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Injectivity of functors to modules and dga'sMarek Golasiński a

a Faculty of Mathematics and Informatics , Nicholas CopernicusUniversity , Chopina 12 / 18, Toruń, 87-100, Poland E-mail:Published online: 27 Jun 2007.

To cite this article: Marek Golasiński (1999) Injectivity of functors to modules and dga's, Communicationsin Algebra, 27:8, 4027-4038, DOI: 10.1080/00927879908826680

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COMh4UNICATIONS IN ALGEBRA, 27(8), 40274038 (1999)

INJECTIVITY OF FUNCTORS T O MODULES AND

Marek Golasiriski

Facul ty of Mathemat ics a n d Informatics Nicholas Copernicus University 87-100 Toruli, Chopina 12/18 P o l a n d e-mail: marekOmat.uni.torun.pl

Abstract . We study injective modules over an EI-category and present their complete de-

scription Given a G-disconnected simplicial set S let O(G, X ) be the EI-category with one object

for each connected component of its fixed point simplicial subsets with respect to all subgroups

of G. We show that the associated O(G,X)-system A> of de Rham algebras is componentwise

injective. Finally, an equivariant de Rham theorem for G-disconnected simplicial sets is derived.

Introduction. Several different approaches to rational homotopy theory have

been devised over the years. Sullivan (19)) introduced his rational de Rham theory

for connected complexes and applied it to show that the de Rham algebra A> of Q-

differential forms on a simply connected complex X determines its rational homotopy

type. Bousfield and Gugenheim ([I]) overcame the naturality problems of Sullivan's

theory by combining the methods of Quillen ([8]) and Sullivan (19)) constructing a

pair of adjoint functors

Copyright O 1999 by Marcel Dekker, Inc

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between the category of commutative differential graded algebras DGAo over the rationals Q and the category of simplicial sets $$. These functors induce inverse

equivalences between the appropriate homotopy categories of connected simplicial

sets and differential graded algebras over Q (Sullivan-de Rham he or em).

We say that a G-simplicia1 complex X with G a finite group is G-connected if

its fixed point subcomplexes XH are connected for all subgroups H C G; otherwise

a G-simplicia1 complex is called to be G-disconnected. As it was observed in [lo],

we have to consider in the G-connected case not only A;, but also the de Rham

algebras Ak1, for all subgroups H C G. It means that we play with a functor Ak on

the category O(G) of canonical orbits and its componentwise injectivity is the key

property. The G-disconnected case was first studied in [5], where the category O(G) has been replaced by an infinite (in general) category O(G,X) , with one object

for each connected component of each fixed point simplicial subcomplex X" of a

G-simplicia1 complex X. for any subgroup H C G. On the other hand, to get a

reasonable homotopy group functor associated with a G-simplicia1 complex X we

have to replace O(G, X ) by the infinite category Oi(G, X ) with one object for each

point of each fixed point simplicial complex of X. Therefore, the used methods in

[5]* can be only applied for G-simplicia1 complexes of finite G-type and finitely G- connected. A generalized and completed version of [5] for G-simplicia1 sets has been

published in the paper [6]. Its main result is based on our (Theorem 2.3) which shows

that for any G-simplicia1 set X the functor A> on the infinite category O(G, X) is

componentwise injective and it can be decomposed into a product of elementary

pieces. We point out that the methods presented in [5, 101 to show that result for

the finite case cannot be extended to the infinite one.

In section 1 we present some necessary algebraic background. The notion of

a (left) module M over a category C (or an RC-module), i.e., a covariant functor

M : 6 -+ R-mod into the category of left R-modules, is a generalization of a (left)

module over a group ring RIG]. We keep all notions as close as possible to this special

case. Modules over a category appear in equivariant obstruction theory (12, 7]), e.g.,

a local coefficient system for a G-simplicia1 set X is an II(G, X)-module over the

fundamental category II(G; X ) . The category of RC-modules inherits the structure

of an abelian category from R-mod, so those notions like exact sequence, injective,

injective resolution, homology are defined. Of course, the category RC-mod possesses

'The author has been informed by B.L. Fine that his equivarinat results work for finite cyclic

groups only.

