351 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
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
The compactness locus of a geometric functor and the formal construction of the Adams isomorphism
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
Stratification in tensor triangular geometry with applications to spectral Mackey functors
We systematically develop a theory of stratification in the context of tensor
triangular geometry and apply it to classify the localizing tensor-ideals of
certain categories of spectral -Mackey functors for all finite groups .
Our theory of stratification is based on the approach of Stevenson which uses
the Balmer-Favi notion of big support for tensor-triangulated categories whose
Balmer spectrum is weakly noetherian. We clarify the role of the
local-to-global principle and establish that the Balmer-Favi notion of support
provides the universal approach to weakly noetherian stratification. This
provides a uniform new perspective on existing classifications in the
literature and clarifies the relation with the theory of Benson-Iyengar-Krause.
Our systematic development of this approach to stratification, involving a
reduction to local categories and the ability to pass through finite \'{e}tale
extensions, may be of independent interest. Moreover, we strengthen the
relationship between stratification and the telescope conjecture. The starting
point for our equivariant applications is the recent computation by
Patchkoria-Sanders-Wimmer of the Balmer spectrum of the category of derived
Mackey functors, which was found to capture precisely the height and height
chromatic layers of the spectrum of the equivariant stable homotopy
category. We similarly study the Balmer spectrum of the category of
-local spectral Mackey functors noting that it bijects onto the height
chromatic layers of the spectrum of the equivariant stable homotopy
category; conjecturally the topologies coincide. Despite our incomplete
knowledge of the topology of the Balmer spectrum, we are able to completely
classify the localizing tensor-ideals of these categories of spectral Mackey
functors.Comment: 61 pages, minor expository change
Stratification and the comparison between homological and tensor triangular support
We compare the homological support and tensor triangular support for `big'
objects in a rigidly-compactly generated tensor triangulated category. We prove
that the comparison map from the homological spectrum to the tensor triangular
spectrum is a bijection and that the two notions of support coincide whenever
the category is stratified, extending work of Balmer. Moreover, we clarify the
relations between salient properties of support functions and exhibit
counter-examples highlighting the differences between homological and tensor
triangular support.Comment: 18 pages, comments welcom
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