EPSRC Reference: 
EP/I034017/1 
Title: 
Lambdastructures in stable categories 
Principal Investigator: 
Guletskii, Dr V 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematical Sciences 
Organisation: 
University of Liverpool 
Scheme: 
Standard Research 
Starts: 
21 May 2012 
Ends: 
20 May 2015 
Value (£): 
97,203

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Motives are objects encoding arithmetic and geometry at the same time. This project is about derived symmetric powers in the MorelVoevodsky motivic stable category over a field.In mathematics it is important to relate additive and multiplicative operations in an algebraic structure, or in a category of geometrical or arithmetical nature. For example, the Kuenneth rule expresses the multiplicative nth power of a sum A + B in terms of the sum of products of multiplicative ith and (ni)th powers of A and B respectively. A categorified lambdastructure in a symmetric monoidal triangulated category T is a set of endofunctors of T, indexed by nonnegative integers, which behave similarly to a usual algebraic lambdastructure in a commutative ring. In particular, the values of the nth endofunctor on the vertices in a distinguished triangle are related by means of a tower (called Kuenneth tower) of morphisms in T whose cones can be computed by the Kuenneth rule. Our first aim is to show that if T is an abstract stable homotopy category, i.e. the homotopy category of symmetric spectra over a nice simplicial symmetric monoidal model category C, then left derived symmetric powers do exist in T, and they give a lambdastructure in the above sense, provided some natural symmetrizability assumption on cofibrations in C. Left derived symmetric powers will be homotopical symmetric powers, i.e. homotopy colimits of the action of symmetric groups on monoidal powers. Lambdastructures of left derived symmetric powers bring a powerful computational tool to compute homotopical symmetric powers in many stable homotopy categories. For example, this works well in topology, when T is the homotopy category of the category of topological symmetric spectra.Being applied in the MorelVoevodsky motivic stable category, such lambdastructures encode deep geometrical and arithmetical properties of algebraic varieties over the ground field, which do not appear in the topological setting. In particular, the relation symmetric powers with the contraction of the affine line to a point, and with operations arising from symmetric powers of algebraic varieties over a field, attract our special attention in this project. Thus, we aim to construct and to study a lambdastructure of left derived symmetric powers in the MorelVoevodsky motivic stable category, and to use it in order to discover completely new phenomena in arithmetic algebraic geometry and motivic theory.

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Organisation Website: 
http://www.liv.ac.uk 