[Paper Review] Invariants of t-structures and classification of nullity classes

Don Stanley|ArXiv.org|Feb 12, 2006

Algebraic structures and combinatorial models8 references4 citations

TL;DR

This paper constructs a complete invariant for t-structures and nullity classes in the derived category of quasi-coherent sheaves on a Noetherian ring, using increasing functions from integers to specialization-closed subsets of the spectrum. It proves that such invariants classify all t-structures in $D_{ ext{qc}}(R)$, and shows that t-structures in $D(\mathbb{Z})$ form a proper class, precluding any finite or set-based classification.

ABSTRACT

We construct an invariant of t-structures on the derived category of a Noetherian ring. This invariant is complete when restricting to the category of quasi-coherent complexes, and also gives a classification of nullity classes with the same restriction. On the full derived category of $\mathbb Z$ we show that the class of distinct t-structures do not form a set.

Motivation & Objective

• To construct a complete invariant for t-structures on the derived category of a Noetherian ring.
• To classify nullity classes in $D_{\text{qc}}(R)$, the derived category of quasi-coherent complexes.
• To show that t-structures in $D(\mathbb{Z})$ do not form a set, indicating the impossibility of a finite or set-theoretic classification.
• To extend Bousfield's classification philosophy from spectra to derived categories of rings.
• To investigate the structural differences between t-structures in $D(R)$ and localizing subcategories, highlighting increased complexity.

Proposed method

• Define an invariant $\phi(\mathcal{A})$ for a nullity class $\mathcal{A}$, mapping $\mathbb{Z}$ to specialization-closed subsets of $\operatorname{Spec}(R)$.
• For each $n$, compute $\phi(\mathcal{A})(n)$ as the support of the thick subcategory generated by $\tau^{\geq -n}\mathcal{A}$.
• Use the Hopkins-Neeman classification of thick subcategories of perfect complexes via supports in $\operatorname{Spec}(R)$.
• Apply the nullification functor $P_E$ to construct aisles from objects $E$, and study their restrictions to $D_{\text{qc}}(R)$.
• Use Shelah's rigid systems of abelian groups to construct uncountably many distinct $P_E$-localizations.
• Prove that distinct rigid systems yield distinct nullity classes via non-vanishing $P_E$-maps, establishing proper class size.

Experimental results

Research questions

• RQ1 Can t-structures on $D_{\text{qc}}(R)$ be completely classified using spectral invariants?
• RQ2 What conditions must a function $\phi: \mathbb{Z} \to \{\text{specialization-closed subsets of } \operatorname{Spec}(R)\}$ satisfy to arise from a t-structure?
• RQ3 Is the class of t-structures in $D(\mathbb{Z})$ a proper class, implying no set-theoretic classification is possible?
• RQ4 Do the restrictions of $\overline{C(E)}$-type aisles to $D_{\text{qc}}(R)$ remain aisles, and under what conditions?
• RQ5 Can the invariant $\phi$ be used to classify all nullity classes in $D_{\text{qc}}(R)$, and is it complete?

Key findings

• The invariant $\phi$ is an order-preserving bijection between nullity classes in $D_{\text{qc}}(R)$ and increasing functions from $\mathbb{Z}$ to specialization-closed subsets of $\operatorname{Spec}(R)$.
• This classification implies that $\phi$ is a complete invariant for t-structures in $D_{\text{qc}}(R)$.
• The class of t-structures in $D(\mathbb{Z})$ is a proper class, not a set, due to the existence of uncountably many rigid systems of abelian groups.
• For each rigid system $\{A_\alpha\}$ of abelian groups, the nullity classes $\overline{C(A_\alpha)}$ in $D(\mathbb{Z})$ are pairwise distinct.
• The same construction yields uncountably many distinct nullity classes in the triangulated category of spectra and in topological spaces.
• The conjecture is proposed that a nullity class is an aisle if and only if it satisfies the condition that if $p' \in \phi(\mathcal{A})(n)$ and $p$ is maximal over $p'$, then $p \in \phi(\mathcal{A})(n+1)$.