By Elena Rubei

Algebraic geometry has a sophisticated, tough language. This booklet includes 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 rookies that understand a few, yet no longer all, simple proof of algebraic geometry to keep on with seminars and to learn papers. The dictionary shape makes it effortless and speedy to consult.

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**Extra info for Algebraic Geometry: A Concise Dictionary**

**Example text**

Proposition. Let A be an Abelian category. If ???? : ???? → ???? and ???? : ???? → ???? are two morphisms in ????(A) and ???? is a quasi-isomorphism, then there exist a complex ???? in A and two morphisms ???? : ???? → ???? and ???? : ???? → ???? with ???? quasi-isomorphism such that the following diagram commutes: c ???? e eeee???? ee e ???? c b???? . cc }} } cc } }} ???? ???? cc }} ???? ???? Definition. Let A be an Abelian category. The derived category of A, denoted by ????(A) is the following category: – the objects of ????(A) are the complexes of objects of A; – a morphism ???? : ???? → ???? in ????(A) is a triplet (????, ????, ℎ), where ???? is a third complex, ???? : ???? → ???? and ℎ : ???? → ???? are homotopy equivalence classes of morphisms of complexes and ???? is a quasi-isomorphism.

Analogously, if ???? is a right exact functor, we can define the classical ????-th left derived functor for ????, ???????? ????. , we can take A equal to the category of ????-modules for some commutative ring with unity ????. Let ???? be an object of A. 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 ∇ 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 (???????????? ????)). Compatibility with holomorphic structures and metrics Definition. Let (????, (⋅, ⋅)) be a complex vector bundle with a Hermitian metric. A connection ∇ on ???? is said to be compatible with the metric if for all ????1 , ????2 ∈ ????(????).