aa r X i v : . [ m a t h . A T ] N ov A FAMILY OF ACYCLIC FUNCTORS
ANTONIO D´IAZ RAMOS Introduction and summary
In this paper we focus on the following problem(1) find colim-acyclic objects in Ab C .Here, Ab denote the category of abelian groups and Ab C denote the (abelian) functorcategory for the small category C . The functor colim : Ab C → Ab is the direct limitfunctor and F ∈ Ab C is colim-acyclic if colim i F = 0 for i ≥ F is projective then it is colim-acyclic but, in the same way as not every flat moduleis projective (see, for example, [16, Section 3.2]), we may be missing colim-acyclicobjects if we just consider projective ones.We shall assume the hypothesis that the category C is a graded partially ordered set(a graded poset for short). These are special posets in which we can assign an integerto each object (called the degree of the object) in such a way that preceding elementsare assigned integers which differs in 1. Thus a graded poset can be divided into a setof “layers” (the objects of a fixed degree), and these layers are linearly ordered. Anysimplicial complex (viewed as the poset of its simplices with the inclusions amongthem) and any subdivision category is a graded poset. Also, every CW -complex is(strong) homotopy equivalent to a simplicial complex, and thus to a graded poset.To attack problem (1) we start giving a characterization of the projective objectsin Ab C . Recall that for any small category C (not necessarily a poset), the projectiveobjects in Ab C are well known to be, by the Yoneda Lemma, summands of directsums of representable functors. Moreover, if C is a poset with the descending chaincondition (not necessarily graded) then [3, Corollary 3] the projective objects in Ab C are also direct sums of representable functors (see also [11, Proposition 7] and [4,Theorem 9] for related results). In case C is a graded poset we characterize theprojective functors in Ab C as those functors which satisfy two conditions: Theorem A (Theorem 4.9) . Let C be a bounded below graded poset and let F : C → Ab be a functor. Then F is projective if and only if:(1) for any object i of C Coker F ( i ) is a free abelian group.(2) F is pseudo-projective. Here, Coker F ( i ) is the quotient of F ( i ) by the images of all the non-trivial mor-phisms arriving to i . For the actual definition of pseudo-projectiveness see Definition Date : November 1, 2018. C = Z of the integers. This graded poset does not satisfies the de-scending chain condition, and thus neither is it bounded below. The constant functorof value Z over this poset is projective but it is not a sum of representable func-tors. The constant functor of value Z /n (for some n ≥
1) satisfies both conditions inTheorem A but it is not projective (see Remark 4.10).Theorem A is the first step towards finding colim-acyclic objects in Ab C . Thereason is that the second of the conditions in the theorem, i.e., pseudo-projectiveness,implies colim-acyclicity: Theorem B (Theorem 5.2) . Let F : C → Ab be a pseudo-projective functor over abounded below graded poset C . Then F is colim -acyclic. This is the main result of this work, and it gives a family of functors in Ab C whichare colim-acyclic but not necessarily projective. To show that there exist functors inthis situation consider the functor Z Z × o o × / / Z . This is a pseudo-projective functor which, by Theorem B, is acyclic. Moreover, it doesnot satisfy condition (1) in Theorem A and so it is not projective (see Examples 4.11and 5.3). On the other hand, pseudo-projective functors do not cover all colim-acyclicfunctors: the functor 0 Z o o / / Z . is not pseudo-projective but, as a straightforward computation shows, it is acyclic. Forvector spaces the notion of pseudo-projectiveness becomes identical to projectivenessas condition (1) in Theorem A is unnecessary in the context of functors to k − mod (where k is a field). Even in this favorable case the functor0 k o o / / k shows that there are acyclic functors which are not projective.The main ingredient in the proof of Theorem B is a meticulous use of a spectralsequence built upon the grading of the partially ordered set C : Proposition C (Proposition 3.2) . For a (decreasing) graded poset C and a functor F : C → Ab : • There exists a cohomological type spectral sequence E ∗ , ∗∗ with target colim ∗ F . • There exists a homological type spectral sequence ( E p ) ∗∗ , ∗ with target the column E p, ∗ for each p . Applications of these results to computation of integral cohomology of posets aregiven in [7]. The work also contains the dual version of the above, in which weconsider injective objects in Ab C , the right derived functors of the inverse limit functorlim : Ab C → Ab and the respective lim-acyclic objects.
FAMILY OF ACYCLIC FUNCTORS 3
The paper is structured as follows: in Section 2 we introduce preliminaries aboutgraded partially ordered sets. In Section 3 we build some spectral sequences arisingfrom the grading of a graded poset. Afterwards, in Section 4, we work out thecharacterization of projective objects in Ab C . In Section 5 we prove that pseudo-projectiveness implies colim-acyclicity. We finish with Section 6, where the dualdefinitions and results for lim-acyclicity are stated without proof. Acknowledgements:
I would to thank my Ph.D. supervisor Prof. A. Viruel forhis support during the development of this work. Also, thanks to Prof. C.A. Weibelfor all his fruitful suggestions and comments, in particular for a short proof of Lemma4.2. 2.
Graded posets
In this section we define a special kind of categories: graded partially ordered sets(graded posets for short). We shall think of a poset P as a category in which thereis an arrow p → p ′ if and only if p ≤ p ′ . The notion of graded poset is not new andit was already used in [6, pp. 29-33]. The definition there is weaker than the onegiven here, being the difference that here we ask for every morphism to factor troughmorphisms of degree 1 (some kind of “saturation” condition). Definition 2.1. If P is a poset and p < p ′ then p precedes p ′ if p ≤ p ′′ ≤ p ′ impliesthat p = p ′′ or p ′ = p ′′ . Definition 2.2.
Let P be a poset. P is called graded if there is a function deg :Ob( P ) → Z , called the degree function of P , which is order preserving and thatsatisfies that if p precedes p ′ then deg ( p ′ ) = deg ( p ) + 1. If p is an object of P then deg ( p ) is called the degree of p .Notice that the degree function associated to a graded poset is not unique (considerthe translations deg ′ = deg + c k for k ∈ Z ). According to the definition the degreefunction increases in the direction of the arrows: we say that this degree functionis increasing . If the degree function is order reversing and satisfies the alternativecondition that p precedes p ′ implies deg ( p ′ ) = deg ( p ) −
1, i.e., deg decreases in thedirection of the arrows, then we say that deg is a decreasing degree function. Clearlyboth definitions are equivalent (by taking deg ′ = − deg ). Example 2.3.
The “pushout category” b ← a → c , the “telescope category” a → b → c → ... , and the opposite “telescope category” ... → c → b → a are gradedposets. The integers Z is a graded poset. The rationals Q with the usual order is aposet but it is not a graded poset.If P is a graded poset and p < p ′ then it is straightforward that the number deg ( p ′ ) − deg ( p ) does not depend on the degree function deg . Thus, we can “extend”the degree function deg to the morphisms set Hom( P ) by deg ( p → p ′ ) = | deg ( p ′ ) − deg ( p ) | . Whenever P is a graded poset we denote by Ob n ( P ) the objects of degree n and by Hom n ( P ) the arrows of degree n . ANTONIO D´IAZ RAMOS
Boundedness on graded posets.
