Grothendieck Abelian Category Research Articles

The N-stable category

A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes of projectives, and the singularity category. To adapt this result to N-complexes, one must find an appropriate candidate for the N-analogue of the stable category. We identify this “N-stable category” via the monomorphism category and prove Buchweitz’s theorem for N-complexes over a Grothendieck abelian category. We also compute the Serre functor on the N-stable category over a self-injective algebra and study the resultant fractional Calabi–Yau properties.

T-Structures on stable derivators and Grothendieck hearts

We prove that, given any strong and stable derivator and a t-structure in its base triangulated category D, the t-structure canonically lifts to all the (coherent) diagram categories and each incoherent diagram in the heart uniquely lifts to a coherent one. We use this to show that the t-structure being compactly generated implies that the coaisle is closed under directed homotopy colimits which, in turn, implies that the heart is an (Ab.5) Abelian category. If, moreover, D is a well-generated algebraic or topological triangulated category, then the heart of any accessibly embedded (in particular, compactly generated) t-structure has a generator. As a consequence, it follows that the heart of any compactly generated t-structure of a well-generated algebraic or topological triangulated category is a Grothendieck Abelian category.

Derived, coderived, and contraderived categories of locally presentable abelian categories

For a locally presentable abelian category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived category $\mathsf D(\mathsf B)$ is generated, as a triangulated category with coproducts, by the projective generator of $\mathsf B$. For a Grothendieck abelian category $\mathsf A$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's principle, we prove that the derived category $\mathsf D(\mathsf A)$ is generated, as a triangulated category with products, by the injective cogenerator of $\mathsf A$. More generally, we define the notion of an exact category with an object size function and prove that the derived category of any such exact category with exact $\kappa$-directed colimits of chains of admissible monomorphisms has Hom sets. In particular, the derived category of any locally presentable abelian category has Hom sets.

A Derived Gabriel–Popescu Theorem for t-Structures via Derived Injectives

Abstract We prove a derived version of the Gabriel–Popescu theorem in the framework of dg-categories and t-structures. This exhibits any pretriangulated dg-category with a suitable t-structure (such that its heart is a Grothendieck abelian category) as a t-exact localization of a derived dg-category of dg-modules. We give an original proof based on a generalization of Mitchell’s argument in A quick proof of the Gabriel-Popesco theorem that involves derived injective objects. As an application, we provide a short proof of the fact that derived categories of Grothendieck abelian categories have a unique dg-enhancement.

Uniqueness of enhancements for derived and geometric categories

AbstractWe prove that the derived categories of abelian categories have unique enhancements—all of them, the unbounded, bounded, bounded above and bounded below derived categories. The unseparated and left completed derived categories of a Grothendieck abelian category are also shown to have unique enhancements. Finally, we show that the derived category of complexes with quasi-coherent cohomology and the category of perfect complexes have unique enhancements for quasi-compact and quasi-separated schemes.

On perfectly generated weight structures and adjacent t-structures

This paper is dedicated to the study of smashing weight structures (these are the weight structures "coherent with coproducts"), and the application of their properties to t-structures. In particular, we prove that the hearts of compactly generated t-structures are Grothendieck abelian categories; this statement strengthens earlier results of several other authors. The central theorem of the paper is as follows: any perfect (as defined by Neeman) set of objects of a triangulated category generates a weight structure; we say that weight structures obtained this way are perfectly generated. An important family of perfectly generated weight structures are (the opposites to) the ones right adjacent to compactly generated t-structures; they give injective cogenerators for the hearts of the latter. Moreover, we establish the following not so explicit result: any smashing weight structure on a well generated triangulated category (this is a generalization of the notion of a compactly generated category that was also defined by Neeman) is perfectly generated; actually, we prove more than that. Furthermore, we give a classification of compactly generated torsion theories (these generalize both weight structures and t-structures) that extends the corresponding result of D. Pospisil and J. Šťovíček to arbitrary smashing triangulated categories. This gives a generalization of a t-structure statement due to B. Keller and P. Nicolas.

CONNECTIVITY AND PURITY FOR LOGARITHMIC MOTIVES

AbstractThe goal of this article is to extend the work of Voevodsky and Morel on the homotopyt-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel’s connectivity theorem and show a purity statement for$({\mathbf {P}}^1, \infty )$-local complexes of sheaves with log transfers. The homotopyt-structure on${\operatorname {\mathbf {logDM}^{eff}}}(k)$is proved to be compatible with Voevodsky’st-structure; that is, we show that the comparison functor$R^{{\overline {\square }}}\omega ^*\colon {\operatorname {\mathbf {DM}^{eff}}}(k)\to {\operatorname {\mathbf {logDM}^{eff}}}(k)$ist-exact. The heart of the homotopyt-structure on${\operatorname {\mathbf {logDM}^{eff}}}(k)$is the Grothendieck abelian category of strictly cube-invariant sheaves with log transfers: we use it to build a new version of the category of reciprocity sheaves in the style of Kahn-Saito-Yamazaki and Rülling.

