# EP/I034017/1 - Lambda-structures in stable categories

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Research Perspectives grant details from EPSRC portfolio

Principal Investigator - Mathematical Sciences, University of Liverpool

Start Date

05/2012

End Date

05/2015

Value

£97,203

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Motives are objects encoding arithmetic and geometry at the same time. This project is about derived symmetric powers in the Morel-Voevodsky 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 Knneth rule expresses the multiplicative n-th power of a sum A + B in terms of the sum of products of multiplicative i-th and (n-i)-th powers of A and B respectively. A categorified lambda-structure in a symmetric monoidal triangulated category T is a set of endofunctors of T, indexed by non-negative integers, which behave similarly to a usual algebraic lambda-structure in a commutative ring. In particular, the values of the n-th endofunctor on the vertices in a distinguished triangle are related by means of a tower (called Knneth tower) of morphisms in T whose cones can be computed by the Knneth 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 lambda-structure 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. Lambda-structures 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 Morel-Voevodsky motivic stable category, such lambda-structures 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 lambda-structure of left derived symmetric powers in the Morel-Voevodsky 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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Name: Lambda-structures in stable categories - EP/I034017/1

Start Date: 2012-05-21T00:00:00+00:00

End Date: 2015-05-20T00:00:00+00:00

Organization: University of Liverpool

Description: Motives are objects encoding arithmetic and geometry at the same time. This project is about derived symmetric powers in the Morel-Voevodsky motivic stable category over a field.In mathematics it is important to relate additive and multiplicative operation ...