By David Mumford
From the studies: "Although numerous textbooks on smooth algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the crimson e-book of sorts and schemes, now as ahead of essentially the most very good and profound primers of contemporary algebraic geometry. either books are only actual classics!" Zentralblatt
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Extra info for Algebraic Geometry I: Complex Projective Varieties
This defines the map tjd:OnsdRn -+ R[I],, so t+bd homogenizes all elements of degree I d. , we specialize I to 1. 21’. In particular, H is right exact. g. g. graded R[I]-module M with H M = N . If=, Proof of Claim 2: Write N z R'"'/K where K = Rxi. , xi,,) in R'"), we take d greater than the maximum of the degrees of all the x i j , and define x f = ($&lr.. ,$&in) E (R[IId)('). Letting K' = R [ R ] x i , a graded submodule of R[I]'"', we have H(R[A]'"'/K') z H(R[I]("))/HK' z R(")/Kz N since H is right exact.
Euler Characteristic The FFR property has an important tie to topology. 42: Suppose R has IBN. g. free resolution 0 F, + ... + F, + M + 0 is defined as x ( M ) = CY=,(- l)irank(&). 43: x ( M )is independent of the FFR because of the generalized Schanuel lemma. In particular, if 0 + F, + ... + F, + 0 is exact then - l)irank(4) = 0. 44: Suppose S is a left denominator set of R and S-‘R also has IBN. If an R-module has FFR then S - ’ M has FFR in S-’R-&O~,and x ( S - ’ M ) = x ( M ) . ) The name “Euler characteristic” comes from Euler’s observation that #faces - #edges + #vertices of a simplicia1 complex is a topological invariant.
If, on the contrary, M ’ / f K is nonsingular then there is some x in f ” such that (1K)x-l is not large and thus misses a left ideal L of R; in other words, L x (Lx + fK)/fK. Since K is a summand of R(”) for some n we see by means of tensoring by Q that there is a nonzero map h,: K -+ L such that hotkerf) # 0. Writing x = f ’ a for suitable a in N we have a map h: K + N given by hy = (h,y)a. Note f ’ h K = f ‘ ( ( h , K ) a )= ( h 0 K ) f ’ a = h0Kx; thus p ( f ’ h K ) = p ( h 0 K x ) = p ( h 0 K ) > 0.