Timeline for Semisimplicity of the category of coherent sheaves?

8 events

when toggle format	what		by	license	comment
Apr 13, 2019 at 15:40	history	edited	user25309	CC BY-SA 4.0	edited body
Apr 13, 2019 at 15:40	comment	added	Leonid Positselski		Oh, yes. Then you are right.
Apr 13, 2019 at 15:40	history	edited	user25309	CC BY-SA 4.0	added 174 characters in body
Apr 13, 2019 at 15:38	comment	added	user25309		The issue is maybe the correct definition of "semisimple". If I look at ncatlab.org/nlab/show/semisimple+category or en.wikipedia.org/wiki/Semi-simplicity , the definition is that every object is a direct sum of finitely many simple objects. If we remove the condition "finitely many", I agree with your comments.
Apr 13, 2019 at 15:24	comment	added	Leonid Positselski		... So, in particular, the structure sheaf over such a scheme $X$ is the infinite direct sum, and at the same time the infinite product, of the one-dimensional (skyscraper) sheaves $k_x$ sitting at the points $x\in X$. These skyscraper sheaves are simple objects.
Apr 13, 2019 at 15:19	comment	added	Leonid Positselski		No, the category of coherent sheaves over an infinite disjoint union of reduced points is semisimple abelian, in fact. It is equivalent to the infinite Cartesian product of the categories of finite-dimensional vector spaces over the related fields. Every object in it is naturally the direct sum of its components sitting at the points, and at the same time it is the infinite product of the same components.
Apr 13, 2019 at 15:10	vote	accept	CommunityBot
Apr 13, 2019 at 15:05	history	answered	user25309	CC BY-SA 4.0