Mixed Tate motives with finite coefficients
Как было обещано, переношу с бумаги в компьютер набросок плана статьи.
0. Introduction
[Not sure what should go here. Perhaps some discussion of Beilinson's conjectures emphasizing the abelian category of motives, t-structures of the derived type/silly filtrations/K(π,1)-conjecture, abelian vs. exact cores/subcategories, combinations of characteristics of the basic field and coefficients, presence/absence of l-roots of unity, the role of Koszulity in these cases, what else?]
1. Toy example: the graded = t-structure case
[Given an exceptional sequence, when the related t-structure exists?; assuming the Ext-s are on the diagonal, when this t-structure is of the derived type?]
2. Exact categories of filtered objects
[The exact category of mixed Tate motives with finite coefficients is equivalent to the category of filtered Galois-modules without any assumptions beyond the existence of the etale realization.]
3. Filtered bar construction
[Computation of Ext's in the exact category of filtered Galois-modules. Silly filtrations for mixed Tate motives with finite coefficients + MBK-conjecture <=> Koszulity conjecture + low-degree part of MBK.]
Appx A?. Exact categories
Appx B. Exact subcategories of triangulated categories
Appx C. Silly filtrations
[Abstract definition and general results not bases on any assumptions about vanishing of the negative Ext's.]
Appx D. Triangulated categories of algebraic origin.
[Generalization of the applications of Beilinson's filtered triangulated categories business from the t-structures case to the exact subcategories case.]
+ связь? глупых фильтраций с K(π,1)-гипотезой в формулировке через бар-конструкцию DG-алгебры, вычисляющей мотивные когомологии (вопрос Блоха)
+ глупые фильтрации для мотивов Артина-Тейта, связанных с фиксированным циклическим расширением и теорема Гильберта 90 для Милноровских K-групп
К Section 2: два обобщения в разные стороны
- мотивы Артина-Тейта
- мотивы Z/l^k(i) над Z/l^m
0. Introduction
[Not sure what should go here. Perhaps some discussion of Beilinson's conjectures emphasizing the abelian category of motives, t-structures of the derived type/silly filtrations/K(π,1)-conjecture, abelian vs. exact cores/subcategories, combinations of characteristics of the basic field and coefficients, presence/absence of l-roots of unity, the role of Koszulity in these cases, what else?]
1. Toy example: the graded = t-structure case
[Given an exceptional sequence, when the related t-structure exists?; assuming the Ext-s are on the diagonal, when this t-structure is of the derived type?]
2. Exact categories of filtered objects
[The exact category of mixed Tate motives with finite coefficients is equivalent to the category of filtered Galois-modules without any assumptions beyond the existence of the etale realization.]
3. Filtered bar construction
[Computation of Ext's in the exact category of filtered Galois-modules. Silly filtrations for mixed Tate motives with finite coefficients + MBK-conjecture <=> Koszulity conjecture + low-degree part of MBK.]
Appx B. Exact subcategories of triangulated categories
Appx C. Silly filtrations
[Abstract definition and general results not bases on any assumptions about vanishing of the negative Ext's.]
Appx D. Triangulated categories of algebraic origin.
[Generalization of the applications of Beilinson's filtered triangulated categories business from the t-structures case to the exact subcategories case.]
+ связь? глупых фильтраций с K(π,1)-гипотезой в формулировке через бар-конструкцию DG-алгебры, вычисляющей мотивные когомологии (вопрос Блоха)
+ глупые фильтрации для мотивов Артина-Тейта, связанных с фиксированным циклическим расширением и теорема Гильберта 90 для Милноровских K-групп
К Section 2: два обобщения в разные стороны
- мотивы Артина-Тейта
- мотивы Z/l^k(i) над Z/l^m