Becker Coderived Category Overview

• Becker coderived category is an exotic unbounded derived framework defined using complexes of injectives and coacyclicity conditions.
• It utilizes an injective model structure where all complexes are cofibrant, yielding a Verdier quotient equivalent to the homotopy category of injective complexes.
• Under local coherence, its compact objects link to the bounded derived category of finitely presentable objects, bridging injective homotopy theory with classical derived methods.

Searching arXiv for core and follow-up papers on Becker coderived categories. arXiv search: Becker coderived category Grothendieck abelian category injective model structure compactly generated The Becker coderived category is an “exotic” unbounded derived category attached to an abelian, exact, or DG-enhanced context in which complexes of injectives are taken as the fundamental fibrant objects and equivalences are defined by annihilation against injectives rather than by ordinary quasi-isomorphism alone. In its basic Grothendieck-abelian form, it is the Verdier quotient of the homotopy category by the thick subcategory of Becker-coacyclic complexes, and it is canonically equivalent to the homotopy category of all complexes of injective objects. For locally coherent Grothendieck categories, this category admits a particularly sharp finiteness theory: it is compactly generated, and its compact objects are identified with a bounded derived category of finitely presentable objects (Stovicek, 2014). Later work extended this picture to arbitrary Grothendieck abelian categories and to Grothendieck abelian DG-categories, where graded-injective models and absolute derived categories play the analogous role (Positselski et al., 2021, Positselski et al., 2022).

1. Definition in Becker’s sense

Let $\mathsf A$ be a Grothendieck abelian category, $C(\mathsf A)$ its category of unbounded complexes, and $\mathsf K(\mathsf A)$ the corresponding homotopy category. In Becker’s formulation, a complex $X^\bullet$ is coacyclic if it is right-orthogonal to all complexes of injectives. This orthogonality is expressed either in the homotopy category by

$$\operatorname{Hom}_{\mathsf K(\mathsf A)}(X^\bullet,J^\bullet)=0 \quad\text{for all }J^\bullet\in \mathsf K(\mathsf{Inj}(\mathsf A)),$$

or, in the abelian model-categorical realization, by an Ext$^1$-orthogonality condition in the category of complexes. The Becker coderived category is then defined as the Verdier quotient

$$\mathsf D^{\mathrm{co}}(\mathsf A)=\mathsf K(\mathsf A)/\mathsf K(\mathsf A)^{\mathrm{co}},$$

where $\mathsf K(\mathsf A)^{\mathrm{co}}$ denotes the full thick subcategory of coacyclic complexes (Positselski et al., 2021, Stovicek, 2014).

A decisive structural feature is the existence of an injective abelian model structure on $C(\mathsf A)$. In this model structure, all complexes are cofibrant, the fibrant objects are precisely the complexes of injectives, and the weak equivalences are exactly those morphisms whose cones are coacyclic. Consequently, the homotopy category of this model structure is simultaneously the Becker coderived category and the homotopy category of complexes of injectives: $$\mathsf D^{\mathrm{co}}(\mathsf A)\simeq \mathsf K(\mathsf{Inj}(\mathsf A)).$$ This equivalence is the central concrete realization of Becker’s notion in both the 2014 locally coherent setting and the later general Grothendieck-abelian formulation (Stovicek, 2014, Positselski et al., 2021).

This construction is larger than the ordinary unbounded derived category. The reason is classical in unbounded homological algebra: an acyclic complex of injectives need not be contractible, so $C(\mathsf A)$0 generally contains more information than $C(\mathsf A)$1. The Becker coderived category is designed precisely to retain that information while still admitting a tractable model-categorical description (Stovicek, 2014).

2. The locally coherent theorem

The most influential structural theorem in this direction concerns a locally coherent Grothendieck category $C(\mathsf A)$2, meaning that $C(\mathsf A)$3 is locally finitely presentable and its full subcategory $C(\mathsf A)$4 of finitely presentable objects is abelian. In this setting, the homotopy category of complexes of injectives is compactly generated, and its full subcategory of compact objects is canonically equivalent to the bounded derived category of finitely presentable objects: $C(\mathsf A)$5 More concretely, the functor sending a bounded complex in $C(\mathsf A)$6 to a chosen injective resolution induces an equivalence from $C(\mathsf A)$7 onto the compact objects of $C(\mathsf A)$8 (Stovicek, 2014).

This theorem generalizes Krause’s earlier locally noetherian result to the coherent, not necessarily noetherian, context. Its significance is twofold. First, it shows that the Becker coderived category is not merely a formal enlargement of $C(\mathsf A)$9, but a compactly generated triangulated category with a completely explicit compact core. Second, it identifies that compact core with the expected bounded finite-presentation data, thereby linking a genuinely unbounded injective homotopy theory to the classical derived category of finitely presentable objects (Stovicek, 2014).