Often we will restrict to:
Definition 2.4.
A graded poset P with increasing degree function deg is boundedbelow ( bounded above ) if the set deg ( P ) ⊂ Z has a lower bound (an upper bound).If the degree function deg of P is decreasing then P is bounded below ( boundedabove ) if and only if deg ( P ) ⊂ Z has an upper bound (a lower bound). If P isbounded below and over then N def = max ( deg ( P )) − min ( deg ( P )) exists and it isfinite, and it does not depend on the degree function deg . We call it the dimensionof P , and we say that P is N dimensional. Example 2.5.
The “pushout category” b ← a → c is 1-dimensional, the “telescopecategory” a → b → c → .. is bounded below but it is not bounded over. The opposite“telescope category” .. → c → b → a is bounded over but it is not bounded below.Notice that in a bounded above (below) graded poset there are maximal (minimal)elements, but that the existence of maximal (minimal) objects does not guaranteeboundedness. Also it is clear that, in general, neither dcc posets are graded norgraded posets are dcc. 3. A spectral sequence
In this section we shall construct spectral sequences with targets colim i F and lim i F for F : C →
Ab with C a graded poset. Some conditions for (weak) convergence shallbe given. We build the spectral sequences starting from filtered differential modules(see [15], where the notion of weak convergence we use is also given).Recall that (see [8, Appendix II.3], [1, XII.5.5], [1, XI.6.2] or [9, p.409ff.]) thereis, for any small category C and covariant functor F : C →
Ab, a concrete chain(cochain) complex C ∗ ( C , F ) ( C ∗ ( C , F )) whose homology groups (cohomology groups)are precisely the left derived functors colim i (right derived functors lim i ).Let N C denote the nerve of the small category C whose n -simplices are chain ofcomposable morphisms in C : σ = σ α / / σ α / / ... α n − / / σ n − α n / / σ n . Then C n ( C , F ) = M σ ∈ N C n F σ , where F σ = F ( σ o ). Moreover, C ∗ ( C , F ) is a simplicial abelian group with face anddegeneracy maps induced by those of the nerve N C . The chain complex ( C ∗ ( C , F ) , d )with differential of degree − d = P ni =0 ( − i d i satisfies(2) colim i F = H i ( C ∗ ( C , F ) , d ) . For the inverse limit lim : Ab C → Ab there is a cosimplicial abelian group withsimplices C n ( C , F ) = Y σ ∈ N C n F σ , FAMILY OF ACYCLIC FUNCTORS 5 where F σ = F ( σ n ). This cosimplicial object gives rise to a cochain complex ( C ∗ ( C , F ) , d )with differential of degree 1 d = P n +1 i =0 ( − i d i . It is well known that(3) lim i F = H i ( C ∗ ( C , F ) , d ) . Remark 3.1.
We can apply the Dold-Kan correspondence (see [16, 8.4]) to thesimplical and cosimplicial abelian groups constructed above. This means that we shalluse the normalized chain (cochain) complex to compute the homology (cohomology)in Equation (2) ( Equation (3)).There is a decreasing filtration of the chain complex ( C ∗ ( C , F ) , d ) given by L p C n ( C , F ) = M σ ∈ N C n ,deg ( σ n ) ≥ p F σ . It is straightforward that the triple ( C ∗ ( C , F ) , d, L ∗ ) is a filtered differential graded Z -module, so it yields a spectral sequence ( E ∗ , ∗ r , d r ) of cohomological type whosedifferential d r has bidegree ( r, − r ). The E ∗ , ∗ page is given by E p,q ≃ H p + q ( L p C/L p − C ) . The differential graded Z -module L p C/L p − C is in fact a simplicial abelian groupbecause the face operators d i and the degeneracy operators s i respect the filtration L ∗ . The n -simplices are ( L p C/L p − C ) n = M σ ∈ N C n ,deg ( σ n )= p F σ . Moreover, for each p , L p C/L p − C can be filtered again by the condition deg ( σ ) ≤ p ′ to obtain a homological type spectral sequence. Then arguing as above we obtain: Proposition 3.2.
For a (decreasing) graded poset C and a functor F : C → Ab : • There exists a cohomological type spectral sequence E ∗ , ∗∗ with target colim ∗ F . • There exists a homological type spectral sequence ( E p ) ∗∗ , ∗ with target the column E p, ∗ for each p . Notice that the column E p, ∗ is given by the cohomology of the simplicial abeliangroup formed by the simplices that end on objects of degree p , the column ( E p ) p ′ , ∗ is given by the homology of the simplicial abelian group formed by the simplicesthat end on degree p and begin on degree p ′ , and all the differentials in the spectralsequences above are induced by the completely described differential of ( C ∗ ( C , F ) , d ).As S p L p C n = C n and T p L p C n = 0 for each n the spectral sequence E ∗ , ∗∗ convergesweakly to its target. In case the map deg has a bounded image, i.e., when C is N dimensional, the filtration L ∗ is bounded below and over, and so E ∗ , ∗∗ collapses aftera finite number of pages. The same assertions on weak converge and boundednesshold for the spectral sequences ( E p ) ∗∗ , ∗ .If we proceed in reverse order, i.e., filtrating first by the degree of the beginningobject and later by the degree of the ending object, we obtain: Proposition 3.3.
For a (decreasing) graded poset C and a functor F : C → Ab : ANTONIO D´IAZ RAMOS
Complex Degree First Second First ss. Second ss.function filtration filtration C ∗ ( C , F ) decreasing deg ( σ n ) ≥ deg ( σ ) ≤ cohomol. type homol. type C ∗ ( C , F ) decreasing deg ( σ ) ≤ deg ( σ n ) ≥ homol. type cohomol. type C ∗ ( C , F ) increasing deg ( σ n ) ≤ deg ( σ ) ≥ homol. type cohomol. type C ∗ ( C , F ) increasing deg ( σ ) ≥ deg ( σ n ) ≤ cohomol. type homol. type C ∗ ( C , F ) decreasing deg ( σ n ) ≤ deg ( σ ) ≥ homol. type cohomol. type C ∗ ( C , F ) decreasing deg ( σ ) ≥ deg ( σ n ) ≤ cohomol. type homol. type C ∗ ( C , F ) increasing deg ( σ n ) ≥ deg ( σ ) ≤ cohomol. type homol. type C ∗ ( C , F ) increasing deg ( σ ) ≤ deg ( σ n ) ≥ homol. type cohomol. type Table 1.
