For a morphism of schemes X\to Y, let \Delta: X\to X\times_Y X be the
diagonal morphism, which is a
closed immersion if X is
separated over Y. Let \mathcal I be the ideal sheaf of X in X\times_Y X. Then the sheaf of
differentials \Omega^1_{X/Y} can be defined as the pullback \Delta^*\mathcal I of \mathcal I to X. Sections of this sheaf are called
1-forms on X over Y, and they can be written locally on X as finite sums \textstyle\sum f_j\, dg_j for regular functions f_j and g_j. If X is locally of finite type over a field k, then \Omega^1_{X/k} is a coherent sheaf on X. If X is
smooth over k, then \Omega^1 (meaning \Omega^1_{X/k}) is a vector bundle over X, called the
cotangent bundle of X. Then the
tangent bundle TX is defined to be the dual bundle (\Omega^1)^*. For X smooth over k of dimension n everywhere, the tangent bundle has rank n. If Y is a smooth closed subscheme of a smooth scheme X over k, then there is a short exact sequence of vector bundles on Y: :0\to TY \to TX|_Y \to N_{Y/X}\to 0, which can be used as a definition of the
normal bundle N_{Y/X} to Y in X. For a smooth scheme X over a field k and a natural number i, the vector bundle \Omega^i of
i-forms on X is defined as the i-th
exterior power of the cotangent bundle, \Omega^i = \Lambda^i \Omega^1. For a smooth
variety X of dimension n over k, the
canonical bundle K_X means the line bundle \Omega^n. Thus sections of the canonical bundle are algebro-geometric analogs of
volume forms on X. For example, a section of the canonical bundle of affine space \mathbb A^n over k can be written as :f(x_1,\ldots,x_n) \; dx_1 \wedge\cdots\wedge dx_n, where f is a polynomial with coefficients in k. Let R be a
commutative ring and n a natural number. For each integer j, there is an important example of a line bundle on projective space \mathbb P^n over R, called \mathcal O(j). To define this, consider the morphism of R-schemes :\pi: \mathbb A^{n+1}-0\to \mathbb P^n given in coordinates by (x_0,\ldots,x_n) \mapsto [x_0,\ldots,x_n]. (That is, thinking of projective space as the space of 1-dimensional linear subspaces of affine space, send a nonzero point in affine space to the line that it spans.) Then a section of \mathcal O(j) over an open subset U of \mathbb P^n is defined to be a regular function f on \pi^{-1}(U) that is homogeneous of degree j, meaning that :f(ax)=a^jf(x) as regular functions on (\mathbb A^{1} - 0) \times \pi^{-1}(U). For all integers i and j, there is an isomorphism \mathcal O(i) \otimes \mathcal O(j) \cong \mathcal O(i+j) of line bundles on \mathbb P^n. In particular, every
homogeneous polynomial in x_0,\ldots,x_n of degree j over R can be viewed as a global section of \mathcal O(j) over \mathbb P^n. Note that every closed subscheme of projective space can be defined as the zero set of some collection of homogeneous polynomials, hence as the zero set of some sections of the line bundles \mathcal O(j). This contrasts with the simpler case of affine space, where a closed subscheme is simply the zero set of some collection of regular functions. The regular functions on projective space \mathbb P^n over R are just the "constants" (the ring R), and so it is essential to work with the line bundles \mathcal O(j).
Serre gave an algebraic description of all coherent sheaves on projective space, more subtle than what happens for affine space. Namely, let R be a Noetherian ring (for example, a field), and consider the polynomial ring S = R[x_0,\ldots,x_n] as a
graded ring with each x_i having degree 1. Then every finitely generated graded S-module M has an
associated coherent sheaf \tilde M on \mathbb P^n over R. Every coherent sheaf on \mathbb P^n arises in this way from a finitely generated graded S-module M. (For example, the line bundle \mathcal O(j) is the sheaf associated to the S-module S with its grading lowered by j.) But the S-module M that yields a given coherent sheaf on \mathbb P^n is not unique; it is only unique up to changing M by graded modules that are nonzero in only finitely many degrees. More precisely, the abelian category of coherent sheaves on \mathbb P^n is the
quotient of the category of finitely generated graded S-modules by the
Serre subcategory of modules that are nonzero in only finitely many degrees. The tangent bundle of projective space \mathbb P^n over a field k can be described in terms of the line bundle \mathcal O(1). Namely, there is a short exact sequence, the
Euler sequence: : 0\to \mathcal O_{\mathbb P^n}\to \mathcal O(1)^{\oplus \; n+1}\to T\mathbb P^n\to 0. It follows that the canonical bundle K_{\mathbb P^n} (the dual of the
determinant line bundle of the tangent bundle) is isomorphic to \mathcal O(-n-1). This is a fundamental calculation for algebraic geometry. For example, the fact that the canonical bundle is a negative multiple of the
ample line bundle \mathcal O(1) means that projective space is a
Fano variety. Over the complex numbers, this means that projective space has a
Kähler metric with positive
Ricci curvature.
Vector bundles on a hypersurface Consider a smooth degree-d hypersurface X \subseteq \mathbb{P}^n defined by the homogeneous polynomial f of degree d. Then, there is an exact sequence :0 \to \mathcal O_X(-d) \to i^*\Omega_{\mathbb{P}^n} \to \Omega_X \to 0 where the second map is the pullback of differential forms, and the first map sends : \phi \mapsto d(f\cdot \phi) Note that this sequence tells us that \mathcal O(-d) is the conormal sheaf of X in \mathbb P^n. Dualizing this yields the exact sequence : 0 \to T_X \to i^*T_{\mathbb{P}^n} \to \mathcal O(d) \to 0 hence \mathcal O(d) is the normal bundle of X in \mathbb P^n. If we use the fact that given an exact sequence :0 \to \mathcal E_1 \to \mathcal E_2 \to \mathcal E_3 \to 0 of vector bundles with ranks r_1,r_2,r_3, there is an isomorphism :\Lambda^{r_2}\mathcal E_2 \cong \Lambda^{r_1}\mathcal E_1\otimes \Lambda^{r_3}\mathcal E_3 of line bundles, then we see that there is the isomorphism :i^*\omega_{\mathbb P^n} \cong \omega_X\otimes \mathcal O_X(-d) showing that :\omega_X \cong \mathcal O_X(d - n -1) == Serre construction and vector bundles ==