arXiv subjects
Publications and source records attributed to Michal Hrbek.
We study the transfer of (co)silting objects in derived categories of module categories via the extension functors induced by a morphism of commutative rings. It is proved that the extension functors preserve (co)silting objects of (co)finite type. In many cases the bounded silting property descends along faithfully flat ring extensions. In particular, the notion of bounded silting complex is Zariski local.
We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D(Mod-A) of a ring A. To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in A, which is a natural extension of the construction of compactly generated t-structures from chains of subsets of the Zariski spectrum known for the commutative noetherian case. We also provide constructions of silting and cosilting objects in D(Mod-A). This leads us to classification results over some classes of commutative rings and over finite dimensional hereditary algebras.
We show that compactly generated t-structures in the derived category of a commutative ring $R$ are in a bijection with certain families of compactly generated t-structures over the local rings $R_\mathfrak{m}$ where $\mathfrak{m}$ runs through the maximal ideals in the Zariski spectrum $\mathrm{Spec}(R)$. The families are precisely those satisfying a gluing condition for the associated sequence of Thomason subsets of $\mathrm{Spec}(R)$. As one application, we show that the compact generation of a homotopically smashing t-structure can be checked locally over localizations at maximal ideals. In combination with a result due to Balmer and Favi, we conclude that the $\otimes$-Telescope Conjecture for a quasi-coherent and quasi-separated scheme is a stalk-local property. Furthermore, we generalize the results of Trlifaj and Şahinkaya and establish an explicit bijection between cosilting objects of cofinite type over $R$ and compatible families of cosilting objects of cofinite type over all localizations $R_\mathfrak{m}$ at maximal primes.
We show that any homotopically smashing t-structure in the derived category of a commutative noetherian ring is compactly generated. This generalizes the validity of the telescope conjecture for commutative noetherian rings due to Neeman. As another consequence, we obtain a cofinite type result for pure-injective cosilting objects. We also give a formulation of telescope conjecture for homotopically smashing t-structures in underlying triangulated categories of certain Grothendieck derivators.
In the setting of the unbounded derived category D(R) of a ring R of weak global dimension at most one we consider t-structures with a definable coaisle. The t-structures among these which are stable (that is, the t-structures which consist of a pair of triangulated subcategories) are precisely the ones associated to a smashing localization of the derived category. In this way, our present results generalize those of [BŠ17] to the non-stable case. As in the stable case [BŠ17], we confine for the most part to the commutative setting, and give a full classification of definable coaisles in the local case, that is, over valuation domains. It turns out that unlike in the stable case of smashing subcategories, the definable coaisles do not always arise from homological ring epimorphisms. We also consider a non-stable version of the telescope conjecture for t-structures and give a ring-theoretic characterization of the commutative rings of weak global dimension at most one for which it is satisfied.
We classify all compactly generated t-structures in the unbounded derived category of an arbitrary commutative ring, generalizing the result of [ATLJS10] for noetherian rings. More specifically, we establish a bijective correspondence between the compactly generated t-structures and infinite filtrations of the Zariski spectrum by Thomason subsets. Moreover, we show that in the case of a commutative noetherian ring, any bounded below homotopically smashing t-structure is compactly generated. As a consequence, all cosilting complexes are classified up to equivalence.
It is well-known that a class of all modules, which are torsion-free with respect to a set of ideals, is closed under injective envelopes. In this paper, we consider a kind of a dual to this statement - are the divisibility classes closed under flat covers? - and argue that this is seldom the case. More precisely, we show that the class of all divisible modules over an integral domain R is closed under flat covers if and only if R is almost perfect. Also, we show that if the class of all s-divisible modules, where s is a regular element of a commutative ring R, is closed under flat covers then the quotient ring R/sR satisfies some rather restrictive properties. The question is motivated by the recent classification [11] of tilting classes over commutative rings.
A classic result by Raynaud and Gruson says that the notion of an (infinite dimensional) vector bundle is Zariski local. This result may be viewed as a particular instance (for n = 0) of the locality of more general notions of quasi-coherent sheaves related to (infinite dimensional) n-tilting modules and classes. Here, we prove the latter locality for all n and all schemes. We also prove that the notion of a tilting module descends along arbitrary faithfully flat ring morphisms in several particular cases (including the case when the base ring is noetherian).
A weak basis of a module is a generating set of the module minimal with respect to inclusion. A module is said to be regularly weakly based provided that each of its generating sets contain a weak basis. In the paper we study (1) rings over which all modules are regularly weakly based, refining results of Nashier and Nichols, (2) regularly weakly based modules over Dedekind domains.
We classify all tilting classes over an arbitrary commutative ring via certain sequences of Thomason subsets of the spectrum, generalizing the classification for noetherian commutative rings by Angeleri-Pospíšil-Šťovíček-Trlifaj. We show that the n-tilting classes can be equivalently expressed as classes of all modules vanishing in the first n degrees of one of the following homology theories arising from a finitely generated ideal: Tor_*(R/I,-), Koszul homology, Čech homology, or local homology (even though in general none of those theories coincide). Cofinite-type n-cotilting classes are described by vanishing of the corresponding cohomology theories. For any cotilting class of cofinite type, we also construct a corresponding cotilting module, generalizing the construction of Šťovíček-Trlifaj-Herbera. Finally, we characterize cotilting classes of cofinite type amongst the general ones, and construct new examples of n-cotilting classes not of cofinite type, which are in a sense hard to tell apart from those of cofinite type.
We classify 1-tilting classes over an arbitrary commutative ring. As a consequence, we classify all resolving subcategories of finitely presented modules of projective dimension at most 1. Both these collections are in 1-1 correspondence with faithful Gabriel topologies of finite type, or equivalently, with Thomason subsets of the spectrum avoiding a set of primes associated in a specific way to the ring. We also provide a generalization of the classical Fuchs and Salce tilting modules, and classify the equivalence classes of all 1-tilting modules. Finally we characterize the cases when tilting modules arise from perfect localizations.
Tilting modules over commutative rings were recently classified in [12]: they correspond bijectively to faithful Gabriel topologies of finite type. In this note we extend this classification by dropping faithfulness. The counterpart of an arbitrary Gabriel topology of finite type is obtained by replacing tilting with the more general notion of a silting module.