Filtrations and spectral sequences obtained • There exists a homological type spectral sequence E ∗∗ , ∗ with target colim ∗ F . • There exists a cohomological type spectral sequence ( E p ) ∗ , ∗∗ with target the col-umn E p, ∗ for each p . If the degree function we take is increasing then the appropriate conditions for thefiltrations are deg ( σ n ) ≤ p and deg ( σ ) ≥ p ′ , and the spectral sequences obtained inPropositions 3.2 and 3.3 are of homological (cohomological) type instead of cohomo-logical (homological) type.For the case of the cochain complex ( C ∗ ( C , F ) , d ) the choices for the filtrationsare deg ( σ n ) ≤ p and deg ( σ ) ≥ p ′ for a decreasing degree function and deg ( σ n ) ≥ p and deg ( σ ) ≤ p ′ for an increasing one. Analogously we obtain spectral sequenceswith target lim i F which columns in the first page are computed by another spectralsequence.Table 1 shows a summary of the types of the spectral sequences for all the cases.The statements on weak convergence and boundedness apply to any of the spectralsequences of the table. Remark 3.4.
It is straightforward that normalizing (see Remark 3.1) the simplicial(cosimplicial) abelian groups that computes the page 1 of the spectral sequencesabove has the same effect as considering the spectral sequences of the normalizationsof C ∗ ( C , F ) ( C ∗ ( C , F )). 4. Projective objects in Ab P . Consider the abelian category Ab P for some graded poset P . In this section weshall determine the projective objects in Ab P . Recall that in Ab the projective objectsare the free abelian groups. Along the rest of the section P denotes a graded poset.Suppose F ∈ Ab P is projective. How does F look? Consider an object i of P . Weshow that the quotient of F ( i ) by the images of the non-identity morphisms arrivingto i is free abelian. To prove it, write Definition 4.1. Im F ( i ) = P i α → i ,α =1 i Im F ( α ) (or Im F ( i ) = 0 if the index set ofthe sum is empty) and Coker F ( i ) = F ( i ) / Im F ( i ). FAMILY OF ACYCLIC FUNCTORS 7
It is straightforward that for a fixed object i of P there is a functorCoker · ( i ) : Ab P → Abwhich maps F to Coker F ( i ). This functor is left adjoint to the skyscraper functorAb → Ab P which maps the abelian group A to the functor A : P →
Ab with values A ( i ) = (cid:26) A for i = i i = i on objects, and values A ( α ) = (cid:26) A for α = 1 i α = 1 i on morphisms. As the skyscraper functor is exact we obtain by [16, Proposition2.3.10] that Coker . ( i ) preserves projective objects: Lemma 4.2.
Let F : P → Ab be a projective functor over a graded poset P . Then Coker F ( i ) is free abelian for every object i of P . This means that we can write F ( i ) = Im F ( i ) ⊕ Coker F ( i )with Coker F ( i ) free abelian for every object i of P , and also that Example 4.3.
For the category P with shape · → · the functor F : P →
Ab with values Z × n → Z is not projective as Coker F on the right object equals the non-free abelian group Z /n .Now that we know a little about the values that a projective functor F : P →
Abtakes on objects we can wonder about the values F ( α ) for α ∈ Hom( P ). Do theyhave any special property? Recall that a feature of graded posets is that there is atmost one arrow between any two objects, and also that Remark 4.4. If P is graded then for any object i of P Im F ( i ) = X i α → i ,deg ( α )=1 Im F ( α )because every morphism factors as composition of morphisms of degree 1.We prove that the following property holds for F : Definition 4.5.
Let F : P →
Ab be a functor over a graded poset P with degreefunction deg . Given d ≥ F is d -pseudo-projective if for any object ANTONIO D´IAZ RAMOS i of P and k different objects i j in P , arrows α j : i j → i with deg ( α j ) = d , and x j ∈ F ( i j ) j = 1 , .., k such that X j =1 ,..,k F ( α j )( x j ) = 0we have that x j ∈ Im F ( i j ) j = 1 , .., k . If F is d -pseudo-projective for each d ≥ F pseudo-projective . Remark 4.6.
In case k = 1 and Im F ( i ) = 0 the condition states that F ( α ) is amonomorphism. Notice that any functor is 0-pseudo-projective as the identity is amonomorphism.Before proving that projective functors F over a graded poset verify this propertywe define two functors Coker F and Coker ′ F and natural transformations σ and π thatfit in the diagram F σ (cid:11) (cid:19) Coker ′ F π + Coker F + F : P →
Ab with P a graded poset. We begin defining Coker F .Because for every α : i → i holds that F ( α )(Im F ( i )) ≤ Im F ( i ) we can factor F ( α )as in the diagram F ( i ) (cid:15) (cid:15) (cid:15) (cid:15) F ( α ) / / F ( i ) (cid:15) (cid:15) (cid:15) (cid:15) Coker F ( i ) F ( α ) / / Coker F ( i ) . In fact, if α = 1 i , then F ( α ) ≡ i cannotbe factorized (by non-identity morphisms) in a graded poset then we have a functorCoker F with value Coker F ( i ) on the object i of P and which maps the non-identitymorphisms to zero. Coker F is a kind of “discrete” functor. Also it is clear that thereexists a natural transformation σ : F ⇒ Coker F with σ ( i ) the projection F ( i ) ։ Coker F ( i ).Now we define Coker ′ F from Coker F in a similar way as free diagrams are con-structed. Let Coker ′ F be defined on objects byCoker ′ F ( i ) = M α : i → i Coker F ( i ) . For β ∈ Hom( P ), β : i → i , Coker ′ F ( β ) is the only homomorphism which makescommute the diagram Coker ′ F ( i ) Coker ′ F ( β ) / / Coker ′ F ( i )Coker F ( i ) ?(cid:31) O O / / Coker F ( i ) ?