42 Lecture 9: March 4 Local coordinates on algebraic varieties. Let X be an algebraic variety over afieldk, with structure sheaf OX . More precisely, X is a scheme of finite type over k, meaning that for every affine open subset U X, the ring of functions Γ(U, OX ) is a finitely generated k-algebra, or in other words,✓ a quotient of a polynomial ring. We say that X is nonsingular of dimension n if, at each closed point x X,the stalk 2 OX,x =limΓ(U, OX ) U x 3 is a regular local ring of dimension n; in other words, if mx OX,x denotes the maximal ideal, then ✓ 2 dimOX,x/mx mx/mx = n =dimOX,x. When the field k is perfect (which is always the case in characteristic zero), an 1 equivalent condition is that the sheaf of K¨ahler di↵erentials ⌦X/k is locally free of rank n. Since we are going to need this in a moment, let me briefly review derivations and K¨ahler di↵erentials. Let A be a finitely generated k-algebra. A derivation from A into an A-module M is a k-linear mapping D : A M such that δ(fg)= ! fδ(g)+gδ(f) for every f,g A. We denote by Derk(A, M) the set of all such derivations; this is an A-module2 in the obvious way. In the special case M = A, we use the notation Derk(A) for the derivations from A to itself. In view of the formula δ(fg)=fδ(g)+gδ(f), such a derivation is the algebraic analogue of a vector field, acting on the set of functions in A.WehaveDerk(A) Endk(A), and one can check that if δ ,δ Der (A), then their commutator ✓ 1 2 2 k [δ ,δ ]=δ δ δ δ End (A) 1 2 1 ◦ 2 − 2 ◦ 1 2 k is again a derivation. It is the analogue of the Lie bracket on complex manifolds. The module of K¨ahler di↵erentials ⌦1 represents the functor M Der (A, M), A/k 7! k in the sense that one has a functorial isomorphism 1 Derk(A, M) ⇠= HomA ⌦A/k,M . In other words, ⌦1 is an A-module, together with a derivation d: A ⌦1 ,such A/k ! A/k that every derivation δ Der (A, M) factors uniquely as δ = δ˜ d for a unique 2 k ◦ A-linear map δ˜:⌦1 M. Concretely, ⌦1 can be constructed by taking the A/k ! A/k free A-module on the set of generators df , for f A, and imposing the relations d(fg)=fdg + gdf and d(f + g)=df + dg for every2 f,g A, and df = 0 for every f k. By construction, one has 2 2 1 Derk(A) ⇠= HomA(⌦A/k,A), which makes the module of K¨ahler di↵erentials dual to the module of derivations. 1 Globally, ⌦X/k is a coherent sheaf of OX -modules, such that for every affine open subset U X, one has Γ(U, ⌦1 )=⌦1 ,whereA =Γ(U, O ). There is again ✓ X/k A/k X a universal derivation d: O ⌦1 . Think of ⌦1 as an algebraic analogue of X ! X/k X/k the sheaf of holomorphic one-forms on a complex manifold. The tangent sheaf T = om (⌦1 , O ) X H OX X/k X is defined as the dual of the sheaf of K¨ahler di↵erentials; on affines, one has Γ(U, TX )=Derk(A), using the notation from above. This is an algebraic ana- logue of the sheaf of holomorphic tangent vector fields on a complex manifold. 1 Now suppose that X is nonsingular of dimension n, or equivalently, that ⌦X/k is locally free of rank n. At every closed point x X, one can choose local coordinates 2 43 in the following way: there is an affine open neighborhood U of x, together with n regular functions x ,...,x Γ(U, O ), such that 1 n 2 X n ⌦1 = O dx . X/k U ⇠ X U · i i=1 M Dually, we have derivations @ ,..., @ Der Γ(U, O ) , such that 1 n 2 k X n T = O @ . X U ⇠ X U · i i=1 M This says that df = @ (f) dx + + @ (f) dx for every f Γ(U, O ), and so 1 · 1 ··· n · n 2 X the derivation @i plays the role of the partial derivative operator @/@xi. One can choose the functions x1,...,xn Γ(U, OX ) in such a way that they generate the 2 n maximal ideal mx OX,x. Keep in mind that the morphism U Ak defined by the local coordinates✓ is ´etale, but not usually