Philosophy of contraherent cosheaves
Nov. 4th, 2023 08:11 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
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Contraherent cosheaves are module objects over algebraic varieties defined by gluing using the colocalization functors. Contraherent cosheaves are designed to be used for globalizing contramodules and contraderived categories for the purposes of Koszul duality and semi-infinite algebraic geometry. One major technical problem associated with contraherent cosheaves is that the colocalization functors, unlike the localizations, are not exact. The reason is that, given a commutative ring homomorphism R → S arising in connection with a typical covering in algebraic geometry, the ring S is usually a flat, but not a projective R-module. We argue that the difference between projective and flat modules, from the standpoint of homological algebra, is generally not that big. It becomes even smaller if one observes that the ring S is often a very flat R-module.
Мораль: тридцать лет прошли. Я по-прежнему размышляю про кошулеву двойственность. Дошел в этих размышлениях уже до того, что написал недавний препринт про accessible categories, например. И это еще не все.
Contraherent cosheaves are module objects over algebraic varieties defined by gluing using the colocalization functors. Contraherent cosheaves are designed to be used for globalizing contramodules and contraderived categories for the purposes of Koszul duality and semi-infinite algebraic geometry. One major technical problem associated with contraherent cosheaves is that the colocalization functors, unlike the localizations, are not exact. The reason is that, given a commutative ring homomorphism R → S arising in connection with a typical covering in algebraic geometry, the ring S is usually a flat, but not a projective R-module. We argue that the difference between projective and flat modules, from the standpoint of homological algebra, is generally not that big. It becomes even smaller if one observes that the ring S is often a very flat R-module.
Мораль: тридцать лет прошли. Я по-прежнему размышляю про кошулеву двойственность. Дошел в этих размышлениях уже до того, что написал недавний препринт про accessible categories, например. И это еще не все.
