Starting with a Grothendieck category $\mathcal{G}$ and a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in $\mathcal G$, we study the local finite presentability and local coherence of the heart $\mathcal{H}_{\mathbf{t}}$ of the associated Happel-Reiten-Smalø $t$-structure in the derived category $\mathrm{Der} (\mathcal{G})$. We start by showing that, in this general setting, the torsion pair $\mathbf t$ is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those $\mathbf t$ for which $\mathcal{H}_{\mathbf{t}}$ is locally finitely presented, obtaining a complete answer under some additional assumptions on the ground category $\mathcal{G}$, which are general enough to include all locally coherent categories, all categories of modules and several categories of quasi-coherent sheaves over schemes. The third problem that we tackle is that of local coherence. In this direction we characterize those torsion pairs $\mathbf t=(\mathcal T,\mathcal F)$ in a locally finitely presented $\mathcal G$ for which $\mathcal{H}_{\mathbf{t}}$ is locally coherent in two cases: when the tilted t-structure in $\mathcal{H}_{\mathbf{t}}$ is assumed to restrict to finitely presented objects, and when $\mathcal F$ is cogenerating. In the last part of the paper we concentrate on the case when $\mathcal G$ is a category of modules over a small preadditive category, giving several examples and obtaining very neat (new) characterizations even in this more classical setting, also underlying connections with the notion of an elementary cogenerator.
We show that, under particular conditions, if a t-structure in the unbounded derived category of a locally coherent Grothendieck category restricts to the bounded derived category of its category of finitely presented objects, then its heart is itself a locally coherent Grothendieck category. Those particular conditions are always satisfied when the Grothendieck category is arbitrary and one considers the t-structure associated to a torsion pair in the category of finitely presented objects. They are also satisfied when one takes any compactly generated t-structure in the derived category of a…
If $X$ is a quasi-compact and quasi-separated scheme, the category $Qcoh(X)$ of quasi-coherent sheaves on $X$ is locally finitely presented. Therefore categorical flat quasi-coherent sheaves naturally arise. But there is also the standard definition of flatness in $Qcoh(X)$ from the stalks. So it makes sense to wonder the relationship (if any) between these two notions. In this paper we show that there are plenty of locally finitely presented categories having no other categorical flats than the zero object. As particular instance, we show that $Qcoh(\mathbf{P}^n(R)))$ has no other categorical…
We give an explicit description of the generator of finitely presented objects of the coslice of a locally finitely presentable category under a given object, as consisting of all pushouts of finitely presented maps under this object. Then we prove that the comma category under the direct image part of a morphism of locally finitely presentable category is still locally finitely presentable, and we give again an explicit description of its generator of finitely presented objects. We finally deduce that 2-category $\LFP$ has comma objects computed in $\Cat$.
In the papers of Nakaoka, he introduced the notion of hearts of (twin) cotorsion pairs on triangulated categories and showed that they have structures of (semi-) abelian categories. We study in this article a twin cotorsion pair (S,T),(U,V) on an exact category B with enough projectives and injectives and introduce a notion of the heart. First we show that its heart is preabelian. Moreover we show the heart of a single cotorsion pair is abelian. These results are analog of Nakaoka's results in triangulated categories. We also consider special cases where the heart has nicer structure. By our r…
In this paper we consider the locally self-injective property of the product FI$^m$ of the category FI of finite sets and injections. Explicitly, we prove that the external tensor product commutes with the coinduction functor, and hence preserves injective modules. As corollaries, every projective FI$^m$-modules over a field of characteristic 0 is injective, and the Serre quotient of the category of finitely generated FI$^m$-modules by the category of finitely generated torsion FI$^m$-modules is equivalent to the category of finite dimensional FI$^m$-modules.
Any Thomason filtration of a commutative ring yields (at least) two t-structures in the derived category of the ring, one of which is compactly generated [Hrb20,HHZ21]. We study the hearts of these two t-structures and prove that they coincide in case of a weakly bounded below filtration. Prompted by [SS20], in which it is proved that the heart of a compactly generated t-structure in a triangulated category with coproduct is a locally finitely presented Grothendieck category, we study the local coherence of the hearts associated with a weakly bounded below Thomason filtration, achieving a usef…
In this paper, we answer a question of Dwyer, Greenlees, and Iyengar by proving a local ring $R$ is a complete intersection if and only if every complex of $R$-modules with finitely generated homology is proxy small. Moreover, we establish that a commutative noetherian ring $R$ is locally a complete intersection if and only if every complex of $R$-modules with finitely generated homology is virtually small.
For finitary regular monads T on locally finitely presentable categories we characterize the finitely presentable objects in the category of T-algebras in the style known from general algebra: they are precisely the algebras presentable by finitely many generators and finitely many relations.
We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type $\text{FP}_n$ and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type $\text{FP}_n$ categories as a generalization of locally finitely generated and locally finitely presented categories. We also define and study the injective objects that are Ext-orthogonal to the class of objects of type $\text{FP}_n$, called $\text{FP}_n$-injective objects, which will be the right half of a complete cotorsion pair. As a generalizatio…
We introduce a new class of categories generalizing locally presentable ones. The distinction does not manifest in the abelian case and, assuming Vopenka's principle, the same happens in the regular case. The category of complete partial orders is the natural example of a nearly locally finitely presentable category which is not locally presentable.
Are all subcategories of locally finitely presentable categories that are closed under limits and $λ$-filtered colimits also locally presentable? For full subcategories the answer is affirmative. Makkai and Pitts proved that in the case $λ=\aleph_0$ the answer is affirmative also for all iso-full subcategories, \emph{i.\thinspace e.}, those containing with every pair of objects all isomorphisms between them. We discuss a possible generalization of this from $\aleph_0$ to an arbitrary $λ$.
We show that the homotopy category of complexes K(B) over any finitely accessible additive category B is locally well generated. That is, any localizing subcategory L in K(B) which is generated by a set is well generated in the sense of Neeman. We also show that K(B) itself being well generated is equivalent to B being pure semisimple, a concept which naturally generalizes right pure semisimplicity of a ring R for B = Mod-R.
This paper contributes to a theory of the behaviour of "finite-state" systems that is generic in the system type. We propose that such systems are modeled as coalgebras with a finitely generated carrier for an endofunctor on a locally finitely presentable category. Their behaviour gives rise to a new fixpoint of the coalgebraic type functor called locally finite fixpoint (LFF). We prove that if the given endofunctor preserves monomorphisms then the LFF always exists and is a subcoalgebra of the final coalgebra (unlike the rational fixpoint previously studied by Adámek, Milius and Velebil). Mor…
