Homological epimorphisms, homotopy epimorphisms and acyclic maps

Joseph Chuang; Andrey Lazarev

doi:10.1515/forum-2019-0249

Open Access Published by De Gruyter July 16, 2020

Homological epimorphisms, homotopy epimorphisms and acyclic maps

Joseph Chuang and Andrey Lazarev

From the journal Forum Mathematicum

https://doi.org/10.1515/forum-2019-0249

Abstract

We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

Keywords: Homotopy epimorphism; homological epimorphism; plus-construction; acyclic map

MSC 2010: 16E45; 55R35

1 Introduction

The notion of epimorphism exists in any category 𝒞: a morphism X→Y is an epimorphism if for any object Z of 𝒞 the induced map of sets Hom𝒞⁡(Y,Z)→Hom𝒞⁡(X,Z) is injective. Assuming the existence of pushouts in 𝒞, this is equivalent to requiring that the codiagonal map Y*XY→Y is an isomorphism; cf. for example [19, Proposition 2.1] where this and other easy equivalent reformulations are given. Epimorphisms of sets or groups are just surjections; however already for rings the situation is more interesting, e.g., the inclusion ℤ→ℚ (or any localization of rings) is an epimorphism. The tensor product B⊗AB realizes the pushout B*AB for commutative rings. On the other hand, in the category of (not necessarily commutative) rings, B⊗AB is not a pushout, as it is not even a ring in general. However it is still true that A→B is an epimorphism if and only if the multiplication map B⊗AB→B is an isomorphism [11, Proposition 4.1.1].

If 𝒞 is a category with an additional structure allowing one to do homotopy theory in it (such as the category of topological spaces or simplicial sets or, more generally, a closed model category), there is a similar notion of a homotopy epimorphism: it is a map X→Y such that Y*X𝕃Y→Y is an isomorphism in the homotopy category of 𝒞 where Y*X𝕃Y stands for a homotopy pushout. It is known [22] that homotopy epimorphisms of connected topological spaces are precisely acyclic maps, i.e. maps whose homotopy fibers have zero integral homology groups in positive degrees.

We investigate the notion of homotopy epimorphism in the category of differential graded (dg) algebras, possibly noncommutative. It is known [5, Proposition 3.17] that derived localizations of dg algebras are homotopy epimorphisms. On the other hand, derived localizations are also homological epimorphisms, i.e. maps A→B such that the multiplication map B⊗A𝕃B→B is a quasi-isomorphism. The property of being a homological epimorphism has many nice implications for the induced functors on derived categories and it is natural to ask (especially having in mind the corresponding non-homotopy result) whether homotopy and homological epimorphisms are the same thing. Our first main result is that this is, indeed, true.

Next, we consider topological applications. Given a connected space X we denote its based loop space by GX; then its chain complex C*⁢(G⁢X,𝐤) (simplicial or singular) with coefficients in a commutative ring 𝐤 is a dg 𝐤-algebra. Our second main result is a characterization of 𝐤-acyclic maps f:X→Y as those maps for which the map of dg algebras C*(GX,𝐤),→C*(GY,𝐤) is a homological epimorphism. Equivalently, f is acyclic if an only if it induces a Verdier localization of the triangulated categories of cohomologically locally constant sheaves (also known as infinity local systems) on X and Y .

Finally, we consider, for a given connected space X, its p-plus-construction Xp+; then the canonical map X→X+ is a universal 𝔽p-acyclic map out of X where 𝔽p is the field with p elements. We show that if X is such that π1⁢(X) has a perfect subgroup of finite index, then the map C*⁢(G⁢X,𝔽p)→C*⁢(G⁢Xp+,𝔽p) admits a purely algebraic description as a certain Bousfield localization of the category of dg C*⁢(G⁢X,𝔽p)-modules. Note that H0⁢(G⁢X,𝔽p)≅𝔽p⁢[π1⁢(X)], the 𝔽p–group ring of π1⁢(X). In the case when π1⁢(X) is finite we show that the dg algebra C*⁢(G⁢Xp+,𝔽p) is (quasi-isomorphic to) the derived localization of C*⁢(G⁢X,𝐤) at a certain idempotent of 𝔽p⁢[π1⁢(X)]. In the case when X has no higher homotopy groups, i.e. is the classifying space of a finite group, this is essentially equivalent to the main result [2] (which, however, was formulated without invoking derived localization).

1.1 Notation and conventions

We work in the category of ℤ-graded (dg) modules over a fixed commutative ground ring 𝐤; an object in this category is a pair (V,dV) where V is a graded 𝐤-module and dV is a differential on it; it will always be assumed to be of homological type (so it lowers the degree of a homogeneous element). Unmarked tensor products and homomorphisms will be understood to be taken over 𝐤; we will abbreviate “differential graded” to “dg”. The suspension of a graded vector space V is the graded vector space Σ⁢V so that (Σ⁢V)i=Vi+1. Quasi-isomorphisms and isomorphisms will be denoted by ≃ and ≅, respectively.

A dg algebra is an associative monoid in the dg category of dg vector spaces with respect to the standard monoidal structure given by the tensor product. Given a map A→B of dg algebras (not necessarily central) we will refer to B as a dg A-algebra. A dg vector space V is a (left) dg module over a dg algebra A if it is supplied with a dg map A⊗V→V satisfying the usual conditions of associativity and unitality; a right dg module is defined similarly.

The categories of dg algebras and dg modules over a dg algebra admit structures of closed model categories; cf. [5] for an overview. We will denote by DGA𝐤 the category of dg 𝐤-algebras, and for a dg algebra A we write D⁢(A) for its derived category. Recall that objects of D⁢(A) are cofibrant left dg A-modules and morphisms are chain homotopy classes of dg module maps. For a dg algebra A, a left dg A-module M and a right dg A-module N their tensor product N⊗AM and their derived tensor product N⊗A𝕃M are defined. The latter is defined as either N′⊗AM or N⊗AM′ where M′ and N′ are cofibrant replacements of M and N, respectively. Similarly for two left (or right) dg A-modules M,N their derived module of homomorphisms RHomA⁡(M,N) is defined as Hom⁡(M′,N) where M′ is a cofibrant replacement of M.

