WHEN PROJECTIVE DOES NOT IMPLY FLAT,
AND OTHER HOMOLOGICAL ANOMALIES
L. GAUNCE LEWIS, JR.
Abstract.The category MG of Mackey functors for a group G carries a
symmetric monoidal closed structure. The -product providing this struct*
*ure
encodes the Frobenius axiom, which describes the interaction of inductio*
*n and
multiplication in Mackey functor rings. Mackey functors are of interest*
* in
equivariant homotopy theory since good equivariant cohomology theories a*
*re
Mackey functor valued. In this context, the -product is useful not only*
* be-
cause it encodes the interaction between induction and the cup product, *
*but
also because of the role it plays in the not yet fully understood univer*
*sal coef-
ficient and K"unneth formulae. This role makes it important to know whet*
*her
projective objects in MG are flat, and whether the -product of projecti*
*ve
objects in MG is projective. In the most familiar symmetric monoidal abe*
*lian
categories, the tensor product obviously interacts appropriately with pr*
*ojec-
tive objects. However, the -product for MG need not be so well behaved.*
* For
example, if G is O(n), projectives need not be flat in MG and the -prod*
*uct
of projective objects need not be projective. This misbehavior complicat*
*es the
search for full strength equivariant universal coefficient and K"unneth *
*formulae.
These questions about the interaction of the tensor product with proje*
*ctive
objects can be regarded as compatibility conditions which may be satisfi*
*ed by a
symmetric monoidal closed category M. The primary purpose of this articl*
*e is
to investigate these, and related, conditions. Our focus is on functor c*
*ategories
whose monoidal structures arise in a fashion described by Day. Conditions
are given under which such a structure interacts appropriately with proj*
*ective
objects. Further, examples are given to show that, when these conditions*
* aren't
met, this interaction can be quite bad. These examples were not fabricat*
*ed
to illustrate the abstract possibility of misbehavior. Rather, they are *
*drawn
from the literature. In particular, MG 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 w*
*hich
are widely used in the study of classifying spaces of finite groups.
____________
Date: April 25, 1999.
1991 Mathematics Subject Classification.
Primary 18D10, 18D15, 18G05, 55M35, 55N91, 57S15
Secondary 18E10, 55P91, 57S10.
Department of Mathematics, Syracuse University, Syracuse NY 13244-1150
E-mail address: gaunce@ichthus.syr.edu