## University of Sydney Algebra Seminar

# Amnon Neeman (ANU)

## Friday 7 September, 12-1pm, Place: Carslaw 375

## Approximable triangulated categories

We will begin the talk with five new theorems in algebraic geometry, statements about the derived categories $D^b_{coh}(X)$ and $D^{perf}(X)$ for schemes $X$. Each of these theorems represents a major improvement over what was known. For each of the five theorems special cases go back to results of Bondal and Van den Bergh, Rickard and Rouquier. The short summary is that the old results all assumed equal characteristic, the new results are the first to work in mixed characteristic. The more detailed story is that, even in equal characteristic, our results are a sharp improvement over what was known.

It turns out that all these results are relatively straightforward corollaries of the theorem that the category $D_{qc}(X)$ is approximable. This is a new notion we will explain, and then illustrate its power.