(cid:31) O O FAMILY OF ACYCLIC FUNCTORS 9 for each α : i → i . In the bottom row of the diagram, the direct summands Coker F ( i )of Coker ′ F ( i ) and Coker ′ F ( i ) correspond to α : i → i and to the composition i α → i β → i respectively.Then there exists a candidate to natural transformation π : Coker ′ F ⇒ Coker F which value π ( i ) is the projection π ( i ) : Coker ′ F ( i ) ։ Coker F ( i ) onto the directsummand corresponding to 1 i : i → i . Thus, π is a natural transformation if for every β : i → i with i = i the following diagram is commutativeCoker ′ F ( i ) π ( i ) (cid:15) (cid:15) (cid:15) (cid:15) Coker ′ F ( β ) / / Coker ′ F ( i ) π ( i ) (cid:15) (cid:15) (cid:15) (cid:15) Coker F ( i ) / / Coker F ( i ) . It is clear that this square commutes if the identity 1 i cannot be factorized (bynon-identity morphisms), and this holds in a graded poset.Now we have the commutative triangle F σ (cid:11) (cid:19) ρ u } r r r r rr r r r r Coker ′ F π + Coker F + ρ exists because F is projective. To prove that F is d -pseudo-projective for some d ≥ i of P , k objects i , .., i k , arrows α j : i j → i with deg ( α j ) = d and elements x j ∈ F ( i j ) for j = 1 , .., k such that X j =1 ,..,k F ( α j )( x j ) = 0 . To visualize what is going on consider the diagram above near i for k = 2 F ( i ) u u j j j j j j j j j j j j j j j j j j j j j (cid:15) (cid:15) (cid:15) (cid:15) F ( α ) ' ' OOOOOOOOOOO F ( i ) u u j j j j j j j j j j j j j j j j j j j j j (cid:15) (cid:15) (cid:15) (cid:15) F ( α ) w w ooooooooooo F ( i ) u u j j j j j j j j j j j j j j j j j j j j j (cid:15) (cid:15) (cid:15) (cid:15) Coker ′ F ( i ) Coker ′ F ( α ) ' ' NNNNNNNNNNN
Coker ′ F ( i ) Coker ′ F ( α ) w w ppppppppppp Coker F ( i ) ' ' NNNNNNNNNNN
Coker F ( i ) w w ppppppppppp Coker ′ F ( i ) π ( i ) / / / / Coker F ( i ) where π is not drawn completely for clarity. Recall that we are supposing that { x , .., x k } is such that P j =1 ,..,k F ( α j )( x j ) = 0. Then0 = ρ ( i )(0) = X j =1 ,..,k ρ ( i )( F ( α j )( x j )) = X j =1 ,..,k Coker ′ F ( α j )( ρ ( i j )( x j )) . Now consider the projection p j for j ∈ { , .., k } from Coker ′ F ( i ) onto the directsummand Coker F ( i j ) ֒ → Coker ′ F ( i ) which corresponds to α j : i j → i Coker ′ F ( i ) p j ։ Coker F ( i j ) . Then(4) 0 = p j (0) = p j ( ρ ( i )(0)) = X j =1 ,..,k p j (Coker ′ F ( α j )( ρ ( i j )( x j ))) . For any y = L α : i → i j y α ∈ Coker ′ F ( y j ) p j (Coker ′ F ( α j )( y )) = X α : i → i j ,α j ◦ α = α j y α . So if y j = ρ ( i j )( x j ) = L α : i → i j y j,α ∈ Coker ′ F ( i j ) then p j (Coker ′ F ( α j )( ρ ( i j )( x j ))) = X α : i → i j ,α j ◦ α = α j y j,α . This last sum runs over α : i j → i j such that the following triangle commutes i j α (cid:31) (cid:31) >>>>>>>> α j / / i .i j α j ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Because we are in a graded poset and deg ( i j ) = d for each j = 1 , .., k then the onlychance is i j = i j and α = 1 i j . Because the objects i , .., i k are different this impliesthat j = j too. Thus p j (Coker ′ F ( α j )( ρ ( i j )( x j ))) = (cid:26) y j , ij for j = j j = j and Equation (4) becomes 0 = p j (0) = y j , ij . Notice now that y j , ij is the evaluation of π ( i j ) on y j = ρ ( i j )( x j ) and then0 = y j , ij = π ( i j )( ρ ( i j )( x j )) = σ i j ( x j ) . This last equation means that x j goes to zero by the projection F ( i j ) ։ Coker F ( i j ) = F ( i j ) / Im F ( i j ), and then x i j ∈ Im F ( i j ) . As j was arbitrary this completes the proof of Lemma 4.7.
Let F : P → Ab be a projective functor over a graded poset P . Then F is pseudo-projective. FAMILY OF ACYCLIC FUNCTORS 11
Example 4.8.
For the category P with shape · → · the functor F : P →
Ab with values Z red n → Z /n is not projective as red n is not injective, in spite of the Coker F ’s are Z and 0, whichare free abelian.Till now we have obtained (Lemmas 4.2 and 4.7) that projective functors P →
Abare pseudo-projective and have Coker F ( i ) projective for any object i . In fact, asthe next theorem shows, the restriction we did to graded posets is worthwhile: Theorem 4.9.
Let F : P → Ab be a functor over a bounded below graded poset P .Then F is projective if and only if(1) for any object i of P Coker F ( i ) is a free abelian group.(2) F is pseudo-projective.Proof. It remains to prove that a functor F satisfying the conditions in the statementis projective. We can assume that the degree function deg on P is increasing andtakes values { , , , , ... } , and that Ob ( P ) = ∅ .To see that F is proyective in Ab P , given a diagram of functors with exact rowas shown, we must find a natural transformation ρ : F ⇒ A making the diagramcommutative: F σ (cid:11) (cid:19) ρ { (cid:3) ~ ~ ~ ~~ ~ ~ ~ A π + B + . We define ρ inductively, beginning on objects of degree 0 and successively on objectof degrees 1 , , , .. .So take i ∈ Ob ( P ) of degree 0, and restrict to the diagram in Ab over i . Byhypothesis (1) in the statement, as Im F ( i ) = 0, F ( i ) = Coker F ( i ) is free abelian.So we can close the following triangle with a homomorphism ρ ( i ) F ( i ) σ ( i ) (cid:15) (cid:15) ρ ( i ) { { w w w w w A ( i ) π ( i ) / / B ( i ) / / . As there are no arrows between degree 0 objects we do not worry about ρ being anatural transformation. Now suppose that we have defined ρ on all objects of P of degree less than n ( n ≥ ρ to the full subcategorygenerated by these objects is a natural transformation and verifies π ◦ ρ = σ .The next step is to define ρ on degree n objects. So take i ∈ Ob n ( P ) and considerthe splitting F ( i ) = Im F ( i ) ⊕ Coker F ( i ) where Im F ( i ) = X i α → i ,deg ( α )=1 Im F ( α ) . To define ρ ( i ) such that it makes commutative the diagramIm