# Duality

#### Duality in algebraic geometry

$\newcommand{\calO}{\mathcal{O}}$ $\newcommand{\calF}{\mathcal{F}}$ $\newcommand{\tensor}{\otimes}$ $\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\End}{End}$ $\DeclareMathOperator{\Gal}{Gal}$ Duality between the different cohomology spaces on algebraic varieties. Cohomology of coherent sheaves. Let $X$ be a non-singular algebraic variety of dimension $n$ over an algebraically closed field $k$ and let $\mathcal{L}$ be a locally free sheaf on $X$. Serre's duality theorem states that the finite-dimensional cohomology (vector) spaces $H^i(X,\mathcal{L})$ and $H^{n-i}(X,\check{\mathcal{L}}\otimes\omega_X)$ are mutually dual. Here $\omega_X=\Omega_X^n$ is the sheaf of germs of regular differential forms of degree $n$ on $X$, and $\check{\mathcal{L}}=\operatorname{Hom}(\mathcal{L},\mathcal{O}_X)$ is the locally free sheaf dual to $\mathcal{L}$. If $\mathcal{L}=\mathcal{O}_X(D)$ is the invertible sheaf corresponding to a divisor $D$ on $X$, this theorem establishes the equality

$$\dim H^i(X, \calO_X(D)) = \dim H^{n-i}(X, \calO_X(K - D)),$$ where $K$ is the canonical divisor on $X$. If $n = 1$, a relation equivalent to the above was found as early as the 19th century. There exists a generalization of Serre's theorem to the case of cohomology of arbitrary coherent sheaves on complete algebraic varieties , . In particular, if the variety $X$ is a Cohen–Macaulay subvariety (e.g. a locally complete intersection) of codimension $d$ in a non-singular projective variety $Y$, there is duality between the $k$-space $H^i(X, \calF)$ and the space of global Ext's

$$\Ext^{n-i}(X; \calF, \widetilde \omega_X),$$ where $\calF$ is a coherent sheaf on $X$, $\widetilde \omega_X = \Ext_{\calO_Y}^d (\calO_X, \omega_Y)$ (Grothendieck's dualizing sheaf), while $n = \dim X$. Here, the sheaf $\widetilde \omega_X$ is invertible if and only if $X$ is a Gorenstein scheme (cf. Gorenstein ring).

Etale cohomology. Let $X$ be a complete connected non-singular algebraic variety of dimension $d$ over an algebraically closed field $k$; let $n$ be an integer which is relatively prime to the characteristic of the field $k$; let $\calF$ be a locally free (in the étale topology) sheaf of $\ZZ/n\ZZ$-modules on $X$; and let $\mu_n$ be the sheaf of $n$-th power roots of unity. Then there exists a non-degenerate pairing of $\ZZ/n\ZZ$-modules :

$$H^i(X, \calF) \times H^{2d - i}(X, \Hom(\calF, \mu_n^{\tensor d})) \to \ZZ/n\ZZ.$$ A more general duality theorem concerns smooth, but not necessarily complete, varieties . There exists a non-degenerate pairing of $\ZZ/n\ZZ$-modules

$$H_C^i(X, \calF) \times H^{2d-i}(X, \Hom(\calF, \mu_n^{\tensor d})) \to \ZZ/n\ZZ,$$ where cohomology with compact support is found on the left-hand side. If the field $k$ is the algebraic closure of a field $k'$, $X=X' \tensor_{k'} k$ and $\calF = \calF' \tensor_{k'} k$, then the Galois group $\Gal(k/k')$ acts on $H^i(X, \calF)$ and the preceding pairing is a pairing of $\Gal(k/k')$-modules.

Poincaré's duality theorem is an analogue of the first of the theorems given for $l$-adic cohomology: There exists a non-degenerate pairing of $\ZZ_l$-modules

$$H^i(X, \ZZ_l) \times H^{2d - i}(X, \ZZ_l[d]) \to \ZZ_l,$$ where $\ZZ_l[d]$ is the Tate sheaf, which is non-canonically isomorphic to the sheaf $\ZZ_l$ (cf. $l$-adic cohomology). Hence the isomorphism of $\QQ_l$-spaces

$$H^i(X, \QQ_l) \cong \Hom(H^{2d-i}(X, \QQ_l)(d), \ZZ_l),$$ and, in particular, the equality of the Betti numbers

$$b_i(X; l) = b_{2d-i}(X; l).$$ As in the case of cohomology of coherent sheaves, these results can be generalized to the relative case of a proper scheme morphism, formulated in the language of derived categories .

Other cohomology theories. Analogues of Poincaré's theorem are valid in the theory of crystalline cohomology , and de Rham cohomology over a field of characteristic zero . In number-theoretic applications cohomology of sheaves on the flat Grothendieck topology of arithmetic schemes are important. Duality theorems  are applicable to special cases of such cohomology theories.

How to Cite This Entry:
Duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Duality&oldid=44390
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article