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INJECTIVITY OF FUNCTORS TO MODULES AND DGA'S

sufficiently many projectives and injectives. In (3, 10) there are given complete

descriptions of projective covers and injective envelopes in the category QU(G)-mod,

where & is the field of rationals. For simplicity we can forget a complicated structure

of some geometric categories, but consider any EI-category (i.e., a small category

such that all endomorphisms are isomorphisms). In 17) it is shown how one can

split a projective module over an EI-category C into a direct sum of projective

modules living over the various group rings RAut(C) for C E ObC. We study

injective modules over an EI-category and give their complete description (Theorem

1.2). Then we move to section 2, investigate injective envelopes, the structure of

injective O(G, X)-modules and show that the functor A> of g-polynomial forms on

a G-simplicia1 set X is componentwise injective. Finally, the equivariant de Rham

theorem (Theorem 2.5) for G-disconnected simplicia1 sets is deduced.

Modules over a category and a splitting of injectives. Let R

be a commutative ring with unit. The category of (left) R-modules is denoted by

R-Mod. If C is a small category then a covariant functor C -t R-Mod is called a left

RC-module and the functor category of left RC-modules is denoted by RC-Mod and

called the category of left RC-modules. We also have the category of contravariant

functors C -t R-Mod, alias right RC-modules and denoted by Mod-RC.

The notions submodule, quotient module, kernel, image and cokernel for RC-

modules are defined objectwise. For each object C E ObC we have the right RC-

module

Rc(-, C ) : C --+ R-Mod,

where Rc(C1, C ) is the free R-module with the set of maps c(C', C ) in the category

C as a basis for any object C' E ObC. Similarly, we define the left RC-module

RC(C, -). An injective RC-module is defined by the usual extension property.

Now we describe injective RC-modules for geometrically relevant categories C. In

various categories considered in algebraic topology endomorphisms are isomorphisms.

Therefore, let C be an EI-category, which by definition, is a small category in which

each endomorphism is an isomorphism. Following [7] we define a partial order (which

is crucial for the sequel) on the set Is(c) of isomorphism classes C of objects C in C

by - C 5 77 if C(C, D) # 0.

This induces a partial ordering on the set Is(c) of isomorphism classes of objects,

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since the EI-property ensures that c 5 D and IS 5 ?? implies ?? = D. We write

that c < ti if c 5 ti and c # D. Our aim is to analyse injective RC-modules. It

turns out that they can be constructed from injective modules over group rings. If

C is an object of c with automorphism group Aut(C), we let R[C] = RAut(C) be

the group ring of Aut(C) and write R[C]-Mod for the category of left R[C]-modules.

For fixed C E C we introduce the following covariant functors.

Cosplitting functor Sc : Rc-Mod -+ R[C]-Mod.

If M be an RC-module, let Sc(M) be the R[C]-submodule of M(C) equal to the

intersection of kernels of all R-homomorphisms M(f ) : M(C) -+ M(D) induced by

all non isomorphisms f : C -t D with C as source. Each automorphism g E Aut(C)

induces a map h4(g) : M(C) -+ M(C) which maps Sc(M) into itself. Thus Sc(M)

becomes a left R[C]-module. It is clear how Sc is defined on morphisms.

Restriction functor Resc : RC-Mod -+ R[C]-Mod sends M to M(C) .

Coextenszon functor Ec : R[C]-Mod -+ RC-Mod sends N to HomRrci(RC(-, C), N) Coinclusion functor Ic : R[C]-Mod -+ Rc-Mod assigns to R[C]-module N the

RC-module Ic(N) defined by

We call an RC-module M to be of type T, for T Is(C), if the set {c E Is(c) I M(C) # 0) is contained in T. For any E T choose a representative

C E and fix an RIG]-monomorphism

where Qc is an injective R[C]-module. If M is of type T then we get a monomorphism

of RC-modules

0 + M -+ n EcQc. - CET

In particular, i t follows that any injective RC-module of type T is a direct summand

of an RC-module ncET EcQc, where Qc are injective RIG]-modules for c E T .

