Let
be a map of
–groups, where
denotes the Picard space
of an
–ring spectrum
. We determine the
tensor
of the Thom
––algebra
with a space
; when
is the circle, the tensor with
is topological Hochschild
homology over
. We use the theory
of localizations of
–categories
as a technical tool: we contribute to this theory an
–categorical
analogue of Day’s reflection theorem about closed symmetric monoidal
structures on localizations, and we prove that, for a smashing localization
of the
–category of presentable
–categories, the free
–local presentable
–category on a small
simplicial set
is given by
presheaves on
valued
on the
–localization
of the
–category
of spaces.
If
is a pointed
space, a map
of
–ring spectra
satisfies –basechange if
is the
pushout of
along
.
Building on a result of Mathew, we prove that if
is étale then it satisfies
–base change provided
is connected. We also
prove that
satisfies
–base change provided
the multiplication map of
is an equivalence. Finally, we prove that, under some hypotheses,
the Thom isomorphism of Mahowald cannot be an instance of
–base
change.
