2018 Fiscal Year Research-status Report
Cluster theory through derived categories and self-injective algebras
| Project/Area Number |
18K03238
|
|---|
| Research Institution | Nagoya University |
|---|
Principal Investigator |
ダルポ エリック 名古屋大学, 多元数理科学研究科(国際), 准教授 (00785959)
|
|---|
| Project Period (FY) |
2018-04-01 – 2021-03-31
|
| Keywords | cluster category / derived category / cluster tilting |
|---|
| Outline of Annual Research Achievements |
In the fiscal year 2018, I have studied n-cluster-tilting in cluster categories and derived categories. The conjecture given in the research plan, that every n-cluster-tilting object in the n-cluster category of a herediary algebra comes from a silting object in the corresponding derived category, has been proved.
Additionally, I have investigated functors between cluster categories of Dynkin type. The starting point for this is that indecomposable object is higher cluster categories of type A can be represented by certain diagonals in polygons. If k, l, m and n are integers such that (m-1)/(n-1) = (l+1)/(k+1) and this number is an integer, then the diagonals representing objects in the m-cluster category of type A_k also represent objects in the n-cluster category of type A_l (but not vice versa). This gives an embedding, on the level of objects, from the m-cluster category of A_k to the n-cluster category of type A_l. During the last fiscal year, we have shown that this embedding comes from a functor between the two cluster categories. The functor in questions is not an embedding of categories. However, it can be shown that it factors as composition of two functors, the first of which is an embedding, and the second is a covering functor.
|
| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Substantial results in the direction of the research plan have been obtained.
|
| Strategy for Future Research Activity |
The first goal in the new fiscal year will be to work our the ramifications of the results obtained in the previous year, and to extend the results to higher cluster categories of other (Dynkin) types. After that, work will continue with the next part of the research plan: realising higher cluster categories as stable module categories. In this direction, some recent developments in the field have shown that it is possible to construct triangulated categories with an n-cluster-tilting object as stable categories of Cohen-Macaulay modules of certain Iwanaga-Gorenstein algebras. It seems likely that some of the categories obtained in this way can be used to construct higher cluster categories, generalising the approach described in the research plan.
|
| Causes of Carryover |
Primarily, travel costs were lower than expected during the year, with only one international travel. For the fiscal year 2019, two international conference and research visits are planned, whence the total expenditure for these two years will be approximately in line with the budget.
|
