arXane › Mathematics › Mathematics
Wednesday, 4 November 2026: 3 new papers, 2 cross-lists and 8 replacements in Commutative Algebra (math.AC), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
Let $R$ be a commutative noetherian ring. We ask whether complexes with finitely generated cohomology and a fixed support can be generated by relatively compact objects, using coproducts which exist in the half-bounded derived category. We prove that $\mathbf D^+_{\mathrm{fg},W}(R)=\operatorname{Loc}_{\mathrm{ex}}(\mathbf D^b_{\mathrm{fg},W}(R))$ whenever $W\subseteq\operatorname{Spec} R$ is specialization closed and has finite dimension. The same equality holds for every specialization-closed subset of $\operatorname{Spec} B$ when $B$ is a finite-type algebra over a noetherian domain $A$ of finite character and either $A$ or $B$ is regular. These equalities classify the thick Hom-submodules closed under existing coproducts. We also prove the corresponding bounded-above statement from the support results of Matsui--Takahashi. For an arbitrary noetherian ring, the bounded-below generation conjecture reduces to torsion complexes over quotient domains, with their support unchanged. Finally, over a regular noetherian domain of infinite dimension, we give a bounded-below object which is generated by bounded objects with the same support but has no bounded-below representative by finitely generated modules. The comparison functor from the bounded-below derived category of finite modules is therefore neither full nor essentially surjective.
Let $V$ be a complete valuation domain of rank 1, and let $\Gamma$ be its value group. We investigate the power series ring $V[[t]]$, and finally answer the long standing question of when this ring is coherent. Specifically, we prove that it is coherent if and only if $V$ is almost maximal, and either $\Gamma = \mathbb{R}$ or $\Gamma$ is discrete. Furthermore, we demonstrate that the weak Hilbert-syzygy identity $w. dim V [[X]] = w. dim V + 1$ holds if and only if $V[[X]]$ is a coherent ring.
Motivated by examples such as the local rings of projective integral schemes and the Zariski-Riemann space of a field, we develop a geometry for continuously indexed families of subrings of a commutative ring $R$. The indexing spaces are Stone spaces, and continuity is taken with respect to the Zariski topology on the space of subrings. To each such family we associate a subsheaf of the constant sheaf with value $R$ whose stalks are the indexed subrings. This gives a patch ringed space. Its global sections are locally constant $R$-valued functions whose value at each point belongs to the corresponding subring. The ring of global sections, with this representation, is the associated patch algebra. Our main result establishes an algebra-geometry correspondence between these continuously indexed families of subrings and patch ringed spaces that can be expressed as a category equivalence, with each of the two categories dually equivalent to the category of patch algebras. Thus the indexing space, its sheaf, and the indexed subrings can be recovered functorially from the associated patch algebra, which serves as a coordinate ring for this geometry. We describe kernels and factorizations of patch algebra morphisms in geometric terms and characterize their injectivity and surjectivity. We include applications to Pierce duality and Bhatt-Scholze $w$-localizations of integral separated quasi-compact schemes.
We characterise the commutative rings of weak global dimension at most one whose derived categories can be stratified by the Balmer-Favi support theory. For such rings, this yields a bijective correspondence between the class of localising subcategories of the derived category, and the subsets of the Zariski spectrum. For a large class of rings of weak global dimension at most one including all valuation domains, this occurs if and only if the ring is locally strongly discrete, if and only if the Telescope conjecture holds, and if and only if the module category has Gabriel dimension. However, we also give an example of a valuation domain where all these conditions fail, and yet the residue fields generate its derived category and we can classify all the localising subcategories of $D(R)$. Additionally, in the same example, we show that the Balchin-Stevenson smashing support cannot stratify $D(R)$ either.
We study the tensor triangular geometry of derived categories of valuation domains, obtaining complete ring-theoretic characterizations of their main structural properties. In particular, although valuation domains are almost always non-noetherian, their derived categories always satisfy the local-to-global principle and hence the detection property. We establish precise criteria for homological detection, stratification, and homological stratification, revealing a clear hierarchy among these properties. Notably, we obtain examples of derived categories of commutative rings that are homologically stratified and satisfy the local-to-global principle, but are not stratified. We also determine when the category of modules over a valuation domain has Gabriel dimension, and use this to provide a negative answer to a question of Balchin, G\'omez and Stevenson. Overall, valuation domains provide a rich class of non-noetherian rings whose derived categories can be understood in considerable depth.