A useful way to interpret the theorem is that the coderived category behaves as the correct ambient category for unbounded injective phenomena, while its compact part remembers precisely the finite, bounded homological information. This suggests that coderived methods are not opposed to bounded derived methods; rather, the latter appear inside the former as the compact sector.

3. Purity, fp-injectives, and the mechanism of the proof

The extension from locally noetherian to locally coherent categories is driven by purity. In an additive finitely accessible category $\mathsf K(\mathsf A)$0, a short exact sequence

$\mathsf K(\mathsf A)$1

is pure exact if, for every finitely presentable object $\mathsf K(\mathsf A)$2, the induced sequence

$\mathsf K(\mathsf A)$3

is exact. This pure exact structure gives rise to pure projectives, pure injectives, pure acyclic complexes, and the pure derived category $\mathsf K(\mathsf A)$4 obtained by inverting pure quasi-isomorphisms (Stovicek, 2014).

For additive finitely accessible categories, the pure derived category admits both projective and injective hereditary model structures, and on the triangulated level these fit into a recollement involving the homotopy category and the homotopy category of pure acyclic complexes. A particularly strong consequence is that a pure acyclic complex of pure projectives or of pure injectives is contractible. In the formulation of the 2014 paper, this implies that “pure singularity” phenomena collapse in that pure setting (Stovicek, 2014).

The bridge to the Becker coderived category is provided by fp-injective objects. For a locally coherent Grothendieck category $\mathsf K(\mathsf A)$5, the exact category $\mathsf K(\mathsf A)$6 has enough projectives and enough injectives, and the key comparison theorem identifies the coacyclic complexes of fp-injectives with the pure acyclic complexes of fp-injectives: $\mathsf K(\mathsf A)$7 From this one obtains an equivalence

$\mathsf K(\mathsf A)$8

which is the decisive intermediary in proving compact generation and identifying compact objects (Stovicek, 2014).

This comparison shows that purity is not an auxiliary technicality but the mechanism that makes the coherent case accessible. A plausible implication is that Becker’s coderived category in coherent settings should often be analyzed through exact subcategories controlled by purity rather than through injectives alone.

4. Relation to the ordinary derived category and singularity theory

Under an additional finiteness hypothesis, the coderived category can be placed in a precise relation with the ordinary unbounded derived category. The relevant assumption is that $\mathsf K(\mathsf A)$9 admits a generating set $X^\bullet$0 consisting of finitely presentable objects of finite projective dimension. Under this hypothesis, $X^\bullet$1 is compactly generated, and $X^\bullet$2 is a set of compact generators (Stovicek, 2014).

In this situation, Krause’s recollement exists. The acyclic complexes form a triangulated subcategory $X^\bullet$3, and the derived category $X^\bullet$4 and coderived category $X^\bullet$5 are linked by a localization functor $X^\bullet$6 together with both adjoints. The recollement expresses the coderived category as sitting between the ordinary derived category and a “singular” kernel represented by acyclic injective information (Stovicek, 2014).

The associated singularity category is defined by

$X^\bullet$7

where $X^\bullet$8 denotes the thick subcategory of perfect complexes. The compact objects of the singular part $X^\bullet$9 are then identified with the idempotent completion of this singularity category: $$\operatorname{Hom}_{\mathsf K(\mathsf A)}(X^\bullet,J^\bullet)=0 \quad\text{for all }J^\bullet\in \mathsf K(\mathsf{Inj}(\mathsf A)),$$0 Thus the singularity category appears as the compact shadow of the kernel of the localization from coderived to derived homological algebra (Stovicek, 2014).

This relation clarifies a common source of confusion. The Becker coderived category is not simply another model for $$\operatorname{Hom}_{\mathsf K(\mathsf A)}(X^\bullet,J^\bullet)=0 \quad\text{for all }J^\bullet\in \mathsf K(\mathsf{Inj}(\mathsf A)),$$1. It coincides with $$\operatorname{Hom}_{\mathsf K(\mathsf A)}(X^\bullet,J^\bullet)=0 \quad\text{for all }J^\bullet\in \mathsf K(\mathsf{Inj}(\mathsf A)),$$2 only when the distinction between acyclic and coacyclic becomes trivial. In general it is strictly larger, and the excess over the ordinary derived category is measured by a singular triangulated sector.

5. Generalizations to Grothendieck abelian categories and abelian DG-categories

Subsequent work removed the local coherence restriction from the basic model-categorical existence theory. For any Grothendieck abelian category $$\operatorname{Hom}_{\mathsf K(\mathsf A)}(X^\bullet,J^\bullet)=0 \quad\text{for all }J^\bullet\in \mathsf K(\mathsf{Inj}(\mathsf A)),$$3, there is an injective derived model structure and an injective coderived model structure on $$\operatorname{Hom}_{\mathsf K(\mathsf A)}(X^\bullet,J^\bullet)=0 \quad\text{for all }J^\bullet\in \mathsf K(\mathsf{Inj}(\mathsf A)),$$4. The coderived model has all complexes cofibrant, complexes of injectives fibrant, and weak equivalences given by coacyclic cones; its homotopy category is $$\operatorname{Hom}_{\mathsf K(\mathsf A)}(X^\bullet,J^\bullet)=0 \quad\text{for all }J^\bullet\in \mathsf K(\mathsf{Inj}(\mathsf A)),$$5. In the same work, the derived, coderived, and contraderived categories are shown to be well-generated, and the derived category of any locally presentable abelian category is shown to have Hom sets (Positselski et al., 2021).