Covers and direct limits: a contramodule-based approach

We present applications of contramodule techniques to the Enochs conjecture about covers and direct limits, both in the categorical tilting context and beyond. In the $n$-tilting-cotilting correspondence situation, if $\mathsf A$ is a Grothendieck abelian category and the related abelian category $\mathsf B$ is equivalent to the category of contramodules over a topological ring $\mathfrak R$ belonging to one of certain four classes of topological rings (e.g., $\mathfrak R$ is commutative), then the left tilting class is covering in $\mathsf A$ if and only if it is closed under direct limits in $\mathsf A$, and if and only if all the discrete quotient rings of the topological ring $\mathfrak R$ are perfect. More generally, if $M$ is a module satisfying a certain telescope Hom exactness condition (e.g., $M$ is $\Sigma$-pure-$\operatorname{Ext}^1$-self-orthogonal) and the topological ring $\mathfrak R$ of endomorphisms of $M$ belongs to one of certain seven classes of topological rings, then the class $\mathsf{Add}(M)$ is closed under direct limits if and only if every countable direct limit of copies of $M$ has an $\mathsf{Add}(M)$-cover, and if and only if $M$ has perfect decomposition. In full generality, for an additive category $\mathsf A$ with (co)kernels and a precovering class $\mathsf L\subset\mathsf A$ closed under summands, an object $N\in\mathsf A$ has an $\mathsf L$-cover if and only if a certain object $\Psi(N)$ in an abelian category $\mathsf B$ with enough projectives has a projective cover. The $1$-tilting modules and objects arising from injective ring epimorphisms of projective dimension $1$ form a class of examples which we discuss.

The Tilting–Cotilting Correspondence

AbstractTo a big $n$-tilting object in a complete, cocomplete abelian category ${\textsf{A}}$ with an injective cogenerator we assign a big $n$-cotilting object in a complete, cocomplete abelian category ${\textsf{B}}$ with a projective generator and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of ${\textsf{A}}$ and ${\textsf{B}}$. Under various assumptions on ${\textsf{A}}$, which cover a wide range of examples (for instance, if ${\textsf{A}}$ is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that ${\textsf{B}}$ is the abelian category of contramodules over a topological ring and that the derived equivalences are realized by a contramodule-valued variant of the usual derived Hom functor.

Properties of abelian categories via recollements

A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.

Auslander-Reiten duality for Grothendieck abelian categories

Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor Ext^1(C,-) into modules over the endomorphism ring of C admits a partially defined right adjoint when C is a finitely presented object. This result seems to be new even for module categories. For appropriate schemes over a field, the connection with Serre duality is discussed.

Uniqueness of dg enhancements for the derived category of a Grothendieck category

We prove that the derived category of a Grothendieck abelian category has a unique dg enhancement. Under some additional assumptions, we show that the same result holds true for its subcategory of compact objects. As a consequence, we deduce that the unbounded derived category of quasi-coherent sheaves on an algebraic stack and the category of perfect complexes on a noetherian concentrated algebraic stack with quasi-finite affine diagonal and enough perfect coherent sheaves have a unique dg enhancement. In particular, the category of perfect complexes on a noetherian scheme with enough locally free sheaves has a unique dg enhancement.

Grothendieck Categories as a Bilocalization of Linear Sites

We prove that the 2-category Grt of Grothendieck abelian categories with colimit preserving functors and natural transformations is a bicategory of fractions in the sense of Pronk of the 2-category Site of linear sites with continuous morphisms of sites and natural transformations. This result can potentially be used to make the tensor product of Grothendieck categories from earlier work by Lowen, Shoikhet and the author into a bi-monoidal structure on Grt.

Non-commutative deformations and quasi-coherent modules

We identify a class of quasi-compact semi-separated (qcss) twisted presheaves of algebras $$\mathcal {A}$$ for which well-behaved Grothendieck abelian categories of quasi-coherent modules $$\mathsf {Qch} (\mathcal {A})$$ are defined. This class is stable under algebraic deformation, giving rise to a 1–1 correspondence between algebraic deformations of $$\mathcal {A}$$ and abelian deformations of $$\mathsf {Qch} (\mathcal {A})$$ . For a qcss presheaf $$\mathcal {A}$$ , we use the Gerstenhaber–Schack (GS) complex to explicitly parameterize the first-order deformations. For a twisted presheaf $$\mathcal {A}$$ with central twists, we descibe an alternative category $$\mathsf {QPr} (\mathcal {A})$$ of quasi-coherent presheaves which is equivalent to $$\mathsf {Qch} (\mathcal {A})$$ , leading to an alternative, equivalent association of abelian deformations to GS cocycles of qcss presheaves of commutative algebras. Our construction applies to the restriction $$\mathcal {O}$$ of the structure sheaf of a scheme X to a finite semi-separating open affine cover (for which we have $$\mathsf {Qch} (\mathcal {O}) \cong \mathsf {Qch} (X)$$ ). Under a natural identification of GS cohomology of $$\mathcal {O}$$ and Hochschild cohomology of X, our construction is shown to be equivalent to Toda’s construction from Toda (J Differ Geom 81(1):197–224, 2009) in the smooth case.

