By Charles A. Weibel

A portrait of the topic of homological algebra because it exists at the present time

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**Additional info for An Introduction to Homological Algebra**

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V M O = {u + v : u, v ∈ C} and there is an absolute constant B such that every f ∈ V M O can be written as f = u + v with u, v ∈ C and u ∞ ≤ B f ∗ , v ∞ ≤ B f ∗ . 9 BM O and V M O 37 (k) The conjugation operator (and thus also the Cauchy singular integral operator and the Riesz projection) is bounded on the following pairs of spaces: L∞ → BM O, BM O → BM O, C → V M O, V M O → V M O. (l) We have BM O = {u + P v : u, v ∈ L∞ }, V M O = {u + P v : u, v ∈ C} and f BM O = inf{ u ∞ + v ∞ : u, v ∈ L∞ , u + P v = f } is an equivalent norm in BM O.

Let 1 < p < ∞ and let w be a weight in Lp . 19) I where the supremum is over all subarcs of the circle T and |I| is the arc length of I. 19) is ﬁnite it suﬃces to show that the supremum over all subarcs I satisfying |I| ≤ δ, where δ is any positive number, is ﬁnite. If one asks for a condition on the weight w that ensures that P be bounded o type on both Lp (w) and Lq (w) (1/p + 1/q = 1), then again a Helson-Szeg˝ criterion can be given. The suﬃciency portion of the following result was established by Simonenko [492], its necessary part is due to Krupnik [328].

The simplest local principle is the Gelfand theory of commutative Banach algebras with identity: if A is such an algebra and a ∈ A, then a ∈ GA ⇐⇒ a ∈ /N ∀ N ∈ M (A) ⇐⇒ a(N ) = 0 ∀ N ∈ M (A). Thus, in a sense, invertibility in commutative Banach algebras is local in nature. 35 are what we need from the arsenal of “local techniques” for non-commutative algebras. We want not to go into the history of this topic, but only mention that Simonenko [493], [494], [495] was indisputably the ﬁrst who both realized the local nature of Fredholmness of convolution and related operators and at the same time created a powerful machinery (his local principle) for tackling successfully a whole series of problems.