F ( i ) ⊕ Coker F ( i ) σ ( i ) (cid:15) (cid:15) ρ ( i ) v v m m m m m m m A ( i ) π ( i ) / / B ( i ) / / , we define it on Im F ( i ) and Coker F ( i ) separately. For Coker F ( i ), as it is a freeabelian group, we define it by any homomorphism that makes commutative the dia-gram above when restricted to Coker F ( i ). For Im F ( i ) take x = P j =1 ,..,k F ( α j )( x j )where { i , .., i k } are k different objects, α j : i j → i , deg ( α j ) = 1 and x j ∈ F ( i j ) for j = 1 , .., k (see Remark 4.4). Then define ρ ( i )( x ) = X j =1 ,..,k ( A ( α j ) ◦ ρ ( i j ))( x j ) . To check that ρ ( i )( x ) does not depend on the choice of the i j ’s, α j ’s and x j ’s wehave to prove that X j =1 ,..,k F ( α j )( x j ) = 0 ⇒ X j =1 ,..,k ( A ( α j ) ◦ ρ ( i j ))( x j ) = 0 . So suppose that(5) X j =1 ,..,k F ( α j )( x j ) = 0 . Then using that F is 1-pseudo-projective and Remark 4.4 we obtain objects i j,j ′ ,arrows α j,j ′ of degree 1, and elements x j,j ′ for j = 1 , .., k , j ′ = 1 , .., k j such that(6) X j ′ =1 ,..,k j F ( α j,j ′ )( x j,j ′ ) = x j for every j ∈ { , .., k } . Notice that possibly not all the objects i j,j ′ are different.Replacing Equation (6) in Equation (5) we obtain(7) X j = 1 , .., k , j ′ = 1 , .., k j F ( α j ◦ α j,j ′ )( x j,j ′ ) = 0 . Because in a graded poset there is at most one arrow between two objects, the con-dition i j,j ′ = i j ′ ,j ′′ = i implies α j ◦ α j,j ′ = α j ′ ◦ α j ′ ,j ′′ : i → i . So, considering objects i in P , we can rewrite (7) as(8) X i ∈ Ob( P ) F ( α j ◦ α j,j ′ )( X j,j ′ | i j,j ′ = i x j,j ′ ) = 0 . FAMILY OF ACYCLIC FUNCTORS 13
Call { i ′ , .., i ′ m } = { i j,j ′ | j = 1 , .., k , j ′ = 1 , .., k j } where these sets have m elements.Call β l = α j ◦ α j,j ′ if i ′ l = i j,j ′ and y l = P j,j ′ | i j,j ′ = i ′ l x j,j ′ for l = 1 , .., m . Notice that deg ( β l ) = 2 for each l . Then Equation (8) becomes(9) X l =1 ,..,m F ( β l )( y l ) = 0 . Now we repeat the same argument: applying that F is 2-pseudo-projective andthe Remark 4.4 to Equation (9) we obtain objects i ′ l,l ′ , arrows β l,l ′ of degree 1, andelements y l,l ′ for l = 1 , .., m , l ′ = 1 , .., k ′ l such that(10) X l ′ =1 ,..,k ′ l F ( β l,l ′ )( y l,l ′ ) = y l for every l ∈ { , .., m } . Substituting (10) in (9) X l = 1 , .., m , l ′ = 1 , .., k ′ l F ( β l ◦ β l,l ′ )( y l,l ′ ) = 0 . Now proceed as before regrouping the terms in this last equation.In a finite number of steps, after a regrouping of terms as above, we find objects i ′′ s , arrows γ s , and elements z s of degree 0 for s = 1 , .., r which verify an equation(11) X s =1 ,..,r F ( γ s )( z s ) = 0 . Then pseudo-injectivity gives that z s ∈ Im F ( i ′′ s ) for each s . As deg ( i ′′ s ) = 0 thenIm F ( i ′′ s ) = 0 and so z s = 0 (notice that z s = 0 for s = 1 , .., r does not imply x j = 0for any j ).Recall that we want to prove that(12) X j =1 ,..,k ( A ( α j ) ◦ ρ ( i j ))( x j ) = 0 . Substituting (6) in P j =1 ,..,k ( A ( α j ) ◦ ρ ( i j ))( x j ) we obtain X j =1 ,..,k ( A ( α j ) ◦ ρ ( i j ))( x j )) = X j =1 ,..,k X j ′ =1 ,..,k j ( A ( α j ) ◦ ρ ( i j ) ◦ F ( α j,j ′ ))( x j,j ′ )= X j =1 ,..,k X j ′ =1 ,..,k j ( A ( α j ) ◦ A ( α j,j ′ ) ◦ ρ ( i j,j ′ ))( x j,j ′ )= X j =1 ,..,k X j ′ =1 ,..,k j ( A ( α j ◦ α j,j ′ ) ◦ ρ ( i j,j ′ ))( x j,j ′ ) , as ρ is natural up to degree less than n . Then regrouping terms X j =1 ,..,k X j ′ =1 ,..,k j ( A ( α j ◦ α j,j ′ ) ◦ ρ ( i j,j ′ ))( x j,j ′ ) = X i ∈ Ob( P ) ( A ( α j ◦ α j,j ′ ) ◦ ρ ( i j,j ′ ))( X j,j ′ | i j,j ′ = i x j,j ′ )= X l =1 ,..,m ( A ( β l ) ◦ ρ ( i ′ l ))( y l ) . Then, after a finite number of steps, we obtain X j =1 ,..,k ( A ( α j ) ◦ ρ ( i j ))( x j )) = X s = 1 , .., r ( A ( γ s ) ◦ ρ ( i ′′ s ))( z s ) = 0as z s = 0 for each z = 1 , .., r .So we have checked that ρ ( i )( x ) does not depend on the choice of i j , α j and x j .It is straightforward that ρ ( i ) on Im F ( i ) defined in this way is a homomorphism ofabelian groups.It remains to prove that π ( i ) ρ ( i ) = σ ( i ) when restricted to Im F ( i ). So take x = P j =1 ,..,k F ( α j )( x j ) in Im F ( i ). Then π ( i )( ρ ( i )( x )) = X j =1 ,..,k ( π ( i ) ◦ A ( α j ) ◦ ρ ( i j ))( x j )= X j =1 ,..,k ( B ( α j ) ◦ π ( i j ) ◦ ρ ( i j ))( x j ), π is a natural transformation= X j =1 ,..,k ( B ( α j ) ◦ σ ( i j ))( x j ), by the inductive hypothesis= X j =1 ,..,k ( σ ( i ) ◦ F ( α j ))( x j ), σ is a natural transformation= σ ( i )( x )Defining ρ ( i ) in this way for every i ∈ Ob n ( P ) we have now ρ defined on allobjects of P of degree less or equal than n . Finally, to complete the inductive step wehave to prove that ρ restricted to the full subcategory over these objects is a naturaltransformation. Take α : i → i in this full subcategory. If the degree of i is lessthan n then the commutativity of F ( i ) F ( α ) / / ρ ( i ) (cid:15) (cid:15) F ( i ) ρ ( i ) (cid:15) (cid:15) A ( i ) A ( α ) / / A ( i )is granted by the inductive hypothesis. Suppose that the degree of i is n . Take x ′ ∈ F ( i ). Because P is graded there exists α : i → i of degree 1 and α ′ : i → i such that α = α ◦ α ′ : i α ′ (cid:30) (cid:30) <<<<<<<< α / / i .i α ? ? ~~~~~~~ FAMILY OF ACYCLIC FUNCTORS 15
Write x = F ( α )( x ′ ) = F ( α )( x ) where x = F ( α ′ )( x ′ ). Then, by definition of ρ ( i )on Im F ( i ), ρ ( i )( x ) = ( A ( α ) ◦ ρ ( i ))( x )= ( A ( α ) ◦ ρ ( i ))( F ( α ′ )( x ′ ))= ( A ( α ) ◦ ρ ( i ) ◦ F ( α ′ ))( x ′ )= ( A ( α ◦ α ′ ) ◦ ρ ( i ))( x ′ ), ρ is natural up to degree less than n = ( A ( α ) ◦ ρ ( i ))( x ′ )and so the diagram commutes. (cid:3) Remark 4.10.