an embedding (because! open sets in the Zariski topology are too big). The sheaf of di↵erential operators. Let X be a nonsingular algebraic variety. Our goal is to define the sheaf of di↵erential operators DX , which is a global ana- logue of the Weyl algebra An(k). This will be a quasi-coherent sheaf of OX -modules DX , together with an increasing filtration F DX by coherent OX -modules, where • FjDX consists of di↵erential operators of order j. We start by considering the affine case. So letU X be an affine open subset, ✓ and set A =Γ(U, OX ), which is a finitely generated k-algebra. We are going to define an A-module D(A) Endk(A), whose elements are the algebraic di↵erential operators of finite order on✓A. It will satisfy 1 D(A)= FjD(A), j=0 [ where F D(A) is the submodule of operators of order j. The idea is that operators j of order 0 should be multiplication by elements in A, and that if P FiD(A) and Q F D(A), then their commutator [P, Q]=P Q Q P End2 (A) should 2 j ◦ − ◦ 2 k belong to Fi+j 1D(A). This is consistent with what happens for the Weyl algebra. For an element− f A, we also use the symbol f End (A) to denote the 2 2 k operator of multiplication by f. Observe that P Endk(A) is multiplication by the element P (1) A if and only if P is A-linear if2 and only if [P, f] = 0 for every f A. We can therefore2 define 2 F D(A)= P End (A) [P, f] = 0 for every f A = A. 0 2 k 2 ⇠ We then define FjD(A) recursively by saying that FjD(A)= P Endk(A) [P, f] Fj 1D(A) for every f A . 2 2 − 2 This construction of di↵erential operators is due to Grothendieck. Example 9.1. Let us work out the relationship between F1D(A) and Derk(A). Every derivation δ Der (A) is also a di↵erential operator of order 1, because 2 k [δ, f](g)=δ(fg) fδ(g)=δ(f) g − · for every f,g A, which shows that [δ, f]=δ(f) F D(A). Conversely, suppose 2 2 0 that we have some P F1D(A). By definition, for every f A, there exists some p A such that [P, f2]=p . Concretely, this means that 2 f 2 f P (fg) fP(g)=p g − f · for every f,g A. Taking g = 1, we get p = P (f) fP(1), and so 2 f − P (fg) fP(g) gP(f)+fgP(1) = 0. − − 44 It is then easy to check that P P (1) is a derivation. The conclusion is that − F D(A) = A Der (A) 1 ⇠ ⊕ k with P F D(A) corresponding to the pair P (1),P P (1) . 2 1 − It is easy to see that each FjD(A) is a finitely generated A-module, and that composition in End (A) has the following e↵ect: if P F D(A) and Q F D(A), k 2 i 2 j then P Q Fi+jD(A) and [P, Q] Fi+j 1D(A). With some more work, one can prove the◦ following2 result. 2 − Proposition 9.2. Let A be a finitely generated k-algebra. If A is nonsingular of dimension n, then the following is true: (a) As an A-algebra, D(A) End (A) is generated by Der (A), subject to the ✓ k k relations [δ, f]=δ(f) for every δ Derk(A) and every f A. 2 j 2 (b) One has FjD(A)/Fj 1D(A) ⇠= Sym Derk(A) for j 0. (c) One has an isomorphism− of graded A-algebras ≥ F 1 gr D(A)= FjD(A)/Fj 1D(A) = Sym Derk(A) − ⇠ j=0 M between the associated graded algebra of D(A) and the symmetric algebra on Derk(A). Here, for any A-module M,thej-th symmetric power Symj M is the A-module obtained by quotienting M M by the submodule generated by elements ⌦A ···⌦A of the form m1 mj mσ(1) mσ(j), for all permutations σ Sj.The symmetric algebra⌦···on M is− the graded⌦···A-algebra 2 1 Sym M = Symj M. j=0 M It has the following universal property: if B is any A-algebra, then every morphism of A-modules M B extends uniquely to a morphism of A-algebras Sym M B. ! r ! For example, one has Sym A⊕ ⇠= A[x1,...,xr]. Let us give a concrete description of di↵erential operators in local coordinates. Let U X be an affine open, with local coordinates x ,...,x , and set A = ✓ 1 n Γ(U, OX ).