Then the next result (cf. [7, Lemma 9.311) follows easily from the definitions and

the remark above.

Lemma 1.1. a) The functors Ec and Resc and thefunctors Sc and Ic are adjoint,

ie. , there are natural isomorphisms of R-modules

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INJECTIVITY OF FUNCTORS TO MODULES AND DGA'S

and

Hom~[c](N, S c M ) - -+ H o m ~ c (IcN, M ) .

b) Sc o Ec : R[C]-Mod -+ R[C]-Mod is naturally equivalent to the identity func-

tor. The composition SD o Ec is zero for ?? # n. c) Sc and Ec preserve monomorphisms and injective modules.

Let now M be an RC-module and To IsC. Then the adjunction induces a

natural Rc-map

J c M : M --, Ec o RescM

for any object C in C. The product of these maps J c M running over c E To yields

JTo M : M ---r fl Ec o Resc M. - CETo

Define kerToM as the kernel of JT,M. If M is of type T then kerT,M is of type

T\(T C7 To). We are now in a position to dualize Theorem 9.39 from [7, p.1741.

Theorem 1.2 (Filtration Theorem for injective RC-modules). Let C be an

EI-category and 0 = To c TI . . . c = T a filtration of a subset T C_ Is(c) such

that c E T,, D E Tj , ?? < D implies i > j . If Q is an injective RC-module of type T

then there is a natural filtration b y RC-modules

b) Q(') is injective of type T\Ti.

c) Let in' : Q(') -+ Q be the composition inl o inz 0. . . o h , . Then Scini : ScQ(" -+

ScQ is an isomorphism for E T\T,.

d) There zs a natural exact sequence which splits (not naturally with respect to

Q) o - Q(') Q('-') - TIC~T,~T,-, EC 0 SCQ --+ 0.

Conversely, if an RC-module Q of type T such that ScQ are injective R[C]-

modules for E T satisfies a), c), d) then the modules Q(" are injective.

Proof. We proceed inductively on i = 0,1,. . . ,1 . Let Q(O) = Q and for i > 0 let Q(i) = kerT,~("-'), J, = J % Q ( ~ - ~ ) : Q(i-1) -+ ncETi E c ~ e s c ~ ( i - l ) .

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For i = 0 consider an exact sequence

Then

bl) Q(') is of type T\Tl,

cl) by Lemma 1.1 the induced map Scinl : ScQ(') + ScQ is an isomorphism for -

E T\Ti, dl) there is an exact sequence

since RescQ = ScQ for c E TI.

Let now Q(i) = kerT,Q('-') and assume that

bi) Q(" is of type T\T,,

c,) the induced map Scinh ScQ(" + ScQ is an isomorphism for E T\T,, di) there is an exact sequence

Consider an exact sequence

Then

bi+l) Q('+') is of type T\T,+',

c , + ~ ) by Lemma 1.1 the induced map Scin,+l : ScQ('+') + ScQ(') is an iso-

morphism for c E T\T,+l hence by the inductive hypothesis the induced map

Scin,+l : ScQ('+l) -+ ScQ(') is an isomorphism for C E T\Ti+l. Moreover, from

the construction it follows that Q('+')(C) = 0 for c E Ti+l. By inductive hypothesis