The category of reduced simplicial sets will be denoted by SSet* and the category of simplicial groups by SGp. By “a space” we mean “a simplicial set”; nevertheless all the results in the paper are of homotopy invariant nature and so they make sense and are valid for topological spaces by the well known correspondence between topological spaces and simplicial sets.

2 Derived free products of dg algebras

Let A,B and C be graded algebras with A being flat over 𝐤. In this case we can form the free product B*AC, this is again a graded algebra satisfying an appropriate universal property; if A,B and C are dg algebras, then so is B*AC. The derived version B*A𝕃C is described in [5, Section 2]. This is a homotopy pushout in the closed model category of dg algebras and can be defined concretely as B′*AC′ where B′ and C′ are A-cofibrant replacements of A-algebras B and C, respectively.

For an A-algebra B we denote by B¯ the cokernel of the unit map A→B; it is clearly a dg A-bimodule. Then we have the following important technical result.

Lemma 2.1.

Let A be a dg algebra and let B,C be dg A-algebras. Assume that the unit maps A→B and A→C are cofibrations of left A-modules. Then B*AC has a natural filtration by dg A-bimodules

0=F0⊂B=F1⊂F2⊂⋯

with ⋃nFn=B*AC and Fn⁢Fk⊂Fn+k and such that

F2⁢n+1/F2⁢n≅B⊗A(C¯⊗AB¯)⊗An,

F2⁢n+2/F2⁢n+1≅B⊗A(C¯⊗AB¯)⊗An⊗AC¯

for n≥0.

Proof.

Let F2⁢n,F2⁢n+1⊂B*AC be spanned by the monomials b1⁢c1⁢⋯⁢bn⁢cn and b1⁢c1⁢⋯⁢bn⁢cn⁢bn+1 with bi∈B and ci∈C, respectively. It is clear that this filtration consists of dg A-bimodules, is exhaustive and multiplicative. Since A→B and A→C are cofibrations of left dg A-modules, it follows that B¯ and C¯ are cofibrant as left A-modules, in particular they are A-flat, after forgetting the differential. Now the required conclusion on the associated graded quotients follows from [10, Theorem 4.6]; cf. also [11, p. 206] for a simpler argument in the case when C and B are free as left A-modules. ∎

Remark 2.2.

In fact, the conclusion of Lemma 2.1 holds under the weaker assumptions that A→B and A→C are injections and B¯,C¯ are flat left A-modules since this is what is required for the application of [10, Theorem 4.6]. Moreover, modifying the filtration so that its components are spanned by monomials ending with an element in B rather than beginning with one, one obtains a similar conclusion under the assumption that B¯,C¯ are flat rightA-modules.

Corollary 2.3.

Let A,B,C be as in Lemma 2.1. Then there is a quasi-isomorphism of dg algebras

B*AC≃B*A𝕃C.

Proof.

Let B′,C′ be A-cofibrant replacements of B and C, respectively. Then we have a map

f:B*A𝕃C:=B′*AC′→B*AC.

Consider the filtrations on B*AC and B′*AC′ described in Lemma 2.1. Then the map f induces a map on the associated graded to these filtrations and, since B¯,B′¯,C¯,C′¯ are cofibrant as left A-modules, we conclude that the E1-terms of the corresponding spectral sequences are isomorphic, and the desired conclusion follows. ∎

Remark 2.4.

It is natural to ask whether the conclusion of Corollary 2.3 holds under weaker conditions than A-cofibrancy of B and C; cf. Remark 2.2. Assuming that the unit maps A→B and A→C are injections and that B¯ and C¯ are flat as left A modules (disregarding the differential) allows to identify the associated graded of the appropriate filtration of B*AC. However, in order to ensure that the iterated tensor product of B¯ and C¯ and B computes the derived tensor product, one has to assume, in addition, that B¯ and C¯ are homotopically flat as left dg A-modules. Recall that a left dg A-module M is called homotopically flat if for any right A-module N it holds that N⊗A𝕃M≃N⊗AM, e.g., any cofibrant module is homotopically flat. Thus, B*AC computes the derived free product if the unit maps A→B,A→C are injections and B¯,C¯ are flat left A-modules as well as homotopically flat left A-modules (such dg modules are called semi-flat, cf. for example [7] regarding this nomenclature).

Of course, “left” can be replaced with “right” in the above discussion; cf. Remark 2.2. In particular, the conclusion of Corollary 2.3 holds under the assumption that the unit maps A→B and A→C are cofibrations of right dg A-modules.

3 Modules of relative differentials for dg algebras

In this section we recall the construction of the modules of relative differentials for dg algebras. The treatment of [11] extends to the dg case in an obvious manner.

Definition 3.1.

For a dg A-algebra B the module of relative differentials ΩA⁢(B) is the kernel of the multiplication map:

(3.1)ΩA⁢(B)→B⊗AB→B.

Thus, ΩA⁢(B) is a dg B-bimodule. If A→B′ is an A-cofibrant replacement of the A-algebra B, we define the derived module of relative differentials ΩA𝕃⁢(B):=ΩA⁢(B′). Thus, ΩA𝕃⁢(B) is well defined as an object in the homotopy category of dg B-bimodules.

Suppose that f:A→B is a cofibration of left dg A-modules. The short exact sequence (3.1) is split as in the category of left dg B-modules by a map B→B⊗AB, b↦b⊗1. The cokernel of the latter map is isomorphic as a left dg B-module to B⊗AB¯ (recall that B¯ is the cokernel of f). It follows that B⊗AB¯ can be identified with ΩA⁢(B) as a left dg B-module.

The formation of the module of relative differentials behaves well with respect to free products.

Lemma 3.2.

Let A,B and C be dg algebras.

There is an isomorphism of dg B*AC-bimodules:
ΩA⁢(B*AC)≅((B*AC)⊗BΩA⁢(B)⊗B(B*AC))⊕((B*AC)⊗CΩA⁢(C)⊗C(B*AC)).
The maps ΩA⁢(B)→ΩA⁢(B*AC) and ΩA⁢(C)→ΩA⁢(B*AC) induced by the canonical maps B→B*AC and C→B*AC are the compositions
ΩA⁢(B)→1⊗id⊗1(B*AC)⊗BΩA⁢(B)⊗B(B*AC)↪ΩA⁢(B*AC)
and
ΩA⁢(C)→1⊗id⊗1(B*AC)⊗BΩA⁢(C)⊗C(B*AC)↪ΩA⁢(B*AC).