Deriving Auslander's formula

Auslander's formula shows that any abelian category \mathsf C is equivalent to the category of coherent functors on \mathsf C modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to showing that the homotopy category of injective objects of some appropriate Grothendieck abelian category (the category of ind-objects of \mathsf C ) is compactly generated and that the full subcategory of compact objects is equivalent to the bounded derived category of \mathsf C . The same approach shows for an arbitrary Grothendieck abelian category that its derived category and the homotopy category of injective objects are well-generated triangulated categories. For sufficiently large cardinals \alpha we identify their \alpha -compact objects and compare them.

The homotopy category of injectives

Krause studied the homotopy category K(InjA) of complexes of injectives in a locally noetherian Grothendieck abelian category A. Because A is assumed locally noetherian, we know that arbitrary direct sums of injectives are injective, and hence, the category K(InjA) has coproducts. It turns out that K(InjA) is compactly generated, and Krause studies the relation between the compact objects in K(InjA), the derived category D(A), and the category Kac(InjA) of acyclic objects in K(InjA). We wish to understand what happens in the nonnoetherian case, and this paper begins the study. We prove that, for an arbitrary Grothendieck abelian category A, the category K(InjA) has coproducts and is μ-compactly generated for some sufficiently large μ. The existence of coproducts follows easily from a result of Krause: the point is that the natural inclusion of K(InjA) into K(A) has a left adjoint and the existence of coproducts is a formal corollary. But in order to prove anything about these coproducts, for example the μ-compact generation, we need to have a handle on this adjoint. Also interesting is the counterexample at the end of the article: we produce a locally noetherian Grothendieck abelian category in which products of acyclic complexes need not be acyclic. It follows that D(A) is not compactly generated. I believe this is the first known example of such a thing.

The Popescu–Gabriel theorem for triangulated categories

The Popescu–Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differential graded categories. An analogous result for topological triangulated categories has recently been obtained by A. Heider.

Model category structures on chain complexes of sheaves

The unbounded derived category of a Grothendieck abelian category is the homotopy category of a Quillen model structure on the category of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently by Beke. However, in most cases of interest, such as the category of sheaves on a ringed space or the category of quasi-coherent sheaves on a nice enough scheme, the abelian category in question also has a tensor product. The injective model structure is not well-suited to the tensor product. In this paper, we consider another method for constructing a model structure. We apply it to the category of sheaves on a well-behaved ringed space. The resulting flat model structure is compatible with the tensor product and all homomorphisms of ringed spaces.

Elementary torsion theories and locally finitely presented categories

Locally finitely presented Grothendieck abelian categories are shown to be just the “elementary” localizations of Module categories. All categories V are assumed to be Grothendieck abelian (e.g. see [l, 21 or [5]). 55’ is locally finifeely presented if 9? has a generating family Ce = {Gi: i E I} with each Gi a finitely presented object of %. I have shown elsewhere [4] that in such categories one may define a canonical first-order (many-sorted) logic which behaves well one may prove Los’ Theorem, produce K-Saturated extensions of structures etc. Given a torsion theory (= a hereditary torsion theory in the terminology of [5]) (T, S) on Q and C in %‘, Q$ will be the filter of T-dense subobjects of C (where C’s C is T-dense if C/C’ is a member of the torsion class T). %? will denote the quotient category of V with respect to (9,s) and ( -)swill denote the T-localization functor from %’ to %? Tr will be the functor taking objects C of % to their T-torsion subobjects. We say that C is T-finitely generated (T-fig.) if C contains a finitely generated T-dense subobject. Recall that a torsion theory (9,9) is of finite type if for every Gi E 59 (as above), %$I has a cofinal subset of finitely generated members equivalently (by an easy argument) iff this condition holds on %s for any finitely generated c E %. We shall say that a torsion theory (9,T) is elementary if: (i) (9, 9) is of finite type; (ii) for each Gi, Gi E ‘3 (as above) and r: Gi --, Gi with im r E %z, ker r is T-finitely generated (that is (finitely generated) members of %z are **T-finitely presented” in some sense). It may be shown that this notion does not depend on the choice of ‘3 (directly or as a consequence of the following result). The term “elementary” is justified by the following theorem:

Local and stable homological algebra in Grothendieck abelian categories

We define model category structures on the category of chain complexes over a Grothendieck abelian category depending on the choice of a generating family, and we study their behaviour with respect to tensor products and stabilization. This gives convenient tools to construct and understand triangulated categories of motives.