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Michal Hrbek

Publications and source records attributed to Michal Hrbek.

At least 19 recordsLinked to original sources
arXiv2026

Semi-Bousfield classes and nonmonotone perversities

In the generality of a rigidly-compactly generated tensor triangulated category, we introduce semi-Bousfield classes in terms of the vanishing of the tensor product in positive degrees with respect to a fixed reasonable $t$-structure. We show that semi-Bousfield classes provide a common generalisation of Bousfield classes and compactly generated tensor-compatible $t$-structures. Then we specialise to the setting of the unbounded derived category $\mathcal{D}_{\mathrm{qc}}(X)$ of a Noetherian scheme $X$ and show that the stratification bijection naturally extends to an assignment which takes a (not necessarily monotone) perversity on $X$ to a semi-Bousfield class in $\mathcal{D}_{\mathrm{qc}}(X)$. If $X$ is regular, this assignment constitutes a stratification of the whole semi-Bousfield lattice, while in the singular case, its image consists precisely of those semi-Bousfield classes arising from objects of finite Tor-dimension. Restricting this bijection to monotone perversities recovers the recent classification of compactly generated tensor-compatible $t$-structures of Dubey and Sahoo, (arXiv:2204.05015).

math.CT↗
arXiv2026

Proxy smallness meets $t$-structures

We introduce a notion of proxy smallness for $t$-structures on triangulated categories associated to a Noetherian scheme. Specifically, the theory is developed in the presence of tensor actions. Consequently, our results yield a new characterization of schemes that are locally complete intersections in terms of $t$-structures, as well as a topological classification of preaisles on the bounded derived category of coherent sheaves.

math.AG↗
arXiv2026

On silting complexes associated to n-silting modules

We show that any (n+1)-term silting complex whose intermediate cohomology vanishes gives rise to an n-silting module, as recently introduced by Mao. Specializing to commutative noetherian rings, we show that this assignment induces a bijection on the respective equivalence classes. Furthermore, we prove in the same setting that the n-silting modules always correspond to a tilting complex, that is, the associated t-structure is of derived type. We use this to exhibit new examples of tilting complexes in the setting of Commutative Algebra and also to show that the finite type property for n-silting modules, as formulated by Mao, can in general fail.

math.RT↗
arXiv2025

Telescope conjecture for quiver representations over artinian rings

Let $\mathcal{D}(RC)$ be the derived category of representations of a small category $C$ over a commutative noetherian ring $R$. We study the homotopically smashing t-structures on this category. Specifying our discussion to the stalk categories $Γ_{\mathfrak{p}}\mathcal{D}(RQ)$ for a finite quiver $Q$ and a prime ideal $\mathfrak{p}$ of $R$, we prove the telescope conjecture for the derived category of representations of finite quivers over artinian rings. More generally, we prove the same result also outside of the noetherian context, for representations of finite quivers over commutative perfect rings.

math.RT↗
arXiv2025

Govorov--Lazard and finite deconstructibility for Gorenstein and restricted homological dimensions

Over Cohen--Macaulay rings admitting a pointwise dualizing module, we show that the class of modules of restricted projective dimension bounded by any integer is finitely deconstructible and that the class of modules of restricted flat dimension bounded by any integer satisfies the Govorov-Lazard property. Along the way, we prove the corresponding result for Gorenstein projective and flat dimensions over (locally) Gorenstein rings. Outside of Cohen--Macaulay rings, we consider analogous properties for restricted projective dimension zero and restricted flat dimension zero and establish them for commutative noetherian rings of finite Krull dimension. This has consequences for the corresponding classes of finitely generated modules being preenveloping in certain cases and provides generalizations of Holm's results on structure of balanced big Cohen--Macaulay modules in various directions.

math.AC↗
arXiv2025

Perfectly generated $t$-structures for algebraic spaces

This work studies $t$-structures for the derived category of complexes with quasi-coherent cohomology on algebraic spaces. The main result shows that the relative standard $t$-structures are compactly generated. As an application, we classify compactly generated tensor $t$-structures via Thomason filtrations on the underlying topological space.

math.AG↗
arXiv2025

Mutation and the Gabriel spectrum

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.

math.RT↗
arXiv2024

Cotorsion pairs and Tor-pairs over commutative noetherian rings

For a commutative noetherian ring $R$, we classify all the hereditary cotorsion pairs cogenerated by pure-injective modules of finite injective dimension. The classification is done in terms of integer-valued functions on the spectrum of the ring. Each such function gives rise to a system of local depth conditions which describes the left-hand class in the corresponding cotorsion pair. Furthermore, we show that these cotorsion pairs correspond by explicit duality to hereditary Tor-pairs generated by modules of finite flat dimension.

math.AC↗
arXiv2024

Telescope conjecture via homological residue fields with applications to schemes

For a big tt-category, we give a characterization of the Telescope Conjecture (TC) in terms of definable tensor-ideals generated by homological residue fields. We formulate a stalk-locality property of (TC) and prove that it holds in the case of the derived category of a quasi-compact quasi-separated scheme, strengthening a result \cite{HHZ21}. As an application, we find strong links between (TC) and separation properties of the adic topology on local rings. This allows us to recover known examples and counterexamples of when (TC) holds over a scheme, as well as to construct some new ones.