Proof.

Part (i) of the statement above is [11, Theorem 5.8.8]. Unraveling the proof in op. cit. yields (ii). ∎

4 Equivalence of homotopy and homology epimorphisms

We will now introduce the notions of homotopy and homological epimorphisms for dg algebras and show that they are equivalent.

Definition 4.1.

Let f:A→B be a map of dg algebras making B into a dg A-bimodule.

f is said to be a homological epimorphism if the multiplication map B⊗A𝕃B→B is a quasi-isomorphism.
f is said to be a homotopy epimorphism if the codiagonal map B*A𝕃B→B is a quasi-isomorphism.

Remark 4.2.

Homotopy epimorphism can be defined in any closed model category. The notion of a homological epimorphism is more restrictive as it requires the structure of a closed model category (or something similar) on modules over monoids in a given closed model category. It would be interesting to investigate whether the equivalence between homological and homotopy epimorphism is a general categorical phenomenon (rather than special to dg algebras).

Homological epimorphisms of dg algebras were studied in [21]. Their exceptionally good property is that they induce smashing localization functors on the level of derived categories (indeed they are characterized by this property).

We have the following characterization of homological epimorphisms, whose proof is a straightforward consequence of definitions.

Lemma 4.3.

A map A→B is a homological epimorphism if an only if ΩAL⁢(B) is acyclic.∎

Theorem 4.4.

A dg algebra map A→B is a homological epimorphism if and only if it is a homotopy epimorphism.

Proof.

Without loss of generality we can assume that A→B is a cofibration of dg algebras; then we have B*AB≃B*A𝕃B. Suppose that A→B is a homological epimorphism. It suffices to show that the map id*1:B≅B*AA→B*AB is a quasi-isomorphism. Considering the filtration {Fn} on B*AB constructed in Lemma 2.1 and taking into account that B⊗AB¯≃0 since A→B is a homological epimorphism, we conclude that the associated graded quotients Fn/Fn-1 are acyclic for n>0 and so id*1:B→B*AB is indeed a quasi-isomorphism.

Conversely, suppose that A→B is a homotopy epimorphism; we will show that ΩA𝕃⁢(B)≃ΩA⁢(B) is acyclic. We have the following quasi-isomorphisms of B-bimodules:

ΩA⁢(B)≃ΩA⁢(B*AB)≃(B*AB⊗BΩA⁢(B)⊗BB*AB)⊕(B*AB⊗BΩA⁢(B)⊗BB*AB)

≃ΩA⁢(B)⊕ΩA⁢(B).

Here the second quasi-isomorphism follows from Lemma 3.2 (i) and the third follows since B*AB≃B and taking into account that B*AB is cofibrant both as a left and right B-module. By Lemma 3.2 (ii) the map

B→id*1B*AB

(which is a quasi-isomorphism) induces a quasi-isomorphism ΩA⁢(B)≃ΩA⁢(B)⊕ΩA⁢(B) that is also the inclusion into the left direct summand. This is only possible if ΩA⁢(B)≃0 as required.∎

5 𝐤-acyclic maps

Recall that there is a left Quillen functor G:SSet*↦SGp that is part of a Quillen equivalence between the categories of reduced simplicial sets and simplicial groups; cf. [14]. We will also consider the left Quillen functor C*:SSet*↦DGA𝐤 associating to a simplicial group H its normalized simplicial chain complex supplied with the Pontryagin product C*⁢(H,𝐤)⊗C*⁢(H,𝐤)→C*⁢(H,𝐤) induced by the simplicial group operation H×H→H.

Interpreted topologically, the composite functor C*∘G:SSet*→DGA𝐤 associates to a reduced simplicial set X a dg algebra that is quasi-isomorphic to the chain algebra on the loop space of |X|, the geometric realization of X. An obvious modification allows one to consider it as a functor from the homotopy category of connected (not necessarily reduced) spaces.

Definition 5.1.

A map between connected spaces is 𝐤-acyclic if its homotopy fiber F is 𝐤-acyclic, that is Hn⁢(F,𝐤)=0, n>0.

Remark 5.2.

It is customary to call ℤ-acyclic maps simply acyclic.

Lemma 5.3.

Let f:X→Y be a map between connected spaces. Then f is k-acyclic and induces an isomorphism π1⁢(X)≅π1⁢(Y) if and only if the induced map of dg algebras C*⁢(G⁢X,k)→C*⁢(G⁢Y,k) is a quasi-isomorphism.

Proof.

Suppose first that X and Y are both simply-connected. Let f be 𝐤-acyclic. It follows by considering the Serre spectral sequence of f that the dg coalgebras C*⁢(X,𝐤) and C*⁢(Y,𝐤) are quasi-isomorphic. Then the spectral sequences associated with cobar-constructions of the chain coalgebras C*⁢(X,𝐤) and C*⁢(Y,𝐤) converge strongly to H*⁢(G⁢X,𝐤) and H*⁢(G⁢Y,𝐤), respectively (this is where simple connectivity is needed) and it follows that the dg algebras C⁢(G⁢X,𝐤) and C*⁢(G⁢Y,𝐤) are indeed quasi-isomorphic.

Conversely, if the map of dg algebras C*⁢(G⁢X,𝐤)→C*⁢(G⁢Y,𝐤) is a quasi-isomorphism, then the bar-construction spectral sequence implies that f:X→Y induces an isomorphism on 𝐤-homology and thus, again by simple connectivity and the Serre spectral sequence of f, it is a 𝐤-acyclic map. The desired statement is therefore proved in the simply-connected case.