As the following example shows the condition of lowerboundedness of P in Theorem 4.9 cannot be dropped:Consider the inverse ‘telescope category’ P with shape ... → · → · → · It is a graded poset which is not bounded below. Consider the functor of constantvalue Z /p , c Z /p : P →
Ab: ... → Z /p → Z /p → Z /p It is straightforward that it satisfies the conditions in the theorem as all the cokernelsare zero and all the arrows are injective. But it is not a projective object of Ab P because, in that case, the adjoint pair colim : Ab P ↔ Ab : ∆ would give that Z /p isprojective in Ab ( see [5, 3.2, Ex7] or [16, Proposition 2.3.10]).This theorem yields the following examples. The degree functions deg for thebounded below graded posets appearing in the examples are indicated by subscripts i deg ( i ) on the objects i of P and take values { , , , , ... } . Example 4.11.
For the ‘pushout category’ P with shape a f / / g (cid:15) (cid:15) b c a functor F : P →
Ab is projective if and only if • F ( a ), F ( b ) / Im F ( f ) and F ( c ) / Im F ( g ) are free abelian. • F ( f ) and F ( g ) are monomorphisms.For the ‘telescope category’ P with shape a f / / a f / / a f / / a f / / a ... a functor F : P →
Ab is projective if and only if • F ( a ) is free abelian. • F ( a i ) / Im F ( f i ) is free abelian, F ( f i ◦ f i − ◦ .. ◦ f ) is a monomorphism andKer F ( f i ◦ f i − ◦ .. ◦ f i − d ) ⊆ Im F ( f i − d − ) for d = 0 , , .., i − i =1 , , , , ... . 5. Pseudo-projectivity
Consider a functor F : P →
Ab over a graded poset P . In this section we findconditions on F such that colim i F = 0 for i ≥
1. We fix the following notation
Definition 5.1.
Let P be a graded poset and F : P →
Ab. We say that F iscolim -acyclic if colim i F = 0 for i ≥ Theorem 5.2.
Let F : P → Ab be a pseudo-projective functor over a bounded belowgraded poset P . Then F is colim -acyclic.Proof. We can suppose that the degree function deg on P is increasing and takesvalues { , , , , ... } , and that Ob ( P ) = ∅ . To compute colim t F we use the (nor-malized, Remark 3.4) spectral sequences corresponding to the third row of Table 1in Chapter 3. That is, we first filter by the degree of the end object of each simplexto obtain a homological type spectral sequence E ∗∗ , ∗ . To compute the column E p, ∗ wefilter by the degree of the initial object of each object to obtain cohomological typespectral sequences ( E p ) ∗ , ∗∗ .Fix t ≥
1. Notice that to prove that colim t F = 0 is enough to show that E p,t − p is zero for every p . The contributions to E p,t − p come from ( E p ) p ′ ,p − p ′ − t ∞ for p ′ ≤ p − t (we are using normalized (Remark 3.4) spectral sequences). We prove that( E p ) p ′ ,p − p ′ − tr = 0if r is big enough for each p and p ′ ≤ p − t . This implies that colim t F = 0.Consider the increasing filtration L ∗ of C ∗ ( P , F ) that gives rise to the spectralsequence E ∗∗ , ∗ . The n -simplices are L pn = L p C n ( P , F ) = M σ ∈ N P n ,deg ( σ n ) ≤ p F σ . For each p we have a decreasing filtration M ∗ p of the quotient L p /L p − that gives riseto the spectral sequence ( E p ) ∗ , ∗∗ and which n -simplices are( M p ) p ′ n = M σ ∈ N P n ,deg ( σ ) ≥ p ′ ,deg ( σ n )= p F σ . FAMILY OF ACYCLIC FUNCTORS 17
For p ′ ≤ p − t the abelian group ( E p ) p ′ ,q ′ r at the t = − ( p ′ + q ′ ) + p simplices is givenby( E p ) p ′ ,q ′ r = ( M p ) p ′ t ∩ d − (( M p ) p ′ + rt − ) / ( M p ) p ′ +1 t ∩ d − (( M p ) p ′ + rt − ) + ( M p ) p ′ t ∩ d (( M p ) p ′ − r +1 t +1 )where d is the differential of the quotient L p /L p − restricted to the subgroups of thefiltration ( M p ) ∗ . For r > p − p ′ − ( t −
1) there are not ( t − p ′ + r > p − ( t −
1) and ending in degree p , i.e., ( M p ) p ′ + rt − = 0.Because P is bounded below for r big enough ( M p ) p ′ − r +1 t +1 = ( M p ) t +1 = ( L p /L p − ) t +1 ,i.e., ( M p ) p ′ − r +1 t +1 equals all the ( t + 1)-simplices that end on degree p . Thus there exists r such that(13) ( E p ) p ′ ,q ′ r = ( M p ) p ′ t ∩ d − (0) / ( M p ) p ′ +1 t ∩ d − (0) + ( M p ) p ′ t ∩ d (( M p ) t +1 ) . Fix such an r and take [ x ] ∈ ( E p ) p ′ ,q ′ r where(14) x = M σ ∈ N P t ,deg ( σ ) ≥ p ′ ,deg ( σ t )= p x σ and d ( x ) = 0. Notice that by definition there is just a finite number of summands x σ = 0 in the expression (14) for x . We prove that [ x ] = 0 in three steps: Step 1:
In this first step we find a representative x ′ for [ x ] x ′ = M σ ∈ N P t ,deg ( σ ) ≥ p ′ ,deg ( σ t )= p x ′ σ such that deg( α ) = 1 for every σ = σ α / / σ α / / ... α t − / / σ t − α t / / σ t with x ′ σ = 0.Take σ such that x σ = 0 and suppose that deg ( α ) >
1, i.e., deg ( σ ) < deg ( σ ) −
1. Then, as in a graded poset every morphism factors as composition of degree 1morphisms, there exists an object σ ∗ of degree deg ( σ ) < deg ( σ ∗ ) < deg ( σ ) andarrows β : σ → σ ∗ and β : σ ∗ → σ with α = β ◦ β . σ ∗ β ! ! CCCCCCCC σ α / / β = = |||||||| σ . Call ˜ σ to the ( t + 1)-simplex σ = σ β / / σ ∗ β / / σ α / / ... α t − / / σ t − α t / / σ t and consider the ( t + 1)-chain of ( M p ) t +1 y = i ˜ σ ( − x σ ). Its differential in L p /L p − equals d ( y ) = d ( y ) − d ( y ) + X i =2 ,..,t ( − i d i ( y ) = d ( y ) + i σ ( x σ ) + X i =2 ,..,t ( − i d i ( y ) . Notice that the first morphisms appearing in the simplices d (˜ σ ) and d i (˜ σ ) for i =2 , .., t have degree deg ( β ) and deg ( β ) respectively, which are strictly less than deg ( α ). Also notice that d ( y ) ∈ ( M p ) p ′ t ∩ d (( M p ) t +1 ) (which is zero in Equation(13) ). Taking the (finite) sum of the chains y for each term x σ we find that [ x ] = [ x ′ ]where x ′ = M σ ∈ N P t ,deg ( σ ) ≥ p ′ ,deg ( σ t )= p x ′ σ and the maximum of the degrees of the morphisms α of the simplices σ = σ α / / σ α / / ... α t − / / σ t − α t / / σ t with x ′ σ = 0 is smaller than this maximum computed for x . So repeating this processa finite number of times we find a representative as wished. For simplicity we writealso x for this representative. Step 2:
By Step 1 we can suppose that deg( α ) = 1 for every σ = σ α / / σ α / / ... α t − / / σ t − α t / / σ t with x σ = 0. Now our objective is to find a representative x ′ for [ x ] x ′ = M σ ∈ N P t ,deg ( σ )= p ′ ,deg ( σ t )= p x ′ σ , i.e., such that the expression for x ′ runs over simplices σ with begin in degree p ′ .Begin writing x as x = M i = p ′ ,..,p − t x i where x i = M σ ∈ N P t ,deg ( σ )= i,deg ( σ t )= p x σ . Notice that the index i just goes to p − t (and not to p ) because we are using normalized(Remark 3.4) spectral sequences. Now we prove Claim 5.2.1.
