Homotopy Category Research Articles

Gorenstein algebras and recollements

Let A, , and be artin algebras. We prove that if there is a recollement of the bounded Gorenstein derived category relative to the bounded Gorenstein derived categories and , then A is Gorenstein if and only if so are and . In addition, we prove that a virtually Gorenstein algebra A is Gorenstein if and only if the bounded homotopy category of (finitely generated) projective left A-modules and that of (finitely generated) injective left A-modules coincide.

ON DIGITAL H-GROUPS

In this paper, specific properties of digital H-spaces and digital H-groups are studied. It is shown that there is a contravariant functor from the homotopy category of the pointed digital images to the category of groups and homomorphisms. Then it is proven that a pointed digital image having the same digital homotopy type as a digital H-group is itself a digital H-group.

DEFORMATION RETRACT OF A DIGITAL H-SPACE

In this study we investigate some properties of digital H-spaces and digital H-groups. It is shown that a _-deformation retract of a digital H-group is itself a digital H-group. Also we prove that there is a functor from the homotopy category of the pointed digital images to the category of groups and homomorphisms.

An example of a non-Fourier–Mukai functor between derived categories of coherent sheaves

Orlov’s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier–Mukai functor. In this paper we show that this result is false without the fully faithfulness hypothesis. We also show that our functor does not lift to the homotopy category of spectral categories if the ground field is $${\mathbb Q}$$ .

Applications of cotorsion triples

We study homotopy categories of model categories arising from a cotorsion triple, and the equivalences between corresponding stable categories. We characterize homological dimensions with respect to a cotorsion triple. Then, we lift cotorsion triple to complexes, and get the equivalence of homotopy categories of complexes via Quillen equivalence of model categories. Finally, we specify to Gorenstein cotorsion triple . By Quillen equivalence, it is shown that for a left-Gorenstein ring, there is an equivalence , which restricts to an equivalence (compare to [J. Algebra, 2010, 324:2718-2731]); a new proof for Bennis and Mahdou’s equality of Gorenstein global dimension is also given.

A Note on “The Homotopy Category is a Homotopy Category”

In his paper with the title, “The Homotopy Category is a Homotopy Category”, Arne Strom shows that the category Top of topo- logical spaces satisfies the axioms of an abstract homotopy category in the sense of Quillen. In this study, we show by examples that Quillen’s model structure on Top fails to capture some of the subtleties of classical homotopy theory and also, we show that the whole of classical homo-topy theory cannot be retrieved from the axiomatic approach of Quillen. Thus, we show that model category is an incomplete model of classical homotopy theory.

A note on homotopy categories of FP-injectives

For a locally finitely presented Grothendieck category $\mathcal{A}$, we consider a certain subcategory of the homotopy category of FP-injective objects in $\mathcal{A}$ which we show is compactly generated. In the case where $\mathcal{A}$ is locally coherent, we identify this subcategory with the derived category of FP-injective objects in $\mathcal{A}$. Our results are, in a sense, dual to the ones obtained by Neeman on the homotopy category of flat modules. Our proof is based on extending a characterization of the pure acyclic complexes which is due to Emmanouil.

The Balmer spectrum of the equivariant homotopy category of a finite abelian group

For a finite abelian group A, we determine the Balmer spectrum of $${\mathrm {Sp}}_A^{\omega }$$ , the compact objects in genuine A-spectra. This generalizes the case $$A={\mathbb {Z}}/p{\mathbb {Z}}$$ due to Balmer and Sanders (Invent Math 208(1):283–326, 2017), by establishing (a corrected version of) their $$\hbox {log}_p$$ -conjecture for abelian groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn’s blue-shift theorem for Tate-constructions (Kuhn in Invent Math 157(2):345–370, 2004).

Stable Components of Directed Spaces

In this paper, we introduce the notions of stable future, past and total component systems on a directed space with no loops. Then, we associate the stable component category to a stable (future, past or total) component system. Stable component categories are enriched in some monoidal category, eg. the homotopy category of spaces, and carry information about the spaces of directed paths between particular points. It is shown that the geometric realizations of finite pre-cubical sets with no loops admit unique minimal stable (future/past/total) component systems. These constructions provide a new family of invariants for directed spaces.

Derived Koszul duality and [formula omitted]-homology completion of structured ring spectra

Working in the context of symmetric spectra, we consider algebraic structures that can be described as algebras over an operad O. Topological Quillen homology, or TQ-homology, is the spectral algebra analog of Quillen homology and stabilization. Working in a framework for homotopical descent, the associated TQ-homology completion map appears as the unit map of an adjunction comparing the homotopy categories of O-algebra spectra with coalgebra spectra over the associated comonad via TQ-homology. We prove that this adjunction between homotopy categories can be turned into an equivalence by replacing O-algebras with the full subcategory of 0-connected O-algebras. We also construct the spectral algebra analog of the unstable Adams spectral sequence that starts from the TQ-homology groups TQ⁎(X) of an O-algebra X, and prove that it converges strongly to π⁎(X) when X is 0-connected.

AC-GORENSTEIN RINGS AND THEIR STABLE MODULE CATEGORIES

We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring that is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo–Gillespie–Hovey. It is also compatible with the notion of $n$ -coherent rings introduced by Bravo–Perez. So a $0$ -coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$ -coherent AC-Gorenstein ring is precisely a Ding–Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects that are Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects that are Gorenstein AC-injectives.

The Balmer spectrum of rational equivariant cohomology theories

The category of rational G-equivariant cohomology theories for a compact Lie group G is the homotopy category of rational G-spectra and therefore tensor-triangulated. We show that its Balmer spectrum is the set of conjugacy classes of closed subgroups of G, with the topology corresponding to the topological poset of [7]. This is used to classify the collections of subgroups arising as the geometric isotropy of finite G-spectra. The ingredients for this classification are (i) the algebraic model of free spectra of the author and B. Shipley [14], (ii) the Localization Theorem of Borel–Hsiang–Quillen [21] and (iii) tom Dieck's calculation of the rational Burnside ring [4].

On the relation between K-flatness and K-projectivity

Krause [19] has proved that the homotopy category K(R-PureProj) of pure projective modules over an associative ring R is compactly generated and equivalent to the Verdier quotient K(R)/T(R), where T(R) is the triangulated subcategory of the homotopy category K(R) consisting of the pure acyclic complexes. We provide an alternative proof of this result. When restricted to the homotopy category of projective modules, this equivalence reduces to that obtained by Neeman [26]. As an application of this description of K(R-PureProj), we show that the K-flat complexes defined by Spaltenstein [29] are, up to homotopy equivalence, the filtered colimits of K-projective complexes of pure projective modules.

Triangle equivalences involving Gorenstein FP-injective modules

ABSTRACTRecently, Hu et al. [16] introduced the notion of Gorenstein FP-injective modules and investigated a notion of Gorenstein FP-injective dimension for complexes. Let R be a left coherent ring and Db(R-Mod) the bounded derived category of left R-modules. It is proved that the quotient triangulated category of the subcategory of Db(R-Mod) consisting of complexes with both finite Gorenstein FP-injective dimension and FP-projective dimension by the bounded homotopy category of FP-projective-injective left R-modules is triangle-equivalent to the stable category of the Frobenius category of all Gorenstein FP-injective and FP-projective left R-modules. Note that a similar argument is also valid for the case of Gorenstein injective left R-modules. We extend a triangle equivalence established by Beligiannis involving Gorenstein injective left R-modules from rings with finite left Gorenstein global dimension to arbitrary rings.

DIFFERENTIAL GRADED ENDOMORPHISM ALGEBRAS, COHOMOLOGY RINGS AND DERIVED EQUIVALENCES

AbstractIn this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First, we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Then we give an affirmative answer to a problem of Dugas (A construction of derived equivalent pairs of symmetric algebras, Proc. Amer. Math. Soc. 143 (2015), 2281–2300) in some special case.

A homotopy-theoretic model of function extensionality in the effective topos

We present a way of constructing a Quillen model structure on a full subcategory of an elementary topos, starting with an interval object with connections and a certain dominance. The advantage of this method is that it does not require the underlying topos to be cocomplete. The resulting model category structure gives rise to a model of homotopy type theory with identity types, Σ- and Π-types, and functional extensionality. We apply the method to the effective topos with the interval object ∇2. In the resulting model structure we identify uniform inhabited objects as contractible objects, and show that discrete objects are fibrant. Moreover, we show that the unit of the discrete reflection is a homotopy equivalence and the homotopy category of fibrant assemblies is equivalent to the category of modest sets. We compare our work with the path object category construction on the effective topos by Jaap van Oosten.

On the Weight Lifting Property for Localizations of Triangulated Categories

As we proved earlier, for any triangulated category $$\underline C $$ endowed with a weight structure w and a triangulated subcategory $$\underline D $$ of $$\underline C $$ (strongly) generated by cones of a set of morphism S in the heart $$\underline {Hw} $$ of w there exists a weight structure w' on the Verdier quotient $$\underline {C'} = \underline C /\underline D $$ such that the localization functor $$\underline C \to \underline {C'} $$ is weight-exact (i.e., “respects weights”). The goal of this paper is to find conditions ensuring that for any object of $$\underline {C'} $$ of non-negative (resp. non-positive) weights there exists its preimage in $$\underline C $$ satisfying the same condition; we call a certain stronger version of the latter assumption the left (resp., right) weight lifting property. We prove that that these weight lifting properties are fulfilled whenever the set S satisfies the corresponding (left or right) Ore conditions. Moreover, if $$\underline D $$ is generated by objects of $$\underline {Hw} $$ then any object of $$\underline {Hw'} $$ lifts to $$\underline {Hw} $$ . We apply these results to obtain some new results on Tate motives and finite spectra (in the stable homotopy category). Our results are also applied to the study of the so-called Chow-weight homology in another paper.

Compactly generated triangulated subcategories of homotopy categories induced by cotorsion pairs

In this paper, we investigate the homotopy categories [Formula: see text] and [Formula: see text] with respect to a complete and hereditary cotorsion pair [Formula: see text] in a bicomplete abelian category. We prove that [Formula: see text] is compactly generated provided that [Formula: see text] is compactly generated. We introduce and characterize Gorenstein [Formula: see text]-complexes with respect to [Formula: see text] and show that a complex is a Gorenstein [Formula: see text]-complex if and only if its every term is a Gorenstein [Formula: see text]-object. We also show that the inclusion functors [Formula: see text] and [Formula: see text] have right adjoints respectively when [Formula: see text] is compactly generated.

The Bongartz’s Theorem on the Homotopy Category

Let $$\Lambda $$ be an Artin algebra and $$M\in $$ mod $$\Lambda $$ . It is well-known that the Bongartz’s Theorem plays an important role in the representation theory of Artin algebras. In this paper, we investigate the relative version of the Bongartz’s Theorem in the bounded homotopy category $$K^{b}(\text {add}M)$$ and construct a Bongartz’s completion for an $$\text {add}M$$ -relative presilting complex in $$K^{b}(\text {add}M)$$ such that it becomes an $$\text {add}M$$ -relative silting complex. Finally, we give some applications.

The Projective Leavitt Complex

AbstractFor a finite quiverQwithout sources, we consider the corresponding radical square zero algebraA. We construct an explicit compact generator for the homotopy category of acyclic complexes of projectiveA-modules. We call such a generator the projective Leavitt complex ofQ. This terminology is justified by the following result: the opposite differential graded endomorphism algebra of the projective Leavitt complex ofQis quasi-isomorphic to the Leavitt path algebra ofQop. Here,Qopis the opposite quiver ofQ, and the Leavitt path algebra ofQopis naturally${\open Z}$-graded and viewed as a differential graded algebra with trivial differential.