By David Mumford
From the experiences: "Although a number of textbooks on glossy algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the pink booklet of sorts and schemes, now as sooner than probably the most first-class and profound primers of contemporary algebraic geometry. either books are only actual classics!" Zentralblatt
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Additional info for Algebraic Geometry I: Complex Projective Varieties
SUKOCHEV 2. a(1 + D2 )−1/2 ∈ K(N , τ ) for all a ∈ A. We say that (A, H, D) is even if in addition there is a Z2 -grading such that A is even and D is odd. This means there is an operator γ such that γ = γ ∗ , γ 2 = IdN , γa = aγ for all a ∈ A and Dγ + γD = 0. Otherwise we say that (A, H, D) is odd. Remark. (1) We will write γ in all our formulae, with the understanding that, if (A, H, D) is odd, γ = IdN and of course, we drop the assumption that Dγ + γD = 0. (2) By density, we immediately see that the second condition in the deﬁnition of a semiﬁnite spectral triple, also holds for all elements in the C ∗ -completion of A.
By the deﬁnition of a spectral triple, the integrand is τ -compact, and so is in the compact endomorphisms of our module. The functional calculus yields the norm estimates D2 (ε + λ + D 2 )−1 [D, a](ε + λ + D 2 )−1 b ≤ [D, a] b (ε + λ)−1 , and D(ε + λ + D 2 )−1 [D, a]D(ε + λ + D 2 )−1 b ≤ [D, a] b (ε + λ)−1 . Therefore, the integral above is norm-convergent. Thus, D[(ε + D2 )−1/2 , a]b is τ -compact and [Fε , a]b = D[(ε + D 2 )−1/2 , a]b + [D, a](ε + D 2 )−1/2 b, is τ -compact too. Similarly, a[Fε , b] is τ -compact.
Proof. Since σ preserves both OPr0 and OPr , it suﬃces to prove the claim for r = t = 0. Indeed, for Tr ∈ OPr0 and Ts ∈ OPs , there exist A ∈ OP00 and B ∈ OP0 such that Tr = (1 + D 2 )r/2 A and Ts = (1 + D 2 )s/2 B. Thus, the general case will follow from the case t = s = 0 by writing Tr Ts = (1 + D2 )(r+s)/2 σ −s (A)B. So let T ∈ OP00 and S ∈ OP0 . We need to show that T S ∈ OP00 = B1∞ (D, p). For ∞ this, let T = i=0 T1,i T2,i any representation. We will prove that ∞ T1,i (T2,i S), i=0 is a representation of the product T S.
Algebraic Geometry I: Complex Projective Varieties by David Mumford