## 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.

### 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

## 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.