For each i from i = p − t to i = p ′ there exists a representative x ′ i for [ x ] x ′ i = M σ ∈ N P t ,i ≥ deg ( σ ) ≥ p ′ ,deg ( σ t )= p ( x ′ i ) σ such that (15) ( x ′ i ) σ = 0 and deg ( σ ) < i imply deg ( α ) = 1 . Notice that taking i = p ′ in the claim, the step 2 is finished. The case i = p − t inthe claim is fulfilled taking x ′ p − t = x (by step 1). Suppose the statement of the claimholds for i . Then we prove it for i −
1. We have x ′ i such that x ′ i = M σ ∈ N P t ,i ≥ deg ( σ ) ≥ p ′ ,deg ( σ t )= p ( x ′ i ) σ ,d ( x ′ i ) = 0 and [ x ] = [ x ′ i ]. The differential d on L p /L p − restricts to d : ( M p ) p ′ t → ( M p ) p ′ t − FAMILY OF ACYCLIC FUNCTORS 19 and carries z ∈ F σ ֒ → L σ ∈ N P t ,deg ( σ ) ≥ p ′ ,deg ( σ t )= p F σ = ( M p ) p ′ t to d ( z ) = X j =0 , ,..,t − ( − j d j ( z )with d j ( z ) ∈ F d j ( σ ) ֒ → ( M p ) p ′ t − . Notice that the initial object of d j ( σ ) is σ for j = 0 and σ for j = 1 , .., t −
1. Also notice that the final object of d j ( σ ) is σ t for j = 0 , .., t − d ( x ′ i ) = 0. So for every ǫ ∈ N P t − with deg ( ǫ ) ≥ p ′ and deg ( ǫ t − ) = p we can apply the projection π ǫ : ( M p ) p ′ t − ։ F ǫ and obtain π ǫ ( d ( x ′ i )) = 0. If deg ( ǫ ) > i then the remarks on the differential aboveand condition (15) imply that π ǫ ( d ( x ′ i )) = X σ ∈ N P t ,deg ( σ )= i,d ( σ )= ǫ F ( α )(( x ′ i ) σ )and thus(16) 0 = X σ ∈ N P t ,deg ( σ )= i,d ( σ )= ǫ F ( α )(( x ′ i ) σ )for each ǫ ∈ N P t − with deg ( ǫ ) > i and deg ( ǫ t − ) = p . Notice that each summand( x ′ i ) σ with σ ∈ N P t , deg ( σ ) = i and deg ( σ ) = p appears in one and just one equationas (16) (take ǫ = d ( σ )).Fix an ǫ ∈ N P t − with deg ( ǫ ) > i and deg ( ǫ t − ) = p and consider the associatedEquation (16). Then, as F is ( i − deg ( ǫ ))-pseudo-projective, ( x ′ i ) σ ∈ Im F ( σ ) forevery σ ∈ N P t with deg ( σ ) = i and d ( σ ) = ǫ . This means that for every such a σ there exists k σ objects of degree ( i − i σ , .., i k σ σ , arrows β jσ : i jσ → σ andelements x jσ ∈ F ( i jσ ) for j = 1 , .., k σ such that(17) ( x ′ i ) σ = X j =1 ,..,k σ F ( β jσ )( x jσ ) . Consider the ( t + 1)-simplices for j = 1 , .., k σ σ j = i jσ β jσ / / σ α / / σ α / / ... α t − / / σ t − α t / / σ t and the ( t + 1)-chain of ( M p ) i − t +1 y σ = ⊕ j =1 ,..,k σ i σ j ( x jσ ) . The differential of y σ is d ( y σ ) = d ( y σ ) + X j =1 ,..,t ( − j d j ( y σ )= d ( y σ ) + R σ , where R σ = X j =1 ,..,t ( − j d j ( y σ )= X j =1 ,..,k σ i d ( σ j ) ( F ( β jσ )( x jσ )) + R σ = X j =1 ,..,k σ i σ ( F ( β jσ )( x jσ )) + R σ = i σ ( X j =1 ,..,k σ F ( β jσ )( x jσ )) + R σ = i σ (( x ′ i ) σ ) + R σ where the last equality is due to (17). Notice that R σ lives in the subgroup L σ ∈ N P t ,deg ( σ )= i − ,deg ( σ t )= p F σ ⊆ ( M p ) p ′ t of simplices beginning at degree ( i − σ ∈ N P t with deg ( σ ) = i and d ( σ ) = ǫ weobtain y ǫ = P σ y σ such that d ( y ǫ ) = ⊕ σ ∈ N P t ,deg ( σ )= i,d ( σ )= ǫ ( x ′ i ) σ + R ǫ where R ǫ lives in the subgroup L σ ∈ N P t ,deg ( σ )= i − ,deg ( σ t )= p F σ ⊆ ( M p ) p ′ t . Repeatingthe same argument for every ǫ ∈ N P t − with deg ( ǫ ) > i and deg ( ǫ t − ) = p we obtain y = P ǫ y ǫ such that d ( y ) = ⊕ σ ∈ N P t ,deg ( σ )= i,d ( σ t )= p ( x ′ i ) σ + R where R lives in the subgroup L σ ∈ N P t ,deg ( σ )= i − ,deg ( σ t )= p F σ ⊆ ( M p ) p ′ t . By con-struction y ∈ ( M p ) i − t +1 ⊆ ( M p ) t +1 and d ( y ) ∈ ( M p ) i − t ⊆ ( M p ) p ′ t +1 . Thus d ( y ) ∈ ( M p ) p ′ t ∩ d (( M p ) t +1 ). Then, by (13), [ x ′ i ] = [ x ′ i − d ( y )] = [ x ′ i − ] where x ′ i − = ⊕ σ ∈ N P t ,i>deg ( σ ) ≥ p ′ ,d ( σ t )= p ( x ′ i ) σ + R is a representative that lives in M σ ∈ N P t ,i − ≥ deg ( σ ) ≥ p ′ ,deg ( σ t )= p F σ ⊆ ( M p ) p ′ t as wished. That condition (15) holds is clear from the definition of x ′ i − . Step 3:
By Step 2 we can suppose that x = M σ ∈ N P t ,deg ( σ )= p ′ ,deg ( σ t )= p x σ . Our objective now is to see that there exists y ∈ ( M p ) t +1 with d ( y ) = x . This impliesthat [ x ] = 0 and finishes the proof of the theorem. We need the FAMILY OF ACYCLIC FUNCTORS 21
Claim 5.2.2.
