Intersection Theory Class 19 Ravi
نویسنده
چکیده
We defined the Grothendieck groups KX and K0X. They are vector bundles, respectively coherent sheaves, modulo the relation [E] = [E ] + [E ]. We have a pullback on K: f : KX → KY. KX is a ring: [E] · [F] = [E ⊗ F]. We have a pushforward on K0: f∗[F ] = ∑ i≥0(−1) [Rf∗F ]. We obviously have a homomorphism KX → K0X. K0X is aK X-module: KX⊗K0X → X is given by [E] · [F ] = [E⊗ F ]. Unproved fact: If X is nonsingular and projective, the map KX → K0X is an isomorphism. (Reason: If X is nonsingular, then F has a finite resolution by locally free sheaves.)
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