If X and Y are not simply-connected, denote by X¯ and Y¯ their universal covers and note that the condition that f:X→Y induces an isomorphism on fundamental groups implies that there is a homotopy pullback diagram of spaces:

(5.1)

where the map f¯ is induced by f. Then the homotopy fiber of f¯ is 𝐤-acyclic and since X¯,Y¯ are simply-connected, we obtain by the argument above that the dg algebras C*⁢(G⁢X¯,𝐤) and C*⁢(G⁢Y¯,𝐤) are quasi-isomorphic. But clearly there is a homotopy fiber sequence of simplicial groups

(5.2)G⁢X¯→G⁢X→G⁢B⁢π1⁢(X)≃π1⁢(X),

and similarly

(5.3)G⁢Y¯→G⁢X→G⁢B⁢π1⁢(Y)≃π1⁢(Y),

where B⁢π1⁢(X),B⁢π1⁢(Y) are classifying spaces of π1⁢(X) and π1⁢(Y), respectively. It follows that the maps of dg algebras C*⁢(G⁢X¯,𝐤)→C*⁢(G⁢X,𝐤) and C*⁢(G⁢Y¯,𝐤)→C*⁢(G⁢Y,𝐤) induce homology isomorphisms in positive degrees, and therefore the map C*⁢(G⁢X,𝐤)→C*⁢(G⁢Y,𝐤) also has this property. Finally,

H0⁢(G⁢X,𝐤)≅𝐤⁢[π1⁢(X)]→𝐤⁢[π1⁢(Y)]≅H0⁢(G⁢Y,𝐤)

is also an isomorphism and so the dg algebras C*⁢(G⁢X,𝐤) and C*⁢(G⁢Y,𝐤) are indeed quasi-isomorphic as claimed.

Conversely, suppose that the induced map C*⁢(G⁢X,𝐤)→C*⁢(G⁢Y,𝐤) is a quasi-isomorphism. In particular, it gives an isomorphism H0⁢(G⁢X,𝐤)≅𝐤⁢[π1⁢(X)]→𝐤⁢[π1⁢(Y)]≅H0⁢(G⁢Y,𝐤) which implies that f induces an isomorphism π1⁢(X)→π1⁢(Y). Again, we conclude that diagram (5.1) is a homotopy pullback. Similarly, we conclude from (5.2) and (5.3) that the dg algebras C*⁢(G⁢X¯,𝐤) and C*⁢(G⁢Y¯,𝐤) are quasi-isomorphic, and therefore (because of simple-connectivity of X¯ and Y¯) the map f¯ is a 𝐤-acyclic map. Therefore, by 5.1f is also a 𝐤-acyclic map. ∎

Remark 5.4.

If 𝐤=ℤ, then Lemma 5.3 implies that a map X→Y between connected spaces is a weak equivalence if and only if the induced map C*⁢(G⁢X,ℤ)→C*⁢(G⁢Y,ℤ) is a quasi-isomorphism. Surprisingly, this simple and fundamental fact appears to have been noticed only recently; cf. [8, 23].

Theorem 5.5.

Let f:X→Y be a map between two connected spaces. Then the following are equivalent:

f is a 𝐤-acyclic map.
The induced map of dg algebras C*⁢(G⁢X,𝐤)→C*⁢(G⁢Y,𝐤) is a homotopy epimorphism.
The induced map of dg algebras C*⁢(G⁢X,𝐤)→C*⁢(G⁢Y,𝐤) is a homological epimorphism.

Proof.

In light of Theorem 4.4 it suffices to prove the equivalence of (i) and (ii). Without loss of generality we assume that X and Y are reduced (as opposed to merely connected) simplicial sets. The functor X↦C*⁢(G⁢X,𝐤) is a composition of two left Quillen functors and thus is itself left Quillen, and so it preserves homotopy pushouts. It follows that there is a quasi-isomorphism of dg algebras

C*⁢(G⁢(Y*X𝕃Y),𝐤)≃C*⁢(G⁢Y,𝐤)*C*⁢(G⁢X,𝐤)𝕃C*⁢(G⁢Y,𝐤).

Let f:X→Y be a 𝐤-acyclic map. Then the map Y→Y*X𝕃Y mapping Y to the first (or second) wedge component of Y*X𝕃Y is likewise 𝐤-acyclic and, moreover, by the van Kampen theorem, induces an isomorphism on the fundamental groups. By Lemma 5.3, the corresponding map

C*⁢(G⁢Y,𝐤)→C*⁢(G⁢(Y*X𝕃Y),𝐤)≃C*⁢(G⁢Y,𝐤)*C*⁢(G⁢X,𝐤)𝕃C*⁢(G⁢Y,𝐤)

is a quasi-isomorphism, so C*⁢(G⁢X,𝐤)→C*⁢(G⁢Y,𝐤) is a homotopy epimorphism. This chain of implications is clearly reversible, and so we obtain the desired if and only if statement. ∎

Let 𝐤 be a field. It is known [15] that for a connected simplicial set X the derived category D𝐤⁢(|X|) of cohomologically locally constant sheaves of 𝐤-modules on |X| the topological realization of X (also known as infinity local systems on |X|) is equivalent to D⁢(C*⁢(G⁢X,𝐤)), the derived category of the chain algebra on the based loop space of X. This leads to the following result.

Corollary 5.6.

A map f:X→Y of connected spaces is k-acyclic if an only if the inverse image functor f-1:Dk⁢(|Y|)→Dk⁢(|X|) is fully faithful.

Proof.

By the correspondence between cohomologically locally constant sheaves and modules on the chain algebra of based loop spaces mentioned above, the functor f-1:D𝐤⁢(|Y|)→D𝐤⁢(|X|) is fully faithful if an only if the restriction functor D⁢(C*⁢(G⁢Y,𝐤))→D⁢(C*⁢(G⁢X,𝐤)) is fully faithful. The latter is equivalent by [21, Theorem 3.9 (6)] to the map C*(GY,𝐤))→C*(GX,𝐤) being a homological epimorphism, which, by Theorem 5.5, is equivalent to f:X→Y being a 𝐤-acyclic map. ∎

6 Plus-construction and derived localization

6.1 Recollection on plus-construction, localization and completion of spaces

A 𝐤-plus-construction (called the partial 𝐤-completion in [4]) of a connected space X is a space X𝐤+ supplied with a map X→X𝐤+ that is 𝐤-acyclic and is terminal among homotopy classes of 𝐤-acyclic maps out of X. The space X𝐤+ is local with respect to 𝐤-acyclic spaces, i.e. if Y is 𝐤-acyclic, then any basepointed map Y→X𝐤+ is homotopic to the constant map; and the map X→X𝐤+ is initial among homotopy classes of maps from X into local spaces.

