800 research outputs found
Higher comparison maps for the spectrum of a tensor triangulated category
For each object in a tensor triangulated category, we construct a natural
continuous map from the object's support---a closed subset of the category's
triangular spectrum---to the Zariski spectrum of a certain commutative ring of
endomorphisms. When applied to the unit object this recovers a construction of
P. Balmer. These maps provide an iterative approach for understanding the
spectrum of a tensor triangulated category by starting with the comparison map
for the unit object and iteratively analyzing the fibers of this map via
"higher" comparison maps. We illustrate this approach for the stable homotopy
category of finite spectra. In fact, the same underlying construction produces
a whole collection of new comparison maps, including maps associated to (and
defined on) each closed subset of the triangular spectrum. These latter maps
provide an alternative strategy for analyzing the spectrum by iteratively
building a filtration of closed subsets by pulling back filtrations of affine
schemes.Comment: 31 page
A note on triangulated monads and categories of module spectra
Consider a monad on an idempotent complete triangulated category with the
property that its Eilenberg-Moore category of modules inherits a triangulation.
We show that any other triangulated adjunction realizing this monad is
'essentially monadic', i.e. becomes monadic after performing the two evident
necessary operations of taking the Verdier quotient by the kernel of the right
adjoint and idempotent completion. In this sense, the monad itself is
'intrinsically monadic'. It follows that for any highly structured ring
spectrum, its category of homotopy (a.k.a. naive) modules is triangulated if
and only if it is equivalent to its category of highly structured (a.k.a.
strict) modules.Comment: 5 page
The spectrum of the equivariant stable homotopy category of a finite group
We study the spectrum of prime ideals in the tensor-triangulated category of
compact equivariant spectra over a finite group. We completely describe this
spectrum as a set for all finite groups. We also make significant progress in
determining its topology and obtain a complete answer for groups of square-free
order. For general finite groups, we describe the topology up to an unresolved
indeterminacy, which we reduce to the case of p-groups. We then translate the
remaining unresolved question into a new chromatic blue-shift phenomenon for
Tate cohomology. Finally, we draw conclusions on the classification of thick
tensor ideals.Comment: 34 pages, to appear in Invent. Mat
Grothendieck-Neeman duality and the Wirthm\"uller isomorphism
We clarify the relationship between Grothendieck duality \`a la Neeman and
the Wirthm\"uller isomorphism \`a la Fausk-Hu-May. We exhibit an interesting
pattern of symmetry in the existence of adjoint functors between compactly
generated tensor-triangulated categories, which leads to a surprising
trichotomy: There exist either exactly three adjoints, exactly five, or
infinitely many. We highlight the importance of so-called relative dualizing
objects and explain how they give rise to dualities on canonical subcategories.
This yields a duality theory rich enough to capture the main features of
Grothendieck duality in algebraic geometry, of generalized Pontryagin-Matlis
duality \`a la Dwyer-Greenlees-Iyengar in the theory of ring spectra, and of
Brown-Comenetz duality \`a la Neeman in stable homotopy theory.Comment: 36 pages. Minor revision due to referee's comments. Added Examples
3.27, 4.8 & 4.9. To appear in Compositio Mat
A characterization of finite \'etale morphisms in tensor triangular geometry
We provide a characterization of finite \'etale morphisms in tensor
triangular geometry. They are precisely those functors which have a
conservative right adjoint, satisfy Grothendieck--Neeman duality, and for which
the relative dualizing object is trivial (via a canonically-defined map).Comment: 21 page
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