Algebraic geometry. A concise dictionary by Elena Rubei

By Elena Rubei

Algebraic geometry has a sophisticated, tough language. This ebook incorporates a definition, a number of 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 matters are incorporated. It is helping rookies that understand a few, yet no longer all, simple evidence of algebraic geometry to stick with seminars and to learn papers. The dictionary shape makes it effortless and speedy to refer to.

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Example text

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 ????).

Definition. , the image through ???? of any closed subset (closed in the Zariski topology) is a closed subset. Proposition. An algebraic variety ???? over ℂ is complete if and only if it is compact with the usual topology. Proposition. A projective algebraic variety ???? over an algebraically closed field is complete. There exist complete nonprojective algebraic varieties; see [201]. Chow’s lemma. For any complete variety ???? over an algebraically closed field, there exists a projective algebraic variety ???? and a surjective birational morphism from ???? to ????.

The bijection is given by associating to a pointed covering projection on (????, ????), ???????? : (???????? , ???????? ) → (????, ????), the subgroup ????∗ (????1 (???????? , ???????? )) of ????1 (????, ????). Definition. We say that a covering projection ???? : ???????? → ???? with ???????? path-connected, is universal if the first fundamental group of ???????? is trivial. Let ???? : (???????? , ???????? ) → (????, ????) be a pointed covering projection with ???????? path-connected. Let ???? = ????∗ (????1 (???????? , ???????? )). Obviously the fibre ????−1 (????) is in bijection with the set of lateral classes of ???? in ????1 (????, ????): {????????| ???? ∈ ????1 (????, ????)}.