Algebraic Geometry: A Concise Dictionary by Elena Rubei

By Elena Rubei

Algebraic geometry has a sophisticated, tricky language. This ebook includes a definition, a number of references and the statements of the most theorems (without proofs) for each of the commonest phrases during this topic. a few phrases of similar matters are integrated. It is helping novices that recognize a few, yet now not all, simple proof of algebraic geometry to keep on with seminars and to learn papers. The dictionary shape makes it effortless and fast to consult.

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We denote by ???????? (A) the subcategory of ????(A) whose objects are the bounded complexes, and analogously ????+ (A) and ????− (A). Let ???? : ???? → ???? be the following morphism in ????∗ (A) (where ∗ is one among 0, ????, +, − and so throughout the item): ???? •c cc cc ???? cc ???? ????. ~b ~~ ~ ~~ ~~ ℎ We define the cone ????(????) to be the cone of ℎ. Definition. We define the localizing functor from the homotopy category to the derived category ????A : ????∗ (A) → ????∗ (A) in the following way: it is the identity on the set of the objects and, for any ???? morphism in ????∗ (A), we define ????A (????) to be ???? •d dd dd ???? dd ???? c????   .

If ∇ is a connection on a vector bundle ????, there exists a connection ∇∨ on the dual bundle ????∨ such that ????(????, ???????? ) = (∇????, ???????? ) + (????, ∇∨ ???????? ) for any ???? ∈ ????(????), ???????? ∈ ????(????∨ ) (where (⋅, ⋅) means that we are applying the element of ????(????∨ ) to the element of ????(????)). Bianchi’s identity. Let ∇ be a connection on a vector bundle ????. Let ???? be its curvature; it can be seen as an element of ????2 (???????????? ????). Then ∇???? = 0 (by the remarks above, ∇ defines a connection on ????∨ and thus also on ???? ⊗ ????∨ = ????????????(????) and thus we have a map, we call again ∇, from ????2 (???????????? ????) to ????3 (???????????? ????)).

Let C be an additive category and let ???? be an additive automorphism of C (we call ???? shift operator). A triangle in C is a sextuple (????, ????, ????, ????, ????, ????) of objects ????, ????, ???? in C and morphisms ???? : ???? → ????, ???? : ???? → ????, ???? : ???? → ????(????). It is often denoted ???? ???? ???? ???? ????→ ???? ????→ ???? ????→ ????(????). A morphism of triangles is a commutative diagram ???? ????  ???????? ???? ???????? G???? ????  G ???????? ???? ???????? G???? ℎ  G ???????? ???? ???????? G ????(????) ????(????)  G ????(???????? ) . Definition. We say that an additive category C equipped with an additive automorphism ???? and with a family of triangles, called distinguished triangles, is a triangulated category if the following axioms hold: (1) Every triangle isomorphic to a distinguished triangle is a distinguished triangle.

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