[personal profile] posic
http://avva.livejournal.com/2909192.html

Интересно. За других математиков не скажу, а я писал так:

1. Локальные формулы Плюккера связывают метрики на голоморфной кривой в CPn, индуцированные плюккеровыми вложениями присоединенных кривых, с соответствующими кривизнами. Классические (глобальные) формулы Плюккера получаются из локальных интегрированием по кривой [3].

2. Квадратичная алгебра — это градуированная алгебра с образующими степени 1 и соотношениями степени 2. Классическая квадратичная двойственность сопоставляет квадратичной алгебре А с пространством образующих V и пространством соотношений I ⊂ V⊗V квадратичную алгебру А! с образующими из V* и соотношениями I ⊂ V*⊗V*.

3. Ассоциативная градуированная алгебра А = ⊕i=0 Ai над полем k = A0 называется квадратичной, если она порождена A1 и определяется соотношениями градуировки 2, т.е. изоморфна факторалгебре свободной алгебры, порожденной векторным пространством V = A1, по идеалу, порожденному пространством квадратичных соотношений R ⊂ V⊗V; обозначение: А = {V,R}. Двойственная к А квадратичная алгебра — это A! = {V*,R}.

4. Let F be a field, F be its (separable) algebraic closure, and GF = Gal(F/F) be the absolute Galois group. Let l ≠ char F be a prime number; assume that F contains a l-root of unity ζ.

5. Let k be a field and D be a k-linear triangulated category; we will denote, as usually, Homi(X,Y) = Hom(X,Y[i]) and Hom(X,Y) = ⨁i Homi(X,Y). An object E ∈ Ob D is called exceptional if one has Homs(E,E) = 0 for s ≠ 0 and Hom0(E,E) = k.

6. Let F be an arbitrary field and let GF = Gal(F/F) be the Galois group of its (separable) algebraic closure F over it. Two conjectures about the homological properties of the group GF are widely known.

7. Let F be a field and m ≥ 2 be an integer not divisible by the characteristic of F. Consider the absolute Galois group GF = Gal(F/F), where F denotes the (separable) algebraic closure of F.

8. The subject of this book is Semi-Infinite Algebra, or more specifically, Semi-Infinite Homological Algebra. The term “semi-infinite” is loosely associated with objects that can be viewed as extending in both a “positive” and a “negative” direction, with some natural position in between, perhaps defined up to a “finite” movement.

9. A common wisdom says that difficulties arise in Koszul duality because important spectral sequences diverge. What really happens here is that one considers the spectral sequence of a complex endowed with, typically, a decreasing filtration which is not complete.

10. In the paper [2] published in 1987, A. Beilinson formulated his famous conjectures on the properties of hypothetical categories of mixed motivic sheaves over a scheme. In addition to the classical case of motives with rational coefficients, some conjectures about the category of motives with a finite coefficient ring Z/m were proposed there.

11. CDG-algebras (where “C” stands for “curved”) were introduced in connection with nonhomogeneous Koszul duality in [13]. Several years earlier (what we would now call) A-algebras with curvature were considered in [3] as natural generalizations of the conventional A-algebras.

12. В настоящей работе рассматриваются алгебры замкнутых дифференциальных форм на диске, регулярных вне нескольких выбранных координатных гиперплоскостей и имеющих, самое большее, логарифмические особенности вдоль этих гиперплоскостей, по отношению к операции умножения дифференциальных форм. Такие алгебры возникают при изучении смешанных пучков Ходжа–Тейта на гладких алгебраических многообразиях [2].

13. Let K be a field and l ≠ char K be a prime number. The well-known Milnor–Bloch–Kato conjecture claims that the natural morphism of graded Z/l-algebras, called the Galois symbol, or the norm residue homomorphism,

KM(K)/l → ⊕n Hn(GK, μln)

is an isomorphism.

14. A matrix factorization of an element w in a commutative ring R is a pair of square matrices (Φ,Ψ) of the same size, with entries from R, such that both the products ΦΨ and ΨΦ are equal to w times the identity matrix. In the coordinate-free language, a matrix factorization is a pair of finitely generated free R-modules M0 and M1 together with R-module homomorphisms M0 → M1 and M1 → M0 such that both the compositions M0 → M1 → M0 and M1 → M0 → M1 are equal to the multiplication with w.

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