The Balmer spectrum of pseudo-coherent complexes over a discrete valuation ring

TL;DR

This paper computes the Balmer spectrum of pseudo-coherent complexes over a discrete valuation ring, revealing a complex structure related to asymptotic growth conditions in the derived category.

Contribution

It provides the first explicit computation of the Balmer spectrum for pseudo-coherent complexes over a DVR, linking it to distributive lattices of monotonic sequences and their spectral spaces.

Findings

Spectrum coincides with spectral space of a bounded distributive lattice.

Different generation types relate to asymptotic boundedness in homology.

Spectral space complexity exceeds that of perfect complexes.

Abstract

We study the derived category of pseudo-coherent complexes over a noetherian commutative ring, building on prior work by Matsui-Takahashi. Our main theorem is a computation of the Balmer spectrum of this category in the case of a discrete valuation ring. We prove that it coincides with the spectral space associated to a bounded distributive lattice of asymptotic equivalence classes of monotonic sequences of natural numbers. The proof of this theorem involves an extensive study of generation behaviour in the derived category of pseudo-coherent complexes. We find that different types of generation are related to different asymptotic boundedness conditions on the growth of torsion in homology. Consequently, we introduce certain distributive lattices of (equivalence classes of) monotonic sequences where the partial orders are defined by different notions of asymptotic boundedness. These lattices, and the spectral spaces corresponding to them via Stone duality, may be of independent interest. The complexity of these spectral spaces shows that, even in the simplest nontrivial case, the spectrum of pseudo-coherent complexes is vastly more complicated than the spectrum of perfect complexes. From a broader perspective, these results demonstrate that the spectrum of a rigid tensor-triangulated category can expand tremendously when we pass to a (non-rigid) tensor-triangulated category which contains it.