Download Algebraic Geometry II: Cohomology of Algebraic Varieties. by V. I. Danilov (auth.), I. R. Shafarevich (eds.) PDF

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Now, the Riemann-Roeh formula deseribes the extend to whieh the Chern eharacter eh falls to eommute with direet images. Theorem (Grothendieek). For every a E K(X) eh(fka) ·td(Ty) = f*(eh(a) ·td(Tx)). 6. Principle of the Proof. As it often happens, it is easier to prove a general formula than its special case - the formula helps itself. In the case in question, we are dealing with a morphism instead of a fixed variety. Clearly, if Grothendieek's formula is valid for morphisms f: X ---+ Y and g: Y ---+ Z, then it also valid for the eomposition g o f: X ---+ Z.

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