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

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


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
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[1] ANGÉNIOL (B.) et LEJEUNE (M.) . - Calcul différentiel et classes caractéristiques en géométrie algébrique . - A paraître chez Hermann, Paris, France. Zbl 0749.14008 · Zbl 0749.14008
[2] BOREL (A.) et SERRE (J.-P.) . - Le théorème de Riemann-Roch , Bull. Soc. Math. France, t. 86, 1958 , p. 97-136. Numdam | MR 22 #6817 | Zbl 0091.33004 · Zbl 0091.33004
[3] EL ZEIN (F.) . - Mixed Hodge Structures , Trans. Amer. Math. Soc., t. 275, 1983 , p. 71-106. MR 85g:14010 | Zbl 0511.14003 · Zbl 0511.14003
[4] FULTON (W.) . - Intersection Theory, Ergebnisse der Math . - Springer Verlag. · Zbl 0885.14002
[5] FULTON (W.) and GILLET (H.) . - Riemann-Roch for general algebraic schemes , Bull. Soc. Math. France, t. 111, 1983 , p. 287-300. Numdam | MR 85h:14010 | Zbl 0579.14013 · Zbl 0579.14013
[6] GILLET (H.) . - Homological descent for the K-theory of coherent sheaves , Springer Lectures Notes 1046. MR 86a:14016 | Zbl 0557.14009 · Zbl 0557.14009
[7] GRAYSON (D.) . - Products in K-theory and intersecting algebraic cycles , Invent. Math., t. 47, 1978 , p. 71-84. MR 58 #10890 | Zbl 0394.14004 · Zbl 0394.14004
[8] QUILLEN (D.) . - Higher algebraic K-theory I , Lecture Notes in Math. 341, Springer Verlag. MR 49 #2895 | Zbl 0292.18004 · Zbl 0292.18004
[9] DIEUDONNÉ (J.) et GROTHENDIECK (A.) . - Éléments de Géométrie Algébrique , Publ. Math. IHES 8, 1961 . Numdam · Zbl 0203.23301
[10] BERTHELOT (P.) , GROTHENDIECK (A.) et ILLUSIE (L.) . - Théorie des Intersections et Théorème de Riemann-Roch , 1966 - 1967 , Springer Lecture Notes 225, 1971 . Zbl 0218.14001 · Zbl 0218.14001
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