nLab
lim^1 and Milnor sequences

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Limits and colimits

Contents

Idea

The notation “lim 1\underset{\longleftarrow}{\lim}^1” is common notation for the first derived functor R 1limR^1 \underset{\longleftarrow}{\lim} of the limit functor.

Here we consider the case of limits over sequential diagrams of abelian groups (prop. 2 below). In good cases, this is the only obstruction to a naive limit of homotopy sets being the homotopy classes of the correct homotopy limit. Such a situation is expressed by a short exact Milnor sequence (below).

Definition

Definition

Given a tower A A_\bullet of abelian groups

A 3f 2A 2f 1A 1f 0A 0 \cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0

write

:nA nnA n \partial \;\colon\; \underset{n}{\prod} A_n \longrightarrow \underset{n}{\prod} A_n

for the homomorphism given by

:(a n) n(a nf n(a n+1)) n. \partial \;\colon\; (a_n)_{n \in \mathbb{N}} \mapsto (a_n - f_n(a_{n+1}))_{n \in \mathbb{N}}.
Remark

The limit of a sequence as in def. 1 – hence the group lim nA n\underset{\longleftarrow}{\lim}_n A_n universally equipped with morphisms lim nA np nA n\underset{\longleftarrow}{\lim}_n A_n \overset{p_n}{\to} A_n such that all

lim nA n p n+1 p n A n+1 f n A n \array{ && \underset{\longleftarrow}{\lim}_n A_n \\ & {}^{\mathllap{p_{n+1}}}\swarrow && \searrow^{\mathrlap{p_n}} \\ A_{n+1} && \overset{f_n}{\longrightarrow} && A_n }

commute – is equivalently the kernel of the morphism \partial in def. 1.

Definition

Given a tower A A_\bullet of abelian groups

A 3f 2A 2f 1A 1f 0A 0 \cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0

then lim 1A \underset{\longleftarrow}{\lim}^1 A_\bullet is the cokernel of the map \partial in def. 1, hence the group that makes a long exact sequence of the form

0lim nA nnA nnA nlim n 1A n0, 0 \to \underset{\longleftarrow}{\lim}_n A_n \longrightarrow \underset{n}{\prod} A_n \stackrel{\partial}{\longrightarrow} \underset{n}{\prod} A_n \longrightarrow \underset{\longleftarrow}{\lim}^1_n A_n \to 0 \,,

There is a generalization to groups (not necessarily abelian). (Bousfield-Kan 72, IX.2.1)

Properties

Abstract characterizations

Proposition

The functor lim 1:Ab (,)Ab\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab (def. 2) satisfies

  1. for every short exact sequence 0A B C 0Ab (,)0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 \;\;\; \in Ab^{(\mathbb{N}, \geq)} then the induced sequence

    0lim nA nlim nB nlim nC nlim n 1A nlim n 1B nlