而上个学期他讲的是他的reconstruction theorem,也就是说如果把一个abelian category看成是一个category of quasi coherent sheaf on scheme,那么我们可以geometric realization这个abelian category,得到 geometric center和scheme 同胚。而他说事实上,Grothendieck and Gabriel 早就有这方面的想法,Gabriel的重要工作是...
We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field $k$. We also compute the Grothendieck group of the category of $A$-isotypic abelian varieties, for any simple abelian variety $A$, assuming $k$ has characteristic 0, and for any ...
In Section 4 we prove the higher version of Theorem A. Lastly, in Section 5 we apply our results to the classical setting and define the X-Caldero-Chapoton map. 1.1. Conventions and notation (i) Ab denotes the category of all abelian groups. (ii) By an additive subcategory, we will ...
The Grothendieck group is an interesting invariant of an exact category. It induces a decategorication from the category of essentially small exact categories (whose morphisms are exact functors) to the category of abelian groups. Similarly, the triangulated Grothendieck group induces a decategoricatio...
The category of sheaves of Abelian groups on T is an Abelian category with enough injectives, and Grothendieck defined the cohomology groups H*(T, F) to be the right derived functors of F→ F(X). When X is a topological space and T is the poset of open subspaces then sheaf has its...
A key idea of Grothendieck is that one should not try to define what a motive is. Rather, one should try to define the category of motives.It should be an abelian category with finite dimensional rational vector spaces asHomgroups. Crucially, it should admit a tensor product, needed to sta...
Of course, I thought to just consider the case of the bounded derived categoryDb(A)Db(A)of an Abelian categoryAA, and the heart of the naturaltt-structure first, and then let that inspire the approach. (That is, complexes up to quasi-iso and the heart of complexes concentrated Though...
1. Preliminaries About Abelian Category 2. Abelian Tensor Category 3. The Tor Functors 4. Grothendieck rings and Grothendieck modules References 展开▼ 摘要 The Grothendieck group plays an important role in many fields, such as algebra,number theory and algebraic geometry. It may have a ring struc...
Abstract : For an abelian category C, the Grothendieck group K(O)(C) of C is an abelian group which solves the universal problem of finding 'additive' maps of Obj C into abelian groups. The Whitehead group K(1) C is the abelian group which furnishes a 'universal determinant theory' for...
We want to study the Grothendieck group K,, (mod R) of the category of finitely generated modules over the invariant ring R=A". Auslander and Reiten showed in their fundamental paper [AR] that the Grothendieck group of the ring of invariants of a (not necessarily abelian) group G is ...