Reconstructing Tensor-Triangulated Categories

Tensor-triangulated categories naturally occur in various settings, for example, as the derived category of a commutative ring, the category of (possibly equivariant) cohomology theories, or the stable module category of a Frobenius ring. To any such category, one can assign the so-called Balmer spectrum, which is a categorification of the Zariski spectrum of a ring. I will report on joint work with J.P.C. Greenlees which provides a machinery to build up the unit object of the category from smaller building blocks. These building blocks are formed by taking localised completions of the unit at primes of the Balmer spectrum. Once one has this reconstruction result for the unit object, various models can be considered which fracture the category in question. Not only does this result give an insight into the structure of the category, but it usually provides a convenient setting to do calculations in.