Q(') is of type T\Ti hence Q(')(C) = 0 for c E Ti. Therefore RescQ(') = ScQ(')

for ?? E Ti+, and composition with the isomorphisms Scin' : ScQ(') -, ScQ for - C E Ti+'\Ti yields

d,+l) there is an exact sequence

It is easy to see that the maps Ji are epimorphis~s for Q = ITFET ECQC, where

Qc are injective RIG]-modules. But any injective RC-module Q is a direct sum-

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INJECTIVITY OF FUNCTORS TO MODULES AND DGA'S 4033

mand of such a module and the maps J, are compatible with products. There-

fore J, are epimorphisms for i = 0,1, . . . ,1. In particular the map J I - ~ : Q('-') -+

&,,,,-, EcScQ is an isomorphism and Q('-') is an injective RC-module by Lemma

1.1. Finally, we get that the natural exact sequences

split (not naturally with respect to Q ) and Q(') are injective for i = 0 ,1 , . . . , I .

The proof of the converse implication is evident.

0

Applications. Recall that an injective envelope of an Rc-module M is a monomor-

phism 4 : M -4 Q, where Q is an injective Rc-module, with the property: for any

injective RC-module Q' and a monomorphism 4' : M -4 Q' there is a monomorphism

f : Q -+ Q' such that the diagram

commutes.

Let now M be an RC-module of type T and such that ScM are injective R[C]-

modules for E T. Then we may choose R[C]-projections pc : M(C) --+ ScM for - C E T which determine maps dc : M -+ EcScM, by adjointness. Thus we get the

induced map 4 : M -+ nc,, EcScM.

Proposition 2.1. Let a subset T E: IsC, 4: an EI-category, be given such that there

is a filtration satisfying Theorem 1.2. If M is an RC-module of type T and such that

ScM are injective R[C]-modules for all ?? E T then the map q5 : M 4 nE,, EcScM

is an injective envelope of M .

Proof. First we show that the map 4 : M -+ nEET EcScM is a monomorphism.

Let m E M ( D ) and m # 0. Then E T and let k be the minimal natural number

such that there is a non isomorphism [ : D -+ C with E Tk and M ( [ ) ( m ) # 0.

Then M(C)(m) E SCM and k ( D ) ( m ) ( O = pcM([ ) (m) # 0.

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4034 GOLASINSKI

Let now 4' : M + Q' be a monomorphism, where Q' is injective. Then the

induced maps Scdl : ScM + ScQ' are monomorphisms, ScQ' are injective R[C]-

modules for C E IsC and by Theorem 1.2 there is an isomorphism Q' z nEET EcScQ'.

Put 4; : M -, EcScQJ for the composition of 4' with the projection nEET EcScQf -+

EcScQi. Thus there are R[C]-maps fc : EcScM 4 EcScQ1 such that the diagrams

commute for E T. But Scfc = Sc@, hence the maps fc are monomorphisms.

Thus we get the required n~onomorphism f = fc : ncET ECSM + Q1.

0

Corollary 2.2. Let C be an El-category C such that there is a filtration 8 = To C

TI . . . C Tl = IsC satisfying Theorem 1.2 and assume that the group rings R[C] are

semisimple for all C E IsC. Then every RC-module has an injective resolution of

length 1 - 1.

Proof. Let O-MAQ~-";Q~-... be an injective resolution of M ,

where Q0 is an injective envelope of and Qk an injective envelope of c o k e r ~ k - ~

for k 3 1. By Proposition 2.1, QO(C) = M(C) for C E TI and therefore Q1(C) = 0

for E TI. Similarly, we can show by induction that cokerib(C) = Qk+'(C) for

E Tk+2 and therefore Qk+'(C) = 0 for E Tk+> Finally we obtain that Q" 0

fork 1.

0

For the field of rationals Q, we denote by DGAo the category of commut,ative

graded differential Q-algebras. Given a (connected) simplicia1 set X one can form

an object A; in DGAQ by taking collections of Q-polynomial forms on each simplex

(sums of terms of type w(to,. . . , t,)dt,, A . . . A dtlp, where w is a Q-polynomial) that

agree when restricted to common faces (see e.g., [I] for more details).