math.CT↗
arXiv2024

Product-complete tilting complexes and Cohen-Macaulay hearts

We show that the cotilting heart associated to a tilting complex $T$ is a locally coherent and locally coperfect Grothendieck category (i.e. an Ind-completion of a small artinian abelian category) if and only if $T$ is product-complete. We then apply this to the specific setting of the derived category of a commutative noetherian ring $R$. If $\dim(R)<\infty$, we show that there is a derived duality $\mathcal{D}^b_{fg}(R) \cong \mathcal{D}^b(\mathcal{B})^{op}$ between $\mathrm{mod} R$ and a noetherian abelian category $\mathcal{B}$ if and only if $R$ is a homomorphic image of a Cohen--Macaulay ring. Along the way, we obtain new insights about t-structures in $\mathcal{D}^b_{fg}(R)$. In the final part, we apply our results to obtain a new characterization of the class of those finite-dimensional Noetherian rings that admit a Gorenstein complex.

math.RT↗
arXiv2024

Topological endomorphism rings of tilting complexes

In a compactly generated triangulated category, we introduce a class of tilting objects satisfying certain purity condition. We call these the decent tilting objects and show that the tilting heart induced by any such object is equivalent to a category of contramodules over the endomorphism ring of the tilting object endowed with a natural linear topology. This extends the recent result for n-tilting modules by Positselski and Šťovíček. In the setting of the derived category of modules over a ring, we show that the decent tilting complexes are precisely the silting complexes such that their character dual is cotilting. The hearts of cotilting complexes of cofinite type turn out to be equivalent to the category of discrete modules with respect to the same topological ring. Finally, we provide a kind of Morita theory in this setting: Decent tilting complexes correspond to pairs consisting of a tilting and a cotilting derived equivalence as described above tied together by a tensor compatibility condition.

math.RT↗
arXiv2024

Tilting complexes and codimension functions over commutative noetherian rings

In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the "slice" condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen-Macaulay ring. We also provide dual versions of our results in the cosilting case.

math.AC↗
arXiv2023

The finite type of modules of bounded projective dimension and Serre's conditions

Let $R$ be a commutative noetherian ring. We prove that the class of modules of projective dimension bounded by $k$ is of finite type if and only if $R$ satisfies Serre's condition $(S_k)$. In particular, this answers positively a question of Bazzoni and Herbera in the specific setting of a Gorenstein ring. Applying similar techniques, we also show that the $k$-dimensional version of the Govorov-Lazard Theorem holds if and only if $R$ satisfies the "almost" Serre condition $(C_{k+1})$.

math.AC↗
arXiv2023

Fp-projective periodicity

The phenomenon of periodicity, discovered by Benson and Goodearl, is linked to the behavior of the objects of cocycles in acyclic complexes. It is known that any flat $\mathsf{Proj}$-periodic module is projective, any fp-injective $\mathsf{Inj}$-periodic module is injective, and any $\mathsf{Cot}$-periodic module is cotorsion. It is also known that any pure $\mathsf{PProj}$-periodic module is pure-projective and any pure $\mathsf{PInj}$-periodic module is pure-injective. Generalizing a result of Saroch and Stovicek, we show that every $\mathsf{FpProj}$-periodic module is weakly fp-projective. The proof is quite elementary, using only a strong form of the pure-projective periodicity and the Hill lemma. More generally, we prove that, in a locally finitely presentable Grothendieck category, every $\mathsf{FpProj}$-periodic object is weakly fp-projective. In a locally coherent category, all weakly fp-projective objects are fp-projective. We also present counterexamples showing that a non-pure $\mathsf{PProj}$-periodic module over a regular finitely generated commutative algebra (or a hereditary finite-dimensional associative algebra) over a field need not be pure-projective.

math.CT↗
arXiv2023

Restricted injective dimensions over Cohen-Macaulay rings

We show that the small and large restricted injective dimensions coincide for Cohen-Macaulay rings of finite Krull dimension. Based on this, and inspired by the recent work of Sather-Wagstaff and Totushek, we suggest a new definition of Cohen-Macaulay Hom injective dimension. We show that the class of Cohen-Macaulay Hom injective modules is the right constituent of a perfect cotorsion pair. Our approach relies on tilting theory, and in particular, on the explicit construction of the tilting module inducing the minimal tilting class recently obtained in arXiv:2207.01309.

math.AC↗
arXiv2022

Singular equivalences to locally coherent hearts of commutative noetherian rings

We show that Krause's recollement exists for any locally coherent Grothendieck category such that its derived category is compactly generated. As a source of such categories, we consider the hearts of intermediate and restrictable $t$-structures in the derived category of a commutative noetherian ring. We show that the induced tilting object over such a heart gives rise to an equivalence between the two Krause's recollements, and in particular, to a singular equivalence.

math.AC↗
arXiv2022

Countably generated flat modules are quite flat

We prove that if $R$ is a commutative Noetherian ring, then every countably generated flat $R$-module is quite flat, i.e., a direct summand of a transfinite extension of localizations of $R$ in countable multiplicative subsets. We also show that if the spectrum of $R$ is of cardinality less than $κ$, where $κ$ is an uncountable regular cardinal, then every flat $R$-module is a transfinite extension of flat modules with less than $κ$ generators. This provides an alternative proof of the fact that over a commutative Noetherian ring with countable spectrum, all flat modules are quite flat. More generally, we say that a commutative ring is CFQ if every countably presented flat $R$-module is quite flat. We show that all von Neumann regular rings and all $S$-almost perfect rings are CFQ. A zero-dimensional local ring is CFQ if and only if it is perfect. A domain is CFQ if and only if all its proper quotient rings are CFQ. A valuation domain is CFQ if and only if it is strongly discrete.

math.AC↗