The same pattern persists in the DG-enhanced setting. For a Grothendieck abelian DG-category $$\operatorname{Hom}_{\mathsf K(\mathsf A)}(X^\bullet,J^\bullet)=0 \quad\text{for all }J^\bullet\in \mathsf K(\mathsf{Inj}(\mathsf A)),$$6, the Becker coderived category is defined as the Verdier quotient

$$\operatorname{Hom}_{\mathsf K(\mathsf A)}(X^\bullet,J^\bullet)=0 \quad\text{for all }J^\bullet\in \mathsf K(\mathsf{Inj}(\mathsf A)),$$7

where an object is Becker-coacyclic if it is annihilated by all graded-injective objects. There is an injective abelian model structure on $$\operatorname{Hom}_{\mathsf K(\mathsf A)}(X^\bullet,J^\bullet)=0 \quad\text{for all }J^\bullet\in \mathsf K(\mathsf{Inj}(\mathsf A)),$$8, and the coderived category is equivalent to the homotopy category of graded-injectives: $$\operatorname{Hom}_{\mathsf K(\mathsf A)}(X^\bullet,J^\bullet)=0 \quad\text{for all }J^\bullet\in \mathsf K(\mathsf{Inj}(\mathsf A)),$$9 When $^1$0 is locally coherent, this coderived category is compactly generated by the absolute derived category of finitely presentable objects $^1$1, generalizing the 2014 theorem from ordinary Grothendieck categories to abelian DG-categories (Positselski et al., 2022).

These DG-categorical extensions subsume several familiar examples. Complexes in a Grothendieck category, DG-modules over DG-rings or CDG-rings, and quasi-coherent matrix factorizations all fit into the abelian DG-category framework. In particular, for a curved DG-ring with graded left coherent underlying graded ring, the homotopy category of graded-injective CDG-modules is compactly generated, and its compact objects come, up to direct summands, from the absolute derived category of finitely presentable CDG-modules (Positselski et al., 2022).

6. Variants, applications, and open issues

The Becker coderived category has proved useful well beyond the original coherent Grothendieck setting. In the theory of DG-modules over de Rham DG-algebras, coderived categories of quasi-coherent $^1$2-modules are identified with unbounded derived categories of quasi-coherent $^1$3-modules, and the coderived formalism supports derived direct image, extraordinary inverse image, tensor, and duality functors compatible with the $^1$4-module side (Rybakov, 2013). This suggests that coderived categories are particularly natural when unbounded geometric functoriality is controlled more effectively by injective or flat models than by ordinary quasi-isomorphisms.

In the CDG-module setting, Becker’s coderived category of right CDG-modules is realized as the homotopy category of graded-injective modules, while the Becker contraderived category of left CDG-modules is realized by graded-projectives. Under graded right coherence, the coderived category is compactly generated by the absolute derived category of finitely presented right CDG-modules, and its compact objects are anti-equivalent to the compact objects in the corresponding contraderived category (Positselski et al., 2024). This is a particularly explicit instance of coderived–contraderived duality.

A further relative development replaces the ambient abelian category by a hereditary complete cotorsion pair $^1$5 in a Grothendieck category. In that setting, the Becker coderived category of the left-hand class and the Becker contraderived category of the right-hand class are naturally equivalent, both identified with the homotopy category of complexes in the kernel $^1$6. Nested cotorsion pairs then induce adjunctions between the corresponding coderived and contraderived categories (Positselski, 9 Sep 2025). This suggests that Becker’s construction is not limited to the injective/projective extremal cases, but is adaptable to relative homological algebra.

One conceptual issue remains open in general. Positselski’s original coderived category is defined using the smallest triangulated subcategory closed under coproducts and containing totalizations of short exact sequences, whereas Becker’s uses orthogonality to complexes of injectives. The inclusion from Positselski’s coacyclic subcategory into Becker’s coacyclic subcategory is known, but equality is not known in full generality, including for module categories over arbitrary rings (Positselski et al., 2021). The modern literature therefore treats the phrase “Becker coderived category” as denoting a specific orthogonality- and model-structure-based notion, not merely a stylistic variant of earlier terminology.

Taken together, these results establish the Becker coderived category as a central object in unbounded homological algebra: it is the injective homotopy-theoretic refinement of the derived category, compactly controlled in coherent settings, compatible with singularity-theoretic and purity-theoretic constructions, and extensible to DG, CDG, geometric, and relative cotorsion-pair contexts.