Dualizing line bundle and trace datum of a smooth projective variety

Assume the Axiom of Choice. Let $k$ be a field and let $X$ be a projective $k$-scheme of finite type which is smooth of pure relative dimension $n$: every point of $X$ has an open neighbourhood on which ${\mathrm{\Omega }}_{X\mathrm{/}k}^1$ is locally free of rank $n$, which by the in-run theorem thm-differentials-smooth-locally-free of the flat/smooth/etale A page holds for every smooth $X\to \mathrm{Spec}k$ of pure relative dimension $n$. Here ${\mathrm{\Omega }}_{X\mathrm{/}k}^1$ is the sheaf of relative differentials of Sheaf of relative Kähler differentials and $H^q(X,-)$ is sheaf cohomology as in Sheaf cohomology as right derived global sections.

Dualizing line bundle. The sheaf ${\omega }_X:=\det {\mathrm{\Omega }}_{X\mathrm{/}k}^1:={\bigwedge }^n{\mathrm{\Omega }}_{X\mathrm{/}k}^1$ is the $n$-th exterior power of the locally free sheaf ${\mathrm{\Omega }}_{X\mathrm{/}k}^1$ of rank $n$; it is a locally free ${\mathcal{O}}_X$-module of rank one, called the dualizing line bundle (or canonical bundle) of $X$. Its formation is functorial in the following weak sense: an isomorphism $\phi :X\to X^{\mathrm{\prime }}$ of smooth projective $n$-dimensional $k$-schemes induces a canonical isomorphism ${\phi }^{\ast }{\omega }_{X^{\mathrm{\prime }}}\cong {\omega }_X$.

Projective-space model. For $X={\mathbb{P}}_k^n$ with the standard homogeneous coordinates $x_0,\dots ,x_n$, the dualizing bundle is ${\omega }_{{\mathbb{P}}^n}\cong \mathcal{O}(-n-1)$. To see the bundle identity directly, on $U_i=D_{+}(x_i)$ use the ordered coordinates $x_{\mathrm{\ell }}\mathrm{/}{x}_i$ for $\mathrm{\ell }\mathrm{\ne }i$ and the local generator ${\eta }_i=(-1{)}^i{\bigwedge }_{\mathrm{\ell }\mathrm{\ne }i}d(x_{\mathrm{\ell }}\mathrm{/}{x}_i)$, with the indices in increasing order. The coordinate change on $U_i\cap U_j$ gives ${\eta }_j=(x_j\mathrm{/}{x}_i{)}^{-n-1}{\eta }_i$: its Jacobian has the displayed power, and the signs $(-1{)}^i$ remove the ordering sign. The standard frames $x_i^{-n-1}$ of $\mathcal{O}(-n-1)$ have this same transition, so sending $x_i^{-n-1}$ to ${\eta }_i$ glues to the asserted isomorphism. For $n=0$ the empty wedge is $1$ and both bundles are trivial on ${\mathbb{P}}_k^0=\mathrm{Spec}k$. By Cohomology of O(d) on projective space the group $H^n({\mathbb{P}}_k^n,\mathcal{O}(-n-1))$ is one dimensional with Laurent generator $(x_0\cdots x_n{)}^{-1}$. The Laurent-coefficient residue trace is the $k$-linear map $t_{{\mathbb{P}}^n}:H^n({\mathbb{P}}_k^n,{\omega }_{{\mathbb{P}}^n})⟶k$ which, on that monomial basis, sends the class with Laurent tail $(x_0\cdots x_n{)}^{-1}$ to $1\in k$. Compatibility of this normalisation with cup products is the content of the twisting-sheaf duality theorem proved later on this page.

Normalized Serre trace. Let $X$ be smooth projective of pure dimension $n$ over $k$. A normalized Serre trace for $X$ is a $k$-linear map $t_X:H^n(X,{\omega }_X)⟶k$ such that:

1. after any closed immersion $j:X\hookrightarrow {\mathbb{P}}_k^N$ over $k$ with pure codimension $c=N-n$, the Gysin map $G_j:H^n(X,{\omega }_X)\to H^N({\mathbb{P}}_k^N,{\omega }_{{\mathbb{P}}^N})$ is formed by the adjunction and regular-immersion identification $H^n(X,{\omega }_X)\cong {\mathrm{Ext}}_{{\mathbb{P}}^N}^N(j_{\ast }{\mathcal{O}}_X,{\omega }_{{\mathbb{P}}^N})$ followed by the map on Ext induced contravariantly by the unit ${\mathcal{O}}_{{\mathbb{P}}^N}\to j_{\ast }{\mathcal{O}}_X$ and the identification ${\mathrm{Ext}}_{{\mathbb{P}}^N}^N({\mathcal{O}}_{{\mathbb{P}}^N},{\omega }_{{\mathbb{P}}^N})\cong H^N({\mathbb{P}}_k^N,{\omega }_{{\mathbb{P}}^N})$; the normalization condition is the typed equality $t_X=t_{{\mathbb{P}}^N}\circ G_j$;
2. for every finite locally free ${\mathcal{O}}_X$-module $\mathcal{E}$ the evaluation pairing $H^q(X,\mathcal{E})\times H^{n-q}(X,{\mathcal{E}}^{\vee }\otimes {\omega }_X)⟶k,(\alpha ,\beta )⟼t_X(\alpha \cup \beta ),$ is a perfect pairing of $k$-vector spaces for every $0\le q\le n$.

The existence of a normalized Serre trace, its nondegeneracy and its independence of the chosen embedding $j$ are claims of thm-serre-duality-smooth-projective-variety-locally-free-sheaves proved later on this page; this definition only fixes the data, the sign convention ${\omega }_X=\det {\mathrm{\Omega }}_{X\mathrm{/}k}^1$ and the projective-space normalisation. In particular the trace is not part of the definition of ${\omega }_X$ and no existence statement is made here.