## tt-geometry in Trondheim

- Trondheim, Norway

This is the project page for the project `tt-geometry in Trondheim', supported by the Trond Mohn Foundation. For background reading on tensor-triangulated geometry, a good introduction can be found in the survey papers of Balmer:

A recent online lecture given by Drew Heard can be found here.

The image on the left is due to Balmer and Sanders, and shows a part of a computation of the tensor-triangulated geometry of equivariant homotopy theory.

### Members

Drew Heard, project leader.

Clover May, postdoc (August 2021 - ).

Torgeir AambĂ¸, PhD (August 2021 - ).

### Publications

## Rational local systems and connected finite loop spaces. - ** Drew Heard.** *Glasgow Mathematical Journal * (arXiv link)

Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree \(G\)-spectra. More generally, we show that if \(K\) is a closed subgroup of a compact Lie group \(G\) such that the Weyl group \(W_GK\) is connected, then a certain category of rational \(G\)-spectra `at \(K\)' has an algebraic model. For example, when \(K\) is the trivial group, this is just the category of rational cofree \(G\)-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

### Preprints

## Stratification and the comparison between homological and tensor triangular support Tobias Barthel, ** Drew Heard**, and Beren Sanders.

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.

## Stratification in tensor triangular geometry with applications to spectral Mackey functors - Tobias Barthel, **Drew Heard** and Beren Sanders.

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 G-Mackey functors for all finite groups G. 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 \(0\) and height \(\infty\) chromatic layers of the spectrum of the equivariant stable homotopy category. We similarly study the Balmer spectrum of the category of \(E(n)\)-local spectral Mackey functors noting that it bijects onto the height \(\le n\) 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.

## On conjectures of Hovey–Strickland and Chai - Tobias Barthel, ** Drew Heard**, and Niko Naumann.

We prove the height two case of a conjecture of Hovey and Strickland that provides a \(K(n)\)-local analogue of the Hopkins–Smith thick subcategory theorem. Our approach first reduces the general conjecture to a problem in arithmetic geometry posed by Chai. We then use the Gross &ndash Hopkins period map to verify Chai's Hope at height two and all primes. Along the way, we show that the graded commutative ring of completed cooperations for Morava \(E\)-theory is coherent, and that every finitely generated Morava module can be realized by a \(K(n)\)-local spectrum as long as \(2p-2>n^2+n\). Finally, we deduce consequences of our results for descent of Balmer spectra.