4 Sheaf of di↵erentials and canonical line bundle
The goal of this section is to define the sheaf of (relative) di↵erentials and the canonical line bundle. Di↵erentials, measuring the infinitesimal nature of functions, are intuitive geometric notion which provide a lot of topological/geometric information. We will see how it is described algebraically in terms of K¨ahler di↵erentials and sheaves. In the case of a nonsingular variety over C, the sheaf of di↵erentials is almost same as the cotangent bundle defined in complex di↵erential geometry, which is the dual of the tangent bundle. In other words, it corresponds to the vector bundle composed of holomorphic 1-forms. In contrast to di↵erential/complex geometry, the notion of cotangent sheaf is much more “natural” in algebraic geometry. When A is a local k-algebra, and if we have a maximal ideal m of A,theZariski cotangent space at m is defined to be the k-vector space m/m2. 2 It looks natural than the Zariski tangent space (m/m )_, the dual vector space of the Zariski cotangent space. Hence, this will be a dual picture to the geometric pictures what we learned from di↵erential geometry and complex geometry. There are technical merits of the algebraic construction. First, we are able to construct the sheaf even if X is not “smooth”. The cotangent sheaf may not be locally free in this case, but it will be quasi- coherent. Second, the construction naturally works in a relative case; we may define a sheaf of di↵erentials ⌦ =⌦ on X for a morphism f : X Y . Roughly speaking, f X/Y ! this will measure the cotangent vectors of the fiber of the map so that the cotangent vectors of X are a combination of the cotangent vectors of Y and the cotangent vectors of fibers.
Example 111. Let A = C[z1, ,zn] be a polynomial ring over the complex numbers. ··· Let m =(z , ,z ) be the maximal ideal corresponding to the origin. Let f be a 1 ··· n regular function on Cn, then it admits a power series representation
n f = f(0) + aizi + (terms of order at least 2) . ! Xi=1 @f 2 where ai = (0). Hence, its image in m/m is a1z1 + + anzn; the linearization of @zi ··· f at the origin. If we denote @ be the dual basis of (m/m2) , then we may regard @ @zi _ @zi as a linear functional; it reads o↵the i-th coe cient of the linearization by @ (f)= @zi @f (0) = a . @zi i We first begin with the a ne case: we will see the algebraic theory of K¨ahler di↵eren- tials. Suppose A is a B-algebra, so we have a morphism of rings : B A and the ! corresponding morphism of spectra Spec A Spec B. For any A-module M,wemay ! define a B-derivation as follows.
Definition 112. A B-derivation of A into M is a B-linear map d : A M such that !
(i) (additivity) d(a + a0)=da + da0.
(ii) (Leibniz rule) d(aa0)=ada0 + a0da.
43 Topic 2 – Line bundles, sections, and divisors
(iii) (triviality) db = 0 for every b (B) A. 2 ✓ The notion of B-derivations leads to the module of K¨ahler di↵erentials (or, the module of relative di↵erentials) as follows.
Definition 113. The module of K¨ahler di↵erentials of A over B is an A-module ⌦A/B, together with a B-derivation d : A ⌦ which satisfies the following universal ! A/B property: for any A-module M and a B-derivation d : A M,thereisaunique 0 ! A-module homomorphism f :⌦ M such that d = f d. A/B ! 0 In particular, ⌦ is the “representing object” of the functor of B-derivations M A/B 7! DerB(A, M). One can construct such a module ⌦A/B as the free module generated by the symbols da a A , and take the quotient by the submodule generated by all the { | 2 } expressions of the form (i) d(a + a ) da da ; 0 0 (ii) d(aa ) ada a da; 0 0 0 (iii) db so that d : A ⌦ which sends a to da should be a B-derivation. The pair (⌦ ,d) ! A/B A/B is unique up to a unique isomorphism. Caution. Both A and ⌦ are A-modules, but the map d : A ⌦ is not A-linear. A/B ! A/B Proposition 114. If A is a finitely generated B-algebra, then ⌦A/B is a finitely gener- ated A-module. If A is a finitely presented B-algebra, then ⌦A/B is a finitely presented A-module. Proof. Let x , ,x be a set of generators of A as a B-algebra, and let r be a { 1 ··· n} { j} set of relations among x , that is, r is a polynomial such that r (x , ,x ) = 0. Then i j j 1 ··· n ⌦ is generated by dx , ,dx , subject to the relations dr = 0. In particular, we A/B 1 ··· n j do not need to take every single element or every single relation, but can take generators and generators of the relations.
Example 115. Here we give a few baby examples.
(1) Let A = B/I.Thenthemodule⌦A/B = 0. (2) Let A = B[x , ,x ]. The module ⌦ = Adx Adx Adx ,afree 1 ··· n A/B 1 2 ··· n module of rank n. (3) Let B = k is a field of characteristic = 2, and let A = k[x, y]/(y2 x3 + x). The 6 module ⌦ is generated by dx and dy, subject to the relation 2ydy =(3x2 1)dx. A/B In the locus (y = 0), the generator dy can be expressed in terms of dx,hence,the 6 sheaf ⌦^A/B is generated by dx, and is isomorphic to the structure sheaf. Similarly, in the locus ((3x2 1) = 0), the sheaf ⌦^ is generated by dy, and is isomorphic 6 A/B to the structure sheaf. Since the curve defined by the equation y2 x3 + x =0is covered by those two loci, we conclude that ⌦^A/B is an invertible sheaf.
44 (4) Let A = k[x, y]/(y2 x3). We have ⌦ (Adx Ady)/(2ydy 3x2dx), hence, at A/B ' the origin (x = y = 0), it is of rank 2. Outside of the origin, it is of rank 1. Hence,
the sheaf ⌦^A/B is not a locally free sheaf.
There are two geometrically motivated exact sequences.
Proposition 116 (Relative cotangent sequence). Let C B A be ring homomor- ! ! phisms. There is a natural right exact sequence of A-modules
a db adb da da A ⌦ ⌦ 7! / ⌦ 7! / ⌦ / 0. ⌦B B/C A/C A/B Proposition 117 (Conormal sequence). Let B is a C-algebra, I an ideal of B,and A = B/I. Then there is a natural right exact sequence of A-modules