# When projective does not imply flat, and other homological anomalies

## L. Gaunce Lewis, Jr.

If $\cal M$ is both an abelian category and a symmetric monoidal closed category, then it is natural to ask whether projective objects in $\cal M$ are flat, and whether the tensor product of two projective objects is projective. In the most familiar such categories, the answer to these questions is obviously yes. However, the category $\cal M_G$ of Mackey functors for a compact Lie group $G$ is a category of this type in which projective objects need not be so well-behaved. This category is of interest since good equivariant cohomology theories are Mackey functor valued. The tensor product on $\cal M_G$ is important in this context because of the role it plays in the not yet fully understood universal coefficient and K\"{u}nneth formulae. This role makes the relationship between projective objects and the tensor product especially critical. Unfortunately, if $G$ is, for example, $O(n)$, then projectives need not be flat in $\cal M_G$ and the tensor product of projective objects need not be projective. This misbe haviorcomplicates the search for full strength equivariant universal coefficient and K\"{u}nneth formulae.

The primary purpose of this article is to investigate these questions about the interaction of the tensor product with projective objects in symmetric monoidal abelian categories. Our focus is on functor categories whose monoidal structures arise in a fashion described by Day. Conditions are given under which such a structure interacts appropriately with projective objects. Further, examples are given to show that, when these conditions aren't met, this interaction can be quite bad. These examples were not fabricated to illustrate the abstract possibility of misbehavior. Rather, they are drawn from the literature. In particular, $\cal M_G$ is badly behaved not only for the groups $O(n)$, but also for the groups $SO(n)$, $U(n)$, $SU(n)$, $Sp(n)$, and $Spin(n)$. Similar misbehavior occurs in two categories of global Mackey functors which are widely used in the study of classifying spaces of finite groups. Given the extent of the homological misbehavior in Mackey functor categories described here, it is reasonable to expect that similar problems occur in other functor categories carrying symmetric monoidal closed structures provided by Day's machinery.

Keywords: closed category, symmetric monoidal category, Mackey functor, projective, flat, compact Lie group.

1991 MSC: Primary: 18D10, 18D15, 18G05, 19A22, 55M35, 55N91, 57S15; Secondary: 18E10, 55P91, 57S10.

Theory and Applications of Categories, Vol. 5, 1999, No. 9, pp 202-250.

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