By Elena Rubei

ISBN-10: 3110316226

ISBN-13: 9783110316223

Algebraic geometry has a sophisticated, tough language. This publication 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 similar topics are incorporated. It is helping novices that be aware of a few, yet now not all, uncomplicated evidence 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**

**Sample text**

Connections on tensor products and dual bundles Remark. If ???? and ???????? are two vector bundles and ∇ and ∇???? are two connections respectively on ???? and ???????? , we can define a connection ∇ ⊗ ∇???? on the tensor product ???? ⊗ ???????? in the following way: (∇ ⊗ ∇???? )(???? ⊗ ???????? ) = ∇???? ⊗ ???????? + ???? ⊗ ∇???????? for any ???? ∈ ????(????), ???????? ∈ ????(???????? ). Remark. 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 ????(????)).

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 ????. 48 | Dimension For instance, take ???? = ???????? − 1 for some ???? and ????. Let ???? = O???? and ???? = O(1)???? . Call the coordinates in ℙ???? ????????,???? for ???? = 1, .

Coherent sheaves. ([93], [107], [129], [146], [223]). The most common definition of coherent sheaf is the following. Definition. Let (????, O???? ) be a ringed space (see “Space , ringed -”). – We say that a sheaf F of O???? -modules (see “Sheaves”) is of finite type if for every point ???? ∈ ???? there is an open neighborhood ???? of ???? and a surjective morphism of sheaves of O???? -modules O???????? |???? → F|???? for some ???? ∈ ℕ. 28 | Coherent sheaves – – We say that a sheaf F of O???? -modules is coherent if the following two properties hold: (i) F is of finite type, (ii) for any open subset ???? of ????, any ???? ∈ ℕ, and any morphism of sheaves of O???? modules ???? : O???????? |???? → F|???? , we have that ????????????(????) is of finite type.

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