Download e-book for iPad: Algebraic geometry. A concise dictionary by Elena Rubei

By Elena Rubei

ISBN-10: 3110316226

ISBN-13: 9783110316223

Algebraic geometry has a sophisticated, tricky language. This e-book incorporates a definition, numerous references and the statements of the most theorems (without proofs) for each of the most typical phrases during this topic. a few phrases of comparable topics are integrated. It is helping newcomers that be aware of a few, yet now not all, simple proof of algebraic geometry to stick with seminars and to learn papers. The dictionary shape makes it effortless and fast to refer to.

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A distinguished triangle in ????∗ (A) is defined to be a triangle isomorphic to a triangle of the form ???? ???? ???? ???? ????→ ???? ????→ ????(????) ????→ ????(????), where ???? and ???? are the natural maps ???? → ????(????) and ????(????) → ????(????). Analogously we define the distinguished triangles in ????∗ (A). With these families of distinguished triangles, ????∗ (A) and ????∗ (A) are triangulated categories. Definition. Let A and B be Abelian categories and let ???? : ????∗ (A) → ????(B) be a ????-functor. A right derived functor of ???? is a ????-functor ????∗ ???? : ????∗ (A) → ????(B) together with a morphism of functors from ????(A) to ????(B) ???? : ????B ∘ ???? → ????∗ ???? ∘ ????A with the following universal property: if ???? : ????∗ (A) → ????(B) is a ????-functor and ???? : ????B ∘ ???? → ???? ∘ ????A is a morphism of functors, then there exists a unique morphism ???? : ????∗ ???? → ???? such that ???? = (???? ∘ ????A ) ∘ ????.

Determinantal varieties. ([15], [77], [104], [106], [209]). Let ???? be an algebraic variety (or a manifold) and ???? and ???? be two vector bundles on ???? and let ???? : ???? → ???? be a morphism of vector bundles. For any ???? ∈ ℕ, the set ???????? (????) = {???? ∈ ????| ????????(???????? : ???????? → ???????? ) ≤ ????} is said to be a determinantal variety (or the ????-degeneracy locus of ????). Example. ,???? O(???????? ). Then ???? is given by a matrix ???? × ???? whose entry ????, ???? is a polynomial of degree ???????? − ???????? if ???????? ≥ ???????? and 0 if ???????? < ???????? and ???????? (????) is the zero locus in ℙ???? of the minors (???? + 1) × (???? + 1) of ????.

Let ???? be a holomorphic vector bundle of rank ???? on a complex compact manifold ????. The Chern classes ???????? (????) ∈ ????2???? (????, ℤ) for ???? = 1, . . ,???? ????2???? (????, ℤ) (where 1 ∈ ????0 (????, ℤ) ≅ ℤ) are defined by the following three axioms: Axiom 1: Normalization. if ???? = 1, ????1 is the map ????1 (????, O∗ ) → ????2 (????, ℤ) induced by the exponential sequence on ????, 0 → ℤ → O → O∗ → 0 Chern classes | 23 (the first map is given by the inclusion and the second by ???? ????→ ????2???????????? ; see “Exponential sequence”). Axiom 2: Multiplicativity.

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Algebraic geometry. A concise dictionary by Elena Rubei


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