Sira Helena Gratz 1/4 - Cluster categories and thick subcategories

Finite dimensional algebras may possess few localizations as rings, that is the analogue of the spectrum of a commutative ring does not contain so much information. One can remedy this by instead considering homotopical localizations which give rise to a richer structure. This is equivalent to studying the lattice of thick subcategories of the perfect complexes.

We discuss lattices and where they may arise, eventually focusing on lattices of thick subcategories. Discussing properties of lattices in more detail, we explore which properties are exhibited by lattices of thick subcategories. We discuss examples, including cluster categories, and find out that most possibilities occur, with one major exception: Lattices of thick subcategories are always algebraic, hence a distributive lattice of thick subcategories is automatically a spatial frame.
This leads us to the idea of approximating all lattices of thick subcategories by spaces. As we will have seen at this point, lattices of thick subcategories of triangulated categories coming from representation theory are usually not, on the nose, controlled by a space. Can we still find a space that approximates it in a meaningful way? We discover that the answer is "Yes".

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