There exist chains x i ∈ ( M p ) t for i = p ′ , .., and y i ∈ ( M p ) t +1 for i = p ′ , .., such that (18) d ( y i ) = x i + x i − for i = p ′ , .., with x p ′ = x and x = 0 such that(1) x i lives on L σ ∈ N P t ,deg ( σ )= i,deg ( σ t )= p F σ ⊆ ( M p ) t for i = p ′ , .., .(2) d ( x i ) = 0 for i = p ′ , .., . Notice that the claim finishes Step 3: as x = 0 then x = d ( y ), x = d ( y ) − x = d ( y − y ), x = d ( y ) − x = d ( y − y + y ),.., x = x p ′ = d ( y p ′ ) − x p ′ − = d ( y p ′ − y p ′ − + ... + ( − p ′ +1 y ) where y p ′ − y p ′ − + ... + ( − p ′ +1 y ∈ ( M p ) t +1 .Define x p ′ def = x . Then condition (1) and (2) are satisfied for i = p ′ . We construct y i and x i − from x i recursively beginning on i = p ′ . The arguments are similar to thoseused in step 2.The differential d on L p /L p − restricts to d : ( M p ) t → ( M p ) t − . As d ( x p ′ ) = d ( x ) = 0, for every ǫ ∈ N P t − with deg ( ǫ t − ) = p we can apply theprojection π ǫ : ( M p ) t − ։ F ǫ and obtain π ǫ ( d ( x )) = 0. If deg ( ǫ ) > p ′ then π ǫ ( d ( x )) = X σ ∈ N P t ,d ( σ )= ǫ F ( α )( x σ )and thus(19) 0 = X σ ∈ N P t ,d ( σ )= ǫ F ( α )( x σ )for each ǫ ∈ N P t − with deg ( ǫ ) > p ′ and deg ( ǫ t − ) = p . Notice that each summand x σ with σ ∈ N P t , deg ( σ ) = p ′ and deg ( σ ) = p appears in one and just one equation as(19) (take ǫ = d ( σ )). Using now pseudo-injectivity we build as before y σ , y ǫ = P σ y σ and y = P ǫ y ǫ , where ǫ runs over ǫ ∈ N P t − with deg ( ǫ ) > p ′ and deg ( ǫ t − ) = p ,such that d ( y ) = x + R with R living in L σ ∈ N P t ,deg ( σ )= p ′ − ,deg ( σ t )= p F σ ⊆ ( M p ) t . Call y p ′ def = y and x p ′ − = R .Then Equation (18) is satisfied. Condition (1) for i = p ′ − R and condition (2) for i = p ′ − d ( x p ′ − ) = d ( R ) = d ( d ( y ) − x ) = d ( y ) − d ( x ) = 0 − d is a differential and d ( x ) = 0 by hypothesis. Theconstruction of y i and x i − from x i is totally analogous to the construction of y p ′ and x p ′ − from x p ′ that we have just made.After we have built y and x if we try to build y = P ǫ y ǫ and R from x wefind that, because there are not objects of negative degree (thus if z ∈ Im ( i ′ ) where deg ( i ′ ) = 0 then z = 0), x = 0. (cid:3) The following examples come from Example 4.11. They show the weaker conditionsthat are needed for colim-acyclicity instead of projectiveness.
Example 5.3.
For the “pushout category” P with shape a f / / g (cid:15) (cid:15) b c a functor F : P →
Ab is colim-acyclic if F ( f ) and F ( g ) are monomorphisms.For the “telescope category” P with shape a f / / a f / / a f / / a f / / a ... a functor F : P →
Ab is colim-acyclic if F ( f i ◦ f i − ◦ .. ◦ f ) is a monomorphism andKer F ( f i ◦ f i − ◦ .. ◦ f i − d +1 ) ⊆ Im F ( f i − d ) for d = 1 , , , .., i − i = 2 , , , ... Notice that for this it is enough that F ( f i ) is a monomorphism for each i = 1 , , , .. .6. Dual results for injective objects in Ab P . The appropriate notions to characterize the injective objects in the functor categoryAb P , where P is a graded poset, are the following: Definition 6.1.
Ker F ( i ) = T i α → i,α =1 i Ker F ( α ) (or Ker F ( i ) = F ( i ) if the indexset of the intersection is empty) and Coim F ( i ) = F ( i ) / Ker F ( i ). Definition 6.2.
Let F : P →
Ab be a functor over a graded poset P with degreefunction deg . Fix an integer d ≥
0. If for any object i of P , different objects { i j } j ∈ J of P , arrows α j : i → i j with deg ( α j ) = d and elements x j ∈ Ker F ( i j ) for each j ∈ J ,there is y ∈ F ( i ) with F ( α j )( y ) = x j for each j ∈ J , we call F d -pseudo-injective . If F is d -pseudo-injective for each d ≥ F pseudo-injective.Then we can prove the following Theorem 6.3.
Let P be a bounded above graded poset and F : P → Ab be a functor.Then F is injective if and only if(1) for any object i of P Ker F ( i ) is injective in Ab .(2) F is pseudo-injective. Also in the dual case pseudo-injectiveness is enough for vanishing higher inverselimits:
Definition 6.4.
Let P be a graded poset and F : P →
Ab. We say F is lim -acyclic if lim i F = 0 for i ≥ Theorem 6.5.
Let F : P → Ab be a pseudo-injective functor over a bounded abovegraded poset P . Then F is lim -acyclic. FAMILY OF ACYCLIC FUNCTORS 23
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Department of Mathematical Sciences, King’s College, University of Aberdeen,ABERDEEN AB24 3UE U.K.
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