Let now X be a G-simplicia1 set with G a finite group. Much of the algebraic-

topological information about X is encoded in the form of functors from the category

O(G, X) defined as follows:

the set ObO(G, X ) of objects consists of pairs (G/H, a ) , with H a subgroup of

G and cr a connected component of the fixed point subset X H (i.e., a E 7r0(XH)),

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INJECTIVITY OF FUNCTORS TO MODULES AND DGA'S 4035

morphisms (G/K,P) 4 ( G / H , a ) are G-maps 4 : G / K 4 G / H such that

"&)(a) = P, with $ : XH 4 X K the induced map of the fixed point simpli-

cia1 subsets. In the sequel we will identify an object (G/H,a) of U(G, X) with a

and let X t be the connected component of X H corresponding to a E T O ( X ~ ) .

Let H = Ho, H I , . . . , H,,, be distinct subgroups of G conjugate to H. For oi E

ro (XH) let a k E 7ro(XHk) be the corresponding component of XHk. Then the set of

morphisms O(G, X)((G/K, P), (GIH, a ) ) is in one to one correspondence with the

disjoint union

of morphisms, hence we may identify them in our further investigations. Moreover

U(G,X) is an EI-category and for the isomorphism class 5 of its object cu there is

the largest number d(a) = n such that there is a sequence 5 = ?il < . . . < 6,. Since

the group G is finite, we can define a filtration

(satisfying Theorem 1.2) for the set of isomorphism classes IsU(G, X ) , where T, =

{S ; d ( 5 ) 5 i }. For a G-simplicia1 set X , let Ax be an object in the functor category U(G, X)-

DGAa defined by Ax(G/H, a ) = A>$, and the maps on forms are those induced by

the action of G on the connected components of the fixed point simplicia1 subsets.

Now we may state

Theorem 2.3. If X is a G-simplicia1 set then Ax is an injective QO(G, X)-

module.

Proof. The group G is finite, hence the group rings Q[Q] defined in the previous sec-

tion are semisimple and S,A;, are injective as Q[a]-modules for all 5 E IsU(G, X). In r(i-1) view of Theorem 1.2 we define A$) = kerT,~$-') and let in, : A?) = kerT,Ax 4

A$-') be the injection for i = 0, . . . ,1 . Then A?) are of type T\T, and there is a

natural filtration of A>

determined by exact sequences

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To prove the injectivity of A> we show that the maps

are surjective.

Let the object /3 be determined by the pair (G/K, P) . Then, as it was observed

above, we may make the identification O(G, X)((G/K,P), (G/H, a ) ) = U K E H ~ O(G, - k c 5

X)((G/Hk, ak,) , (G/H, a ) ) , where H = Ho, HI , . . . , H,,, are distinct subgroups of G

conjugate to H. Then X$ 5 Xf and an element of nEET,\T,-l (Eu 0 SuAx)(P)

determines a collection of forms { w f ) on Xf C XF. But for x E X: n x:' with

H # H' the isotropy group G, contains properly H or H' and x E X? C_ X z ~ x Z ' . Therefore, we get a form w on the union U X z C_ X;. By the extensions lemmas of

[I] there is a form 0 on Xf such that J , ( / ~ ) ( G ) = w .

0

Denote by C,(X, Q) the right QO(G, X)-module defined by C,(X, Q)(G/H, a ) =

C,(Xc, Q) for (G /H, a ) E O(G, X ) , where the latter stands for the ordinary chain

complex of Xf with coefficients in Q. Then we get the induced right eO(G, X)-

module H,(X, Q) such that H n ( X , Q)(G/H, a ) = Hn(Xf, Q) for (GIH, a ) E ObO(G, X)

and n 2 0.

Propos i t ion 2.4. If X is a G-simplicia1 set, then the right QO(G,X)-module

C, (X, Q) is projective.

Proof . Let C*(X, e ) / I

be a diagram of right &O(G, X)-modules with an exact row and 0 to be constructed.

