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Residues and differential operators on schemes. (English) Zbl 0962.14010
Let \(X\) be a finite type scheme over a perfect field \(k\) (of any characteristic). The author gives a new construction of the residue complex \(K_X^\bullet\) using Beilinson completion algebras (BCAs). The functorial properties of \({K}_X^\bullet\) with respect to étale, finite or proper morphisms and the duality theorem are proved from properties of BCAs. The construction reveals some new properties of \(K_X^\bullet\).
For an \({\mathcal O}_X\)-module \(M\), let Dual \(M = {{\mathcal H}om}_{{\mathcal O}_X}^\bullet (M, K_X^\bullet)\) be the dual complex. The authors prove the existence of a dual differential operator \(\text{Dual}(d)\): \(\text{Dual }N \rightarrow \text{Dual } M\) for any differential operator \(d:M\to N\) and give explicit formulas for \(\text{Dual}(d)\) in terms of differential operators and residues. These are used in the applications to de Rham homology and intersection cohomology \(D\)-modules. The de Rham residue complex \(F_X^\bullet= \text{Dual }\Omega _{X/k}^\bullet\) is obtained from the de Rham complex \({\Omega} _{X/k}^\bullet\). The construction is generalised to formal schemes also. As an application it is shown that the de Rham homology is a contravariant functor on the small étale site \(X_{\text{et}}\). The final application is a description of the intersection cohomology \(D\)-module \(L(X,Y)\) of an integral curve \(X\) embedded in a smooth variety \(Y\) of dimension \(n>1\) (characteristic 0) obtained using BCAs and algebraic residues instead of complex geometry.

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F40 de Rham cohomology and algebraic geometry
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
13N10 Commutative rings of differential operators and their modules
13N05 Modules of differentials
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