
• Hartshorne, Robin (1966), Residues and Duality, Lecture Notes in Mathematics 20, Berlin, New York: Springer-Verlag, pp. 20–48, doi:10.1007/BFb0080482
• Neeman, Amnon (1996), "The Grothendieck duality theorem via Bousfield's techniques and Brown representability", Journal of the American Mathematical Society, 9 (1): 205–236, doi:10.1090/S0894-0347-96-00174-9, ISSN 0894-0347, MR 1308405
• Verdier, Jean-Louis (1969), "Base change for twisted inverse image of coherent sheaves", Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford University Press, pp. 393–408, MR 0274464
• Hopkins, Glenn, An Algebraic Approach to Grothendieck's Residue Symbol (PDF)