Download e-book for kindle: Algebraic geometry. A concise dictionary by Elena Rubei

By Elena Rubei

ISBN-10: 3110316226

ISBN-13: 9783110316223

Algebraic geometry has a classy, tricky language. This e-book encompasses 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 comparable matters are incorporated. It is helping newcomers that recognize a few, yet no longer all, simple evidence of algebraic geometry to keep on with seminars and to learn papers. The dictionary shape makes it effortless and quickly to refer to.

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Let ???? be an object of C. Let ℎ???? : C → S be the covariant functor defined by ℎ???? (????) = ????????????C (????, ????) Categories | 21 for any ???? object of C and sending an arrow ???? : ???? → ???? to the arrow ????????????C (????, ????) → ????????????C (????, ????) given by the composition with ????. Let ℎ???? : C → S be the controvariant functor defined by ℎ???? (????) = ????????????C (????, ????) for any ???? object of C and sending an arrow ???? : ???? → ???? to the arrow ????????????C (????, ????) → ????????????C (????, ????) given by the composition with ????. Sometimes ℎ???? and ℎ???? are denoted respectively by ????????????(????, −) and ????????????( −, ????).

26 | Cohen–Macaulay, Gorenstein, (arithmetically -,-) One can prove that a Cohen–Macaulay algebraic set is equidimensional: for instance, the union of a line and a plane meeting in a point is not Cohen–Macaulay. , removing a subvariety of codimension ≥ 2 cannot disconnect it. For instance two surfaces meeting in a point in a space of dimension 4 cannot be Cohen–Macaulay. Example of Cohen–Macaulay algebraic sets: (locally) complete intersections. (2) An equidimensional projective algebraic set ???? of codimension ???? in ℙ???????? is said arithmetically Cohen–Macaulay if the minimal free resolution (see “Minimal free resolutions”) of the sheaf O???? has length ???? (which is the minimal possible length): 0 → F???? → F????−1 → ⋅ ⋅ ⋅ → F1 → Oℙ???? → O???? → 0.

Definition. , (???? + ???????? ) ∘ ???? = ???? ∘ ???? + ???????? ∘ ????, – – ???? ∘ (???? + ???????? ) = ???? ∘ ???? + ???? ∘ ???????? for all morphisms ????, ???????? , ????, ???????? . “Zero object”. There exists an object 0 of C such that ????????????C (0, 0) is the zero group. “Direct sum”. For any two objects ????1 , ????2 in C there is an object ???? in C and arrows ???????? : ???????? → ????, ???????? : ???? → ???????? for ???? = 1, 2 such that ???????? ∘ ???????? = ???????????? for ???? = 1, 2, ????2 ∘ ????1 = ????1 ∘ ????2 = 0 and ????1 ∘ ????1 + ????2 ∘ ????2 = ???????? . Definition. Let C and D be two additive categories.

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Algebraic geometry. A concise dictionary by Elena Rubei


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