Download Algebraic Geometry: A Concise Dictionary by Elena Rubei PDF

By Elena Rubei

ISBN-10: 3110316226

ISBN-13: 9783110316223

Algebraic geometry has a classy, tricky language. This booklet features 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 incorporated. It is helping newbies that recognize a few, yet no longer all, simple proof of algebraic geometry to stick with seminars and to learn papers. The dictionary shape makes it effortless and speedy to consult.

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Extra resources for Algebraic Geometry: A Concise Dictionary

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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 is a ????-functor and ???? : ????∗ (A) → ????(B) ???? : ????B ∘ ???? → ???? ∘ ????A is a morphism of functors, then there exists a unique morphism ???? : ????∗ ???? → ???? such that ???? = (???? ∘ ????A ) ∘ ????.

Let A be the category of ????-modules for some commutative ring with unity ???? and let ???? be an ????-module. The classical ????-th left derived functor of the right exact functor ⋅ ⊗???? ???? is ???????????????? (⋅, ????). See “Ext, EXT ” and “Tor, TOR”. 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 ????).

Let ???? be the subcategory of ???????? (A) given by the complexes of of injective objects. Consider the left exact functor ???? = ????????????(????, ⋅) from ????(A) to ????(????????) (where ???????? is the category of Abelian groups). Then there exists the derived functor ???? ????????????(????, ⋅) and ???????? ????????????(????, ⋅) ≅ ???????????????? (????, ⋅). Let A be the category of ????-modules for some commutative ring with unity ???? and let ???? be an ????-module. The classical ????-th left derived functor of the right exact functor ⋅ ⊗???? ???? is ???????????????? (⋅, ????).

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Algebraic Geometry: A Concise Dictionary by Elena Rubei


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