Mutation and the Gabriel spectrum

TL;DR

This work develops a deep link between mutations of pure-injective cosilting objects in compactly generated triangulated categories and the topology of the Gabriel spectrum attached to their hearts. It proves that right cosilting mutation induces a bijection between Gabriel spectra and, on complementary subspaces, yields piecewise homeomorphisms, with the stronger result that in the derived category of a commutative noetherian ring the mutation map is open. The authors connect Gabriel topology, localising subcategories, and HRS-tilting to obtain Matlis-type correspondences and a derived-equivalence uniqueness result for locally noetherian hearts, alongside explicit computations in concrete spectral scenarios. They further classify mutation behavior via discrete and strongly perfect cases, and provide a detailed analysis of intermediate truncated-slice filtrations, linking spectral data to cohomological and localisation-theoretic invariants. Altogether, the paper reveals how mutation acts as a topological bridge between triangulated category mutations and spectral data, with concrete implications for derived categories of commutative noetherian rings and their spectral geometry.

Abstract

Mutations occur in multiple algebraic contexts, often enjoying good combinatorial properties. In this paper we study mutations of pure-injective cosilting objects in compactly generated triangulated categories from a topological point of view. We consider the topologies studied by Gabriel, Burke and Prest on the set of indecomposable injective objects in a Grothendieck abelian category, transfer them to associated cosilting subcategories, and show that, in that context, right mutation induces a homeomorphism on two complementary subspaces. We then improve this result in the context of the derived category of a commutative noetherian ring, showing that right mutation is an open bijection. We end the paper with a detailed analysis of a range of cosilting subcategories over commutative noetherian rings for which the topology is completely known. As a byproduct of this analysis, we obtain that the category of modules over a commutative noetherian ring is the unique locally noetherian Grothendieck category in its derived-equivalence class.

Figures (4)

• Figure 1: The squares represent $\mathop{\mathrm{\mathsf{Spec}}}\nolimits(R)$, with the specialisation-closed subset $V_0$ shaded in a darker gray. The diamonds represent the image of the embedding $\lambda_\ast\colon\mathop{\mathrm{\mathsf{Spec}}}\nolimits((R/\mathfrak{p})_\mathfrak{q})\hookrightarrow \mathop{\mathrm{\mathsf{Spec}}}\nolimits(R)$, depending on the three possible cases: $\mathfrak{p},\mathfrak{q}\in V_0$ (left), $\mathfrak{p}\in V_0^\mathsf{c}$ and $\mathfrak{q}\in V_0$ (middle), $\mathfrak{p},\mathfrak{q}\in V_0^\mathsf{c}$ (right). According to Theorem \ref{['thm:onestep-recipe']}, we have $\mathfrak{p}\preceq\mathfrak{q}$ in the heart $\mathcal{H}$ of the mutation at $V_0$ if and only if the white region is coherent in the diamond. Notice that this is trivially true in the first and third case, in accordance with Lemma \ref{['lemma:bound-topology']}.
• Figure 2: The Gabriel topologies of Examples \ref{['example:twodimtwostep']} and \ref{['example:H1-twodim']}, illustrated by the Hasse quiver of their closure order. We depict the primes layered by height. Moving to the right, we tilt at the specialization closed set consisting of the white points, twice at the maximal prime in this case. After the first step, the primes of height $1$ become open points, while the closure $\mathop{\mathrm{\mathsf{\Lambda}}}\nolimits_\preceq(\mathfrak{m})$ of the maximal prime still contains every minimal prime. After the second tilt, $\{\mathfrak{m}\}$ becomes clopen.
• Figure 3: Illustration of the Gabriel topology of Example \ref{['example:dim3']} (for clarity, in the case of a local domain). In the right tilt at the white points (those of height at least $2$) the prime ideals of height $1$ become open points, while the minimal ideals are still in the closure of every point.
• Figure 4: Here we illustrate how the resulting Gabriel topology differs when right tilting at the Zariski closure of a height one ideal depending on whether the complement is coherent or not in Example \ref{['example:2dim-coherent-noncoherent']} and Example \ref{['example:Nagata-ring']}.

Theorems & Definitions (163)

• Theorem A
• Theorem B
• Theorem C: Theorem \ref{['thm:locally-noetherian-rare']}
• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Corollary 1.3
• proof
• Definition 1.4