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In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory.

The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent. Then according to a general principle, Grothendieck's relative point of view, the theory of Jean-Pierre Serre was extended to a proper morphism; Serre duality was recovered as the case of the morphism of a non-singular projective variety (or complete variety) to a point. The resulting theory is now sometimes called Serre–Grothendieck–Verdier duality, and is a basic tool in algebraic geometry. A treatment of this theory, Residues and Duality (1966) by Robin Hartshorne, became an accessible reference. One concrete spin-off was the Grothendieck residue.

To go beyond proper morphisms, as for the versions of Poincaré duality that are not for closed manifolds, requires some version of the compact support concept. This was addressed in SGA2 in terms of local cohomology, and Grothendieck local duality; and subsequently. The Greenlees–May duality, first formulated in 1976 by Ralf Strebel and in 1978 by Eben Matlis, is part of the continuing consideration of this area.

Adjoint functor point of view

Image functors for sheaves
direct image f∗
inverse image f∗
direct image with compact support f!
exceptional inverse image Rf!
f∗⇆f∗{displaystyle f^{*}leftrightarrows f_{*}}
(R)f!⇆(R)f!{displaystyle (R)f_{!}leftrightarrows (R)f^{!}}
Base change theorems
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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 the Gorenstein ring condition.) In a characteristic turn, Grothendieck reformulated general coherent duality as the existence of a right adjoint functor f^!, called twisted or exceptional inverse image functor, to a higher direct image with compact support functor 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). In case f is proper Rf_! = Rf_∗ is itself a right adjoint, to the inverse image functor f^∗. 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

Rf_!f^! → id,

which is denoted by Tr_f (Hartshorne) or ∫_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, f^! exists as an exact functor from a derived category of quasi-coherent sheaves on Y, to the analogous category on X, whenever

f: X → 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 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 f:X→Y{displaystyle f:Xrightarrow Y} of noetherian schemes of finite dimension, found in Hartshorne (Residues and duality) is the following quasi-isomorphism

Rf∗RHomX(F⋅,f!G⋅)→RHomY(Rf∗F⋅,G⋅){displaystyle Rf_{*}RHom_{X}(F^{cdot },f^{!}G^{cdot })to RHom_{Y}(Rf_{*}F^{cdot },G^{cdot })}

for F^⋅ a bounded above complex of O_X-modules with quasi-coherent cohomology and G^⋅ a bounded below complex of O_Y-modules with coherent cohomology. Here the Hom's are the sheaf of homomorphisms.

Construction of the f!{displaystyle f^{!}} pseudofunctor using rigid dualizing complexes

Over the years, several approaches for constructing the f!{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 k be a commutative ring, and let A be a commutative k-algebra. There is a functor RHomA⊗kLA(A,M⊗kLM){displaystyle RHom_{Aotimes _{k}^{L}A}(A,Motimes _{k}^{L}M)} which takes a cochain complex M to an object RHomA⊗kLA(A,M⊗kLM){displaystyle RHom_{Aotimes _{k}^{L}A}(A,Motimes _{k}^{L}M)} in the derived category over A.^[3][4]

Asumming A is noetherian, a rigid dualizing complex over A relative to k is by definition a pair (R,ρ){displaystyle (R,rho )} where R is a dualizing complex over A which has finite flat dimension over k, and where
ρ:R→RHomA⊗kLA(A,R⊗kLR){displaystyle rho :Rto RHom_{Aotimes _{k}^{L}A}(A,Rotimes _{k}^{L}R)} is an isomorphism in the derived category D(A). If such a rigid dualizing complex exists, then it is unique in a strong sense.^[5]

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

If X is a scheme of finite type over k, one can glue the rigid dualizing complexes that its affine pieces have,^[8][9] and obtain a rigid dualizing complex RX{displaystyle R_{X}}. Once one establishes a global existence of a rigid dualizing complex, given a map f:X→Y{displaystyle f:Xto Y} of schemes over k, one can define f!:=DX∘Lf∗∘DY{displaystyle f^{!}:=D_{X}circ Lf^{*}circ D_{Y}}, where for a scheme X, we set DX:=RHomOX(−,RX){displaystyle D_{X}:=RHom_{{mathcal {O}}_{X}}(-,R_{X})}.

Dualizing Complex Examples

Dualizing Complex for a Projective Variety

The dualizing complex for a projective variety X⊂Pn{displaystyle Xsubset mathbb {P} ^{n}} is given by the complex

ωX∙=RHomPn(OX,ωPn[+n]){displaystyle omega _{X}^{bullet }=mathrm {RHom} _{mathbb {P} ^{n}}({mathcal {O}}_{X},omega _{mathbb {P} ^{n}}[+n])}

^[10]

Plane Intersecting a Line

Consider the projective variety

X=Proj(C[x,y,z,w](x)(y,z))=Proj(C[x,y,z,w](xy,xz)){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 RHomP3(OX,ωP3[+3]){displaystyle mathrm {RHom} _{mathbb {P} ^{3}}({mathcal {O}}_{X},omega _{mathbb {P} ^{3}}[+3])} using a resolution L∙→OX{displaystyle {mathcal {L}}^{bullet }to {mathcal {O}}_{X}} by locally free sheaves. This is given by the complex

0→O(−3)→[z−y]O(−2)⊕O(−2)→[xyxz]O→OX→0{displaystyle 0to {mathcal {O}}(-3){xrightarrow {begin{bmatrix}z\-yend{bmatrix}}}{mathcal {O}}(-2)oplus {mathcal {O}}(-2){xrightarrow {begin{bmatrix}xy&xzend{bmatrix}}}{mathcal {O}}to {mathcal {O}}_{X}to 0}

Since ωP3≅O(−4){displaystyle omega _{mathbb {P} ^{3}}cong {mathcal {O}}(-4)} we have that

ωX∙=RHomP3(L∙,O(−4)[+3])=RHomP3(L∙⊗O(4)[−3],O){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

[O(−4)→[xyxz]O(−2)⊕O(−2)→[z−y]O(−1)][−3]{displaystyle [{mathcal {O}}(-4){xrightarrow {begin{bmatrix}xy\xzend{bmatrix}}}{mathcal {O}}(-2)oplus {mathcal {O}}(-2){xrightarrow {begin{bmatrix}z&-yend{bmatrix}}}{mathcal {O}}(-1)][-3]}

See also

• Verdier duality

Notes

References

• 
  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
  


• 
  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)