A closely related notion is that of a localization with respect to the homology theory H⁢𝐤; for a space X its H⁢𝐤 localization is a space LH⁢𝐤⁢X supplied with a map X→LH⁢𝐤⁢X inducing an isomorphism in homology with coefficients in 𝐤 and terminal among homotopy classes of such maps (note that H⁢𝐤 localization is called semi-𝐤-completion in [4]). The space LH⁢𝐤⁢X is local with respect to H⁢𝐤-homology equivalences, i.e. if f:Y→Z is an H⁢𝐤-homology equivalence, then the induced map on homotopy classes [Z,LH⁢𝐤⁢X]→[Y,LH⁢𝐤⁢X] is an isomorphism. Furthermore, the map X→LH⁢𝐤⁢X is initial among homotopy classes of maps from X into LH⁢𝐤-local spaces. The space LH⁢𝐤⁢X is “farther away” from X than X𝐤+ in the sense that the map X→LH⁢𝐤⁢X always factors through X𝐤+; in favorable cases X𝐤+≃LH⁢𝐤⁢X.

Finally, there is a notion of a 𝐤-completion X→X𝐤∧; the latter map always factors through LH⁢𝐤. Often we have LH⁢𝐤⁢X≃X𝐤∧; in this case X is called 𝐤-good.

From now on, we will assume that 𝐤=𝔽p, although all arguments and statements below are valid for an arbitrary field of characteristic p. We will write Xp+,Xp and Xp∧ for X𝔽p+,LH⁢𝔽p⁢X and X𝔽p∧, respectively.

There is a class of spaces X for which Xp+ is particularly well-behaved. Recall that a group G is p-perfect if H1⁢(G,𝔽p)=0. The maximal p-perfect subgroup of G will be denoted by P⁡(G); it is normal in G and the quotient G/P⁡(G) is p-hypoabelian, i.e. it does not have nontrivial p-perfect subgroups.

Definition 6.1.

A group G is p-reasonable if G contains a p-perfect subgroup of finite index (equivalently, G/P⁡(G) is a finite p-group). A connected space X is p-reasonable if π1⁢(X) is.

Remark 6.2.

Clearly, if G is p-reasonable, then the augmentation ideal in the group ring 𝔽p⁢[G/P⁡(G)] is nilpotent. Conversely, if 𝔽p⁢[G/P⁡(G)] has this property then G/P⁡(G) is a finite p-group by [12], and therefore G is p-reasonable.

Proposition 6.3.

Let X be p-reasonable. Then the following assertions hold:

X is p-good (so that Xp∧≃Xp).
π1⁢(Xp∧)≅π1⁢(X)/P⁡(π1⁢(X)).
There is a weak equivalence Xp≃Xp+.

Proof.

Parts (i) and (ii) of the required statement are proved [1, Part III, Proposition 1.11] and similar arguments also prove (iii). Namely, consider the canonical map f:X→B⁢[π1⁢(X)/P⁡(π1⁢(X))] induced by the quotient map π1⁢(X)→π1⁢(X)/P⁡(π1⁢(X)) and recall from [4, Chapter VII, Section 6.2] that Xp+ is constructed as a fibrewise completion of the map f (converted into a fibration). Denoting the homotopy fiber of f by F, we have therefore a homotopy fiber sequence of spaces

(6.1)Fp∧→Xp+→B⁢[π1⁢(X)/P⁡(π1⁢(X))].

Now apply the functor of p-completion to the homotopy fiber sequence F→X→B⁢[π1⁢(X)/P⁡(π1⁢(X))]. We obtain

(6.2)Fp∧→Xp∧→B⁢[π1⁢(X)/P⁡(π1⁢(X))]p∧

and this is also a homotopy fiber sequence by the p-mod Fibre Lemma of [4, Chapter II, Section 5.1] and taking into account that the p-group π1⁢(X)/P⁡(π1⁢(X)) acts nilpotently on H~*⁢(F,𝔽p). There is a map from (6.1) to (6.2) that is a weak equivalence on end terms (since B⁢[π1⁢(X)/P⁡(π1⁢(X))] is already p-complete) and it follows that it is a weak equivalence on the middle terms as required. ∎

6.2 Derived localization of dg algebras

Let A be a dg algebra and let s∈H0⁢(A) be a zero-dimensional cycle in A. In this situation one can construct [5] another dg algebra Ls⁢A together with a dg algebra map f:A→Ls⁢(A) such that f⁢(s) is invertible in H0⁢(Ls⁢A) and initial among the homotopy classes of maps out of A with this property. Moreover, the map A→Ls⁢A can be interpreted as the Bousfield localization of A as a (left) dg module over itself with respect to the map rs:A→A, rs⁢(a)=a⁢s, a∈A. It is also the nullification of A with respect to A/s, the cofiber of rs; cf. [13, Section 4.10 and Proposition 4.11] regarding this result and terminology. There is a homotopy fiber sequence of left dg A-modules

(6.3)Ls⁢A→A→Ls⁢A

where Ls⁢A is the s-colocalization of A (also known as A/s-cellularization and denoted by CellA/s⁡(A) in [13]). This is a kind of a dualizing complex for left dg A modules relative to A/s and it has a nice interpretation in terms of classical homological algebra (cf. [13, Section 4]):

Ls⁢(A)≃RHomA⁡(A/s,A)⊗REndA⁡(A/s,A/s)𝕃A/s.

The following example taken from [13, Subsection 4.1] is instructive.

Example 6.4.

Let A=ℤ, and let s=p so A/s≅ℤ/p, the cyclic group of prime order p. Then Lp⁢ℤ≃ℤ⁢[1p], and from (6.3) we obtain

Lp(ℤ)≃Σ-1(ℤ[1p]/ℤ)=:Σ-1ℤ/p∞.

