Commutative and Homological Algebra of Incomplete Tambara Functors

Abstract

Equivariant algebra is the study of algebra in the category of Mackey functors. In this setting, Mackey functors play the role of abelian groups and incomplete Tambara functors play the role of commutative rings. The study of incomplete Tambara functors parallels the study of classical commutative algebra in many ways, but there are some striking differences as well. We develop aspects of the homological and commutative algebra of incomplete Tambara functors in this thesis. One of the most notable differences is the fact that free incomplete Tambara functors often fail to be free as Mackey functors, even though free algebras (polynomial rings) are always free as modules. We provide conditions under which a free incomplete Tambara functor is flat as a Mackey functor. When the group is solvable, we show that a free incomplete Tambara functor is flat precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors.