Let X —> S be a morphism of schemes. Let E and F be quasi-coherent O_X modules. Let D : E —> F be a finite order differential operator on X\/S. This means that there exists an integer n and an O_X-linear map<\/p>\n
D’ : p_{n, 2, *}(p_{n, 1}^*E) —> F<\/p><\/blockquote>\n
such that D is given by the composition of D’ with the nonlinear map E —> p_{n, 2, *}(p_{n, 1}^*E). Here p_{n, i} : Delta_n —> X are the two projections of the nth infinitesimal neighbourhood of the diagonal X —> X x_S X. (Unfortunately, this description of finite order differential operators is currently missing from the Stacks project.)<\/p>\n
OK, let omega_{X\/S} be a quasi-coherent O_X-module (we’ll see later the properties we require of this module). Set E* = SheafHom(E, omega_{X\/S}) and similarly for F. When does there exist a dual<\/strong> differential operator<\/p>\n
D* : F* —> E* ?<\/p><\/blockquote>\n
The purpose of this blog post is to analyze what we need about S, X, omega_{X\/S}, F, and E in order to get D*.<\/p>\n
Suppose we have F = (F*)*. Then we can think of D’ as a map<\/p>\n
D’ : p_{n, 2, *}(p_{n, 1}^*E) —> (F*)*<\/p><\/blockquote>\n
which is the same thing as a map<\/p>\n
p_{n, 2, *}(p_{n, 1}^*E ⊗ p_{n, 2}^*(F*)) —> omega_{X\/S}<\/p><\/blockquote>\n
which is the same thing as a map<\/p>\n
p_{n, 1}^*E ⊗ p_{n, 2}^*(F*) —> p_{n, 2}^!omega_{X\/S}<\/p><\/blockquote>\n
by duality. If we have an isomorphism p_{n, 2}^!omega_{X\/S} = p_{n, 1}^!omega_{X\/S} then applying the automorphism of X x_S X which switches the factors and going backwards we see that this is the same thing as a differential operator D* of the form desired.<\/p>\n
What did we use in the above? We need<\/p>\n
\n
- F = (F*)*,<\/li>\n
- p_{n, i} : Delta_n —> X are finite morphisms,<\/li>\n
- p_{n, 2}^!omega_{X\/S} = p_{n, 1}^!omega_{X\/S} where p_{n, 2}^! is the functor used in duality for a finite morphism.<\/li>\n<\/ol>\n
The first condition holds for example if F is finite locally free and O_X = SheafHom(omega_{X\/S}, omega_{X\/S}). The second condition holds if X —> S is of finite type. The third condition really does pin down omega_{X\/S} a lot more.<\/p>\n
If S is Noetherian, F and E are finite locally free, and X —> S is a separated, flat morphism of finite type whose fibres are Cohen-Macaulay and equidimensional of a given dimension d, then we can take omega_{X\/S} the usual relative dualizing sheaf and we have enough theory in the Stacks project to get the third condition above.<\/p>\n
But there is another way to think about the condition p_{n, 2}^!omega_{X\/S} = p_{n, 1}^!omega_{X\/S}. Namely, considering finite order differential operators D : O_X —> omega_{X\/S} and arguing as above one sees that giving a (symmetric) isomorphism p_{n, 2}^!omega_{X\/S} = p_{n, 1}^!omega_{X\/S} is the same thing as a rule D ↦ D* which defines an involution on the sheaf of finite order differential operators D : O_X —> omega_{X\/S} such that<\/p>\n
\n
- (f D)* = D* \\circ f<\/li>\n
- (D \\circ f)* = f D*<\/li>\n<\/ol>\n
where f denotes a local section of O_X.<\/p>\n
When X —> S is smooth of relative dimension d and we take omega_{X\/S} = Omega^d_{X\/S}, we can explicitly construct the rule D ↦ D*. For example if we have local coordinates x_1, …, x_d on X\/S and we use the trivialization of Omega^d_{X\/S} given by d(x_1) ∧ … ∧ d(x_d) then we take the algebra involution on differential operators sending a function to itself and sending ∂\/∂x_i to – ∂\/∂x_i. Of course in this case we can also explicitly describe the rule going from D : E —> F to D* : F* —> E* we obtain from this.<\/p>\n
The second case works without appealing to any theory of duality.<\/p>\n","protected":false},"excerpt":{"rendered":"
Let X —> S be a morphism of schemes. Let E and F be quasi-coherent O_X modules. Let D : E —> F be a finite order differential operator on X\/S. This means that there exists an integer n and … Continue reading