Assume that s=e is an idempotent in H0⁢(A); then Le⁢A is quasi-isomorphic as a left A-module to the Bousfield localization of A with respect to the localizing subcategory generated by the left dg A-module A⁢(1-e) (since A⁢(1-e) and A/e≃A⁢(1-e)⊕Σ⁢A⁢(1-e) generate the same localizing subcategory). Therefore, by [13] there is a quasi-isomorphism of left dg A-modules:

Le⁢(A)≃RHomA⁡(A⁢(1-e),A)⊗REndA⁡(A⁢(1-e),A⁢(1-e))𝕃A⁢(1-e)

(6.4)≃A⁢(1-e)⊗(1-e)⁢A⁢(1-e)𝕃(1-e)⁢A.

The A-bimodule

I:=A⁢(1-e)⊗(1-e)⁢A⁢(1-e)𝕃(1-e)⁢A

can be viewed as a “derived two-sided ideal” generated by the idempotent 1-e in A in the sense that the homotopy cofiber A/I (“derived quotient”) of A by I is quasi-isomorphic to the derived localization Le⁢A. This resembles quotienting out by a two-sided ideal generated by an idempotent in a nonderived context.

6.3 Algebraic description of the loop space of a p-plus-construction

Let X be a connected space; below we will write G⁢Xp+ and G⁢Xp∧ for G⁢(Xp+) and G⁢(Xp∧), respectively. It follows from Theorem 5.5 that the map of dg algebras C*⁢(G⁢X,𝔽p)→C*⁢(G⁢Xp+,𝔽p) is a homology epimorphism, and so it is natural to ask for an algebraic description of this map. We will give such a description for a p-reasonable space; the result is particularly pleasant when π1⁢(X) is a finite group.

Set A:=C*⁢(G⁢X,𝔽p). There is a canonical map C*⁢(G⁢X,𝔽p)→H0⁢(G⁢X,𝔽p)≅𝔽p⁢[π1⁢(X)] in the homotopy category of dg 𝔽p-algebras and, since 𝔽p⁢[π1⁢(X)] is augmented, so is C*⁢(G⁢X,𝔽p). Consider the homology functor H⁢𝔽p:M↦𝔽p⊗A𝕃M on the category of dg A-modules. The notation H⁢𝔽p is designed to invoke an analogy with the homology functor in the stable homotopy category given by smashing with the mod-p Eilenberg–MacLane spectrum (in fact, this is not merely an analogy since the category of dg algebras is Quillen equivalent to the category of algebras over the integral Eilenberg–MacLane spectrum H⁢ℤ, cf. [24]). We will denote by M∧ the Bousfield localization of an A-module M with respect to this homology functor. Thus, we have a canonical map M→M∧ that is an H⁢𝔽p-equivalence (i.e. induces a quasi-isomorphism upon applying H⁢𝔽p) and such that M∧ is H⁢𝔽p-local (i.e. it does not admit homotopy nontrivial maps from H⁢𝔽p-acyclic modules).

Theorem 6.5.

Let X be a connected space. Then the following assertions hold:

If X is p-reasonable, then C*⁢(G⁢Xp+,𝔽p) is quasi-isomorphic to C*⁢(G⁢X,𝔽p)∧ and the canonical map
C*⁢(G⁢X,𝔽p)→C⁢(G⁢Xp+,𝔽p)
is the H⁢𝔽p-localization map on the category of left dg C*⁢(G⁢X,𝔽p)-modules.
If π1⁢(X) is a finite group then there exists an idempotent e in 𝔽p⁢[π1⁢(X)] such that the dg algebra C*⁢(G⁢Xp+,𝔽p) is quasi-isomorphic to the derived localization Le⁢C*⁢(G⁢X,𝔽p) and C*⁢(G⁢X,𝔽p)→C*⁢(G⁢Xp+,𝔽p) is the e-localization map.

Proof.

Let X be p-reasonable and let A:=C*⁢(G⁢X,𝔽p) and L⁢A:=C*⁢(G⁢Xp+,𝔽p). To show that L⁢A≃A∧ we need to show the following:

The map A→L⁢A is an H⁢𝔽p-equivalence.
LA is H⁢𝔽p-local.

Denote by M the homotopy fiber of A→L⁢A; then M⊗A𝕃L⁢A≃0 and so

𝔽p⊗A𝕃M≃𝔽p⊗A𝕃L⁢A⊗L⁢A𝕃M≃0,

which implies (a). Next, note that any dg A-module whose A-action is through the augmentation map A→𝔽p is H⁢𝔽p-local. Indeed, if Y is such an A-module and N is H⁢𝔽p-acyclic (i.e. 𝔽p⊗A𝕃N≃0), then

RHomA⁡(N,Y)≃RHom𝔽p⁡(𝔽p⊗A𝕃N,Y)≃0.

Since X is p-reasonable, H0⁢(L⁢A)≅𝔽p⁢[π1⁢(Xp+)] is the 𝔽p-group ring of a finite p-group and is, thus, a local 𝔽p-algebra with a finite-dimensional nilpotent maximal ideal. Any dg H0⁢(L⁢A)-module has a finite filtration induced by the powers of the maximal ideal in H0⁢(L⁢A) with associated graded quotients being 𝔽p-modules, which implies that any such module is H⁢𝔽p-local. The dg A-module LA can be represented as a homotopy inverse limit of its Postnikov tower {L⁢A⁢[n]:n=0,1,…} (so that L⁢A⁢[n] has the same homology as LA up to and including degree n and zero in higher degrees). The homotopy cofibers L⁢A⁢[n]/L⁢A⁢[n+1] are isomorphic as objects in D⁢(A) to dg H0⁢(L⁢A)-modules, and thus are H⁢𝔽p-local. This implies that LA is H⁢𝔽p-local and part (i) is therefore proved.