Let X ' E X be a collection of representatives for T ~ ( X / G ) . Given x' E X', consider

x' as an element of c , ( x ~ ~ ' , Q) for some component a E ?ro(XGr), where G,g is the

isotropy subgroup of x'. Define 0(G/G,,, a)(xl) E M(G/G,,. a ) to be any element

with $J(G/G,I,~)B(G/G,,, a)(xl) = d(G/G,t, a)(xl). Extend B to the whole orbit by

B(gzJ) = g*B(xJ) for g E G, where g* = M ( j : (G/G,,,,gcr) -, (G/G,(,a)) and

g : (GIG,,,, ga) -+ (GIG,,, a ) is the obvious map in O(G, X ) determined by the

element g E G.

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INJECTIVITY OF FUNCTORS TO MODULES AND DGA'S 4037

If now a: E C,(Xf, Q) represents one of generators then x E x?= X; and H C G,. Hence we get a map 4 : (G/H, p) -t (G/G,, a ) . Thus x = C,(X, Q ) ( ~ ) ( x )

and we define B(G/H, P)(x) = M(e)B(G/G,, a)(x).

0

For any right QO(G, X)-module M we write C;(X, M ) = Homaqc,x)(C*(X, Q), M )

and consider cohomology of X (resp. A;) in two ways.

(1) The QO(G, X)-module H n ( X , Q) (resp. Hn(A>)) such that H n ( X , Q)(G/H, a)

= Hn(X:, Q) (resp. (Hn(A>))(G/H, a) = Hn(A;If)) for (G/H, a ) E ObO(G, X )

and n 2 0.

(2) The cohomology HE(X, M ) = Hn(C*(X, M ) ) with coefficients in a right

QO(G, X)-module M (resp. Hn(A>, N ) with coefficients in a left QO(G, X)-module

N) for n 3 0.

Standard homological algebra arguments yield a spectral sequence ([2])

(resp. Ei9q = ExtP(N, Hq(A;I)) J H $ + ~ ( A ~ , N)).

Notice that the projectivity of C*(X, Q) (resp. injectivity of A; as an QC?(G, X)-

module) is necessary for the convergence of the spectral sequence.

For a right QO(G, X)-module M let M* denote its dual left QO(G, X)-module

defined by Me(G/H, a) = Hom(M(G/H, a ) , Q) for (GIH, a) E ObO(G/H, a ) . Tak-

ing these facts into account and the result of [9, 10) we get the following generalized

equivariant de Rham theorem.

Theorem 2.5. For a G-simplicia1 set X there is an isomorphism

H g ( X , M ) x Hn(A>, M*)

for n > O and a right QO(G, X)-module M provided that the cohomology Hn(Xf , Q)

and M(G/H, a ) are of finite type for any (GIH, a ) E ObC?(G, X ) and n 2 0.

Proof. By integration of forms we get a map

inducing an isomorphism on cohomoiogy

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for n >_ 0. Take a projective resolution M*(*) of M* in ~ o ( G , X ) - m o d and con-

sider the spectral sequences determined by the double complexes Hom(M*(*), A;)

and Hom(M*(*), C8(X, Q)). Then ExtP(M*, Hq(A>)) = ExtP(M*, H q ( X , Q)) and

ExtP(M*, H q ( X , Q)) = ExtP(H,(X, Q), M) because of duality. The first spectral

sequence converges to Hp+q(Ax, M*), the second to H;+'(X, M) and this completes

the proof.

0

References

[I] A.K. Bousfield, V.K.A.M. Guggenheim, O n PL de Rham Theory and Rational Homotopy Type, h4em. A.M.S. 179 (1976).

[2] G.E. Bredon, Equivariant Cohomology Theories, Lecture Notes in Math. 34, Springer-Verlag (1967).

[3] R. Doman, On injective Rational Coeficient Systems, Mh. Math. 106 (1988), 171-178.

[4] A.D. Elmendorf, Systems of Fixed Point Set , Trans. Amer. Math. Soc., 277 (1983), 275-284.

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