Stability conditions on affine Noetherian schemes

TL;DR

This paper establishes that the existence of locally finite stability conditions on the derived category of coherent sheaves on an affine Noetherian scheme is equivalent to the scheme being zero-dimensional, and explores the homotopy equivalence of stability condition spaces.

Contribution

It proves the equivalence between the existence of stability conditions and zero-dimensionality of affine Noetherian schemes, and shows the homotopy equivalence of stability condition spaces on related categories.

Findings

Stability conditions exist iff the scheme has dimension zero.

Spaces of stability conditions on different categories are homotopy equivalent.

Provides a characterization of stability conditions on affine Noetherian schemes.

Abstract

We show that the existence of locally finite stability conditions on the bounded derived category $\mathbf{D}^{b}(X)$ of coherent sheaves on an affine Noetherian scheme $X$ is equivalent to $\dim X=0$. We also study the spaces of stability conditions on the category of morphisms $\mathbf M_{X}$ in the derived category of the scheme $X$ and show that the spaces of stability conditions on $\mathbf{D}^{b}(X)$ and $\mathbf{M}_{X}$ are homotopy equivalent to each other.