Fully Faithful Functors and Dimension

TL;DR

This paper introduces the countable Rouquier dimension for triangulated categories and proves that a fully faithful embedding between derived categories of smooth proper varieties implies an inequality of their dimensions.

Contribution

It defines the countable Rouquier dimension and uses it to establish a dimension inequality for fully faithful embeddings of derived categories.

Findings

Countable Rouquier dimension is a new invariant for triangulated categories.

Fully faithful embeddings of derived categories imply dimension inequalities for varieties.

The result connects categorical embeddings with geometric dimension constraints.

Abstract

We define the countable Rouquier dimension of a triangulated category and use this notion together with Theorem 2 of [Ola21] to prove that if there is a fully faithful embedding $D^b_{coh}(X) \subset D^b_{coh}(Y)$ with $X, Y$ smooth proper varieties, then $\mathrm{dim}(X) \leq \mathrm{dim}(Y)$.