Théorème de Riemann-Roch par désingularisation. (Theorem of Riemann-Roch for desingularization). (French) Zbl 0702.14006

Angéniol, B.; El Zein, F.

Bull. Soc. Math. Fr. 116, No. 4, 385-400 (1988).

Let K be any field with the property that every singular K-variety admits a resolution of singularities. The authors present an easy, and very natural, proof of the Riemann-Roch theorem for any, possibly non- singular, algebraic variety X (locally of finite type and separated), defined on K.
The precise statement is the following: There is a homomorphism \(\tau_ X\) from the Grothendieck group \(K_ 0(X)\) to the rational Chow ring \(A_*(X)\otimes {\mathbb{Q}}\), which is covariant for proper morphisms, and coincide with \(ch\cap Todd(T_ X)\) if X is nonsingular. - The proof is by induction on the dimension of X, using resolutions of singularities, the Chow envelopes of Fulton and Gillet, and standard exact sequences in K-theory.
Given a proper morphism f: \(Y'\to Y\), of regular quasiprojective varieties defined on a field of characteristic 0, and closed immersions \(X\to Y\) and \(X'=f^{-1}(X)\to Y'\) such that f induces an isomorphism \(Y'-X'\to Y-X\), assumed to be open and dense subschemes of \(Y'\) and Y, respectively, the authors also prove the existence of an exact sequence \(0\to K_ i(X)\to K_ i(X')\oplus K_ i(Y)\to K_ i(Y')\to 0.\) This sequence reduces the problem of calculating Quillen’s K-groups for general quasiprojective varieties to the same problem for resolutions with at most normal crossing divisors. This is then used to prove a Riemann-Roch theorem “sans dénominateurs” and to compute Chern classes in the case of normal crossing divisors.

Reviewer: O.A.Laudal

MSC:

14C40	Riemann-Roch theorems
14E15	Global theory and resolution of singularities (algebro-geometric aspects)
14C05	Parametrization (Chow and Hilbert schemes)
14B05	Singularities in algebraic geometry

Keywords:

resolution of singularities; Riemann-Roch theorem; rational Chow ring; K- theory

References:

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