اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدIntroduction to Derived Categories
161
0
0.0
(
0
)
تأليف
Amnon Yekutieli
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the old homological algebra (of derived functors between abelian categories) was established. This new homological algebra, of derived categories and derived functors between them, provides a significantly richer and more flexible machinery than the old homological algebra. For instance, the important concepts of dualizing complex and tilting complex do not exist in the old homological algebra. This paper is an edited version of the notes for a two-lecture minicourse given at MSRI in January 2013. Sections 1-5 are about the general theory of derived categories, and the material is taken from my manuscript A Course on Derived Categories (available online). Sections 6-9 are on more specialized topics, leaning towards noncommutative algebraic geometry.
قيم البحث
اقرأ أيضاً
In nature, one observes that a K-theory of an object is defined in two steps. First a structured category is associated to the object. Second, a K-theory machine is applied to the latter category to produce an infinite loop space. We develop a genera
l framework that deals with the first step of this process. The K-theory of an object is defined via a category of locally trivial objects with respect to a pretopology. We study conditions ensuring an exact structure on such categories. We also consider morphisms in K-theory that such contexts naturally provide. We end by defining various K-theories of schemes and morphisms between them.
We apply the Auslander-Buchweitz approximation theory to show that the Iyama and Yoshinos subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a
result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitzs triangle equivalence from Iwanaga-Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitzs triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga-Gorenstein rings and Gorenstein algebras
We show that an A-infinity algebra structure can be transferred to a projective resolution of the complex underlying any A-infinity algebra. Under certain connectedness assumptions, this transferred structure is unique up to homotopy. In contrast to
the classical results on transfer of A-infinity structures along homotopy equivalences, our result is of interest when the ground ring is not a field. We prove an analog for A-infinity module structures, and both transfer results preserve strict units.
Let $mathcal{H}$ be a hereditary abelian category over a field $k$ with finite dimensional $operatorname{Hom}$ and $operatorname{Ext}$ spaces. It is proved that the bounded derived category $mathcal{D}^b(mathcal{H})$ has a silting object iff $mathcal
{H}$ has a tilting object iff $mathcal{D}^b(mathcal{H})$ has a simple-minded collection with acyclic $operatorname{Ext}$-quiver. Along the way, we obtain a new proof for the fact that every presilting object of $mathcal{D}^b(mathcal{H})$ is a partial silting object. We also consider the question of complements for pre-simple-minded collections. In contrast to presilting objects, a pre-simple-minded collection $mathcal{R}$ of $mathcal{D}^b(mathcal{H})$ can be completed into a simple-minded collection iff the $operatorname{Ext}$-quiver of $mathcal{R}$ is acyclic.
We define an obstruction to the formality of a differential graded algebra over a graded operad defined over a commutative ground ring. This obstruction lives in the derived operadic cohomology of the algebra. Moreover, it determines all operadic Mas
sey products induced on the homology algebra, hence the name of derived universal Massey product.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