For part (ii) let e be a primitive idempotent of 𝔽p⁢[π1⁢(X)] which acts as the identity on the trivial module of 𝔽p⁢[π1⁢(X)] and by 0 on other simple modules; it is unique up to conjugation by a unit of 𝔽p⁢[π1⁢(X)]. Then the localization map f:A→Le⁢A is easily seen to be the H⁢𝔽p-localization, so the statement follows by part (i). Indeed, the homotopy fiber Ae of f has the property that Le⁢Ae≃0, but then

𝔽p⊗A𝕃Ae≃𝔽p⊗A𝕃Le⁢A⊗AAe≃𝔽p⊗A𝕃Le⁢Ae≃0,

so Ae is H⁢𝔽p-acyclic and f is an H⁢𝔽p-local equivalence. Also H0⁢(Le⁢A) is a local ring with residue field 𝔽p so Le⁢A is H⁢𝔽p-local. ∎

The following is a straightforward consequence of Theorem 6.5 (i).

Corollary 6.6.

Let X be a p-reasonable space. The homotopy epimorphism of dg algebras

C*⁢(G⁢X,𝔽p)→C*⁢(G⁢Xp∧,𝔽p)

induces a smashing localization functor on the derived categories

D⁢(C*⁢(G⁢X,𝔽p))→D⁢(C*⁢(G⁢Xp∧,𝔽p))

that coincides with H⁢Fp-localization on the perfect subcategory of D⁢(C*⁢(G⁢Xp∧,Fp)).∎

Remark 6.7.

Note that the H⁢𝔽p-localization functor on the full derived category of C*⁢(G⁢X,𝔽p) is not necessarily smashing, even if it is so on the perfect subcategory.

Corollary 6.8.

Let X be a connected space with π1⁢(X) finite and denote by e an idempotent in Fp⁢[π1⁢(X)] acting as the identity on the trivial π1⁢(X)-module and zero on other simple π1⁢(X)-modules. Then there is a homotopy cofiber sequence of dg C*⁢(G⁢X,Fp)-modules:

C*⁢(G⁢X,𝔽p)⁢(1-e)⊗(1-e)⁢C*⁢(G⁢X,𝔽p)⁢(1-e)𝕃(1-e)⁢C*⁢(G⁢X,𝔽p)→C*⁢(G⁢X,𝔽p)→C*⁢(G⁢Xp+,𝔽p).

Proof.

This follows directly from Theorem 6.5 (ii), taking into account (6.3) and (6.4). ∎

Remark 6.9.

Let us consider the case X=B⁢H where H is a finite group; the homotopy theory of the loop space on B⁢Hp∧≃B⁢Hp+ has been studied extensively, see the review paper [9] and the more recent [2]. From Corollary 6.8 we obtain a homotopy fiber sequence

(6.5)𝔽p⁢[H]⁢(1-e)⊗(1-e)⁢𝔽p⁢[H]⁢(1-e)𝕃(1-e)⁢𝔽p⁢[H]→𝔽p⁢[H]→C*⁢(G⁢(B⁢Hp+),𝔽p).

Next, taking the homology long exact sequence of (6.5), we obtain, for n>1,

Hn⁢(G⁢(B⁢Hp+),𝔽p)≅Torn-1(1-e)⁢𝔽p⁢[H]⁢(1-e)⁡(𝔽p⁢[H]⁢(1-e),(1-e)⁢𝔽p⁢[H]),

while for n=1 we have an exact sequence

0→H1⁢(G⁢(B⁢Hp+),𝔽p)→𝔽p⁢[H]⁢(1-e)⊗(1-e)⁢𝔽p⁢[H]⁢(1-e)(1-e)⁢𝔽p⁢[H]→𝔽p⁢[H]→𝔽p⁢[H/P⁡(H)]→0,

which is the main result of [2] obtained by different methods.

Since the localization functor L:D⁢(C*⁢(G⁢X,𝔽p))→D⁢(C*⁢(G⁢Xp+,𝔽p)) is always smashing, it makes sense to ask whether it is finite, i.e. whether the localizing subcategory Ker⁡(L)⊆D⁢(C*⁢(G⁢X,𝔽p)) is compactly generated. Theorem 6.5 (ii) tells us that it is the case when π1⁢(X) is finite (indeed, in this case it is even a derived localization at a single element). It turns out that Ker⁡L may not be compactly generated, even when X is a classifying space of an abelian group. To see that, let us note first the following useful derived analogue of Nakayama’s lemma; here Perf⁡(A) stands for the perfect derived category of a ring A.

Lemma 6.10.

Let A be an ordinary ring and let S=A/J be the quotient by its Jacobson radical J. Then the functor S⊗AL-:Perf(A)→Perf(S) has the zero kernel.

Proof.

This is [17, Lemma 5.3]. ∎

Example 6.11.

Let G be a locally finite p-group; then 𝔽p⁢[G] is local by [20], and so its augmentation ideal coincides with the Jacobson radical. Assume additionally that G is divisible, e.g., G=ℤ/p∞≅⋃n=1∞ℤ/pn. It follows from the divisibility of G that H1⁢(G,𝔽p)=0, and so G is p-perfect. So, by Theorem 6.5 (i) we have the smashing localization functor L:D⁢(𝔽p⁢[G])→D⁢(LH⁢𝔽p⁢𝔽p⁢[G]); moreover, L coincides with the H⁢𝔽p-localization when restricted to perfect dg 𝔽p⁢[G]-modules by Corollary 6.6. Thus, a perfect dg 𝔽p⁢[G]-module M is in Ker⁡L if and only if

H*⁢(𝔽p⊗𝔽p⁢[G]𝕃M)=0,

and by Lemma 6.10, we have M≃0.

Remark 6.12.

The above example is a modification of Keller’s counterexample to the Telescope Conjecture [16].

Example 6.13.

