Algebraic Geometry: A Concise Dictionary by Elena Rubei

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 commonest phrases during this topic. a few phrases of comparable matters are incorporated. It is helping newcomers that comprehend a few, yet now not all, easy proof of algebraic geometry to keep on with seminars and to learn papers. The dictionary shape makes it effortless and quickly to consult.

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For some ???? and some ideal ????. In particular, if ???? is a smooth point of an algebraic variety ???? of dimension ???? over a field ????, then the completion of O????,???? is isomorphic to ????[[????1 , . . , ???????? ]]. See “Regular rings, smooth points, singular points”. Complexes. Let ???? be a ring. A complex of ????-modules, which is usually written ⋅⋅⋅ ????????−2 G ????????−1 ????????−1 G ???????? ???????? G ????????+1 ????????+1 G ⋅⋅⋅ , is the datum of a sequence of ????-modules ???????? and ????-homomorphisms ???????? : ???????? → ????????+1 such that ????????+1 ∘ ???????? = 0 for any ????.

Remark. Strict complete intersection ⇒ arithmetically Gorenstein ⇒ arithmetically Cohen–Macaulay (the first implication is due to the fact that, if ???? is a strict complete intersection, then its minimal resolution is the Koszul complex). 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.

Let A be an Abelian category (see “Categories”). The homotopy category of A, denoted by ????(A), is the following category: – the objects are the complexes of objects of A; – the morphisms are homotopy equivalence classes of morphisms of complexes (See “Complexes” for the definition of homotopically equivalent morphisms of complexes). We denote by ???????? (A) the subcategory of ????(A) whose objects are the bounded complexes, by ????+ (A) the subcategory whose objects are the complexes bounded below, and by ????− (A) the subcategory whose objects are the complexes bounded above.

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