Let p,q be prime numbers such that q divides p-1. Then ℤ/q acts faithfully on ℤ/p and we can form the semidirect product ℤ/p⋊ℤ/q, clearly it is p-perfect. Let X:=B⁢(ℤ/p⋊ℤ/q) An easy calculation with the Serre–Hochschild spectral sequence associated with the normal subgroup ℤ/p of ℤ/p⋊ℤ/q shows that there is an isomorphism of graded algebras

H*⁢(Xp+,𝔽p)≅H*⁢(ℤ/p⋊ℤ/q,𝔽p)≅𝔽p⁢[x′,y′]

with x′,y′ situated in cohomological degrees 2⁢q-1 and 2⁢q, respectively. Since Xp+ is simply-connected, this implies that its lowest homotopy group is π2⁢q-1⁢Xp+≅ℤ/p. It follows that there exists a homotopy fiber sequence

(S2⁢q-1)p∧→𝑗(S2⁢q-1)p∧→Xp+

where j is the self-map of the p-completed 2⁢q-1-sphere (S2⁢q-1)p∧ of degree 2⁢q-1 (this argument is presented in [4, Chapter VII, Proposition 4.4] for p=3,q=2). Thus, G⁢Xp+ is the homotopy fiber of j; in particular, the latter is a loop space. This is a well-known fact and it is also known that this loop space structure is unique [6, Corollary 1.2].

Now let p>3 or q>2. The homology of G⁢Xp+ can be computed with the help of the cobar spectral sequence whose E2 term is

ExtH*⁢(Xp+,𝔽p)⁡(𝔽p,𝔽p)≅𝔽p⁢[x,y]

with |x|=2⁢q-2 and |y|=2⁢q-1. Taking into account the Hopf algebra structure on this spectral sequence with x,y primitive, we conclude that it collapses and there is an isomorphism of graded algebras

H*⁢(G⁢Xp+,𝔽p)≅𝔽p⁢[x,y].

Note that the dg algebra C*⁢(G⁢Xp+,𝔽p) is not formal, i.e. it is not quasi-isomorphic to its homology. Indeed, if it were, then C*⁢(Xp+,𝔽p) would likewise be formal and quasi-isomorphic to H*⁢(Xp+,𝔽p)≅𝔽p⁢[x′,y′]. However, such a result is incompatible with the minimal model of C*⁢(ℤ/p,𝔽p) computed in [18].

The case p=3,q=3 requires a separate treatment since the Hopf algebra structure on the cobar spectral sequence for G⁢Xp+ is insufficient for deducing the multiplicative structure on H*⁢(G⁢Xp+,𝔽p). Note also that in this case X≅B⁢S3, the classifying space of the symmetric group on three letters. An idempotent in 𝔽3⁢[S3] figuring in the formulation of Theorem 6.5 can be found explicitly; some possible choices are -(12)-1,-(13)-1 and -(23)-1. Let e be any one of these. Then the dg algebra Le⁢𝔽3⁢[S3]≃C*⁢(G⁢X3+,𝔽3) is formal and quasi-isomorphic to the graded algebra (with zero differential) generated over 𝔽3 by two indeterminates x,y with |x|=2 and |y|=3 subject to the relation x⁢y=y⁢x and x3=y2 (note that this is not a graded commutative algebra, although its associated graded with respect to the powers of the maximal ideal (x,y) is isomorphic to the graded commutative polynomial algebra 𝔽3⁢[x,y]).

To obtain this result, we will realize 𝔽3⁢[S3]∧ as a “squeezed resolution”, following Benson [2]. Let α:A⁢e→A⁢(1-e) and β:A⁢(1-e)→A⁢e be nonzero homomorphisms; both are unique up to multiplication by a nonzero scalar. Then the dg A-module P=(Pi,di) given by

Pi={A⁢(1-e)if ⁢i>0,A⁢eif ⁢i=0,0if ⁢i<0, di={α∘βif ⁢i>1,βif ⁢i=1,0if ⁢i<1,

is a left 𝔽3-squeezed resolution for S3, in the sense of [2, Definition 3.2]. The chain maps z⁢(n):P→Σ-n⁢P given by

z⁢(n)i={Σ-nif ⁢i>0,Σ-n∘αif ⁢i=0,0if ⁢i<n,

for n≥2, together with the identity map on P, descend to a basis of

HomD⁢(A)∗⁡(P,P)≅HomD⁢(A)∗⁡(A,P)≅H⁢(P).

Since z⁢(m)∘z⁢(n)=z⁢(m+n) for all m,n≥2, as is easily verified, it follows that Le⁢𝔽3⁢[S3]≃RHomA∗⁡(P,P) is quasi-isomorphic to the subalgebra generated by z⁢(2) and z⁢(3), which is a graded algebra with the claimed presentation.

Remark 6.14.

Some examples of computation of H*⁢(G⁢Xp+,𝔽p) for X being a classifying space of a finite group are presented in [3, Section 13C], in particular when this finite group is ℤ/p⋊ℤ/q. The special case p=3,q=2 was not noticed in op. cit.

Communicated by Frederick R. Cohen

Funding source: Engineering and Physical Sciences Research Council

Award Identifier / Grant number: EP/N015452/1

Award Identifier / Grant number: EP/N016505/1

Funding statement: This work was partially supported by EPSRC grants EP/N015452/1 and EP/N016505/1.

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Received: 2019-09-06

Revised: 2020-02-02

Published Online: 2020-07-16

Published in Print: 2020-11-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Homological epimorphisms, homotopy epimorphisms and acyclic maps

Abstract

1 Introduction

1.1 Notation and conventions

2 Derived free products of dg algebras

Lemma 2.1.

Proof.

Remark 2.2.

Corollary 2.3.

Proof.

Remark 2.4.

3 Modules of relative differentials for dg algebras

Definition 3.1.

Lemma 3.2.

Proof.

4 Equivalence of homotopy and homology epimorphisms

Definition 4.1.

Remark 4.2.

Lemma 4.3.

Theorem 4.4.

Proof.

5 𝐤-acyclic maps

Definition 5.1.

Remark 5.2.

Lemma 5.3.

Proof.

Remark 5.4.

Theorem 5.5.

Proof.

Corollary 5.6.

Proof.

6 Plus-construction and derived localization

6.1 Recollection on plus-construction, localization and completion of spaces

Definition 6.1.

Remark 6.2.

Proposition 6.3.

Proof.

6.2 Derived localization of dg algebras

Example 6.4.

6.3 Algebraic description of the loop space of a p-plus-construction

Theorem 6.5.

Proof.

Corollary 6.6.

Remark 6.7.

Corollary 6.8.

Proof.

Remark 6.9.

Lemma 6.10.

Proof.

Example 6.11.

Remark 6.12.

Example 6.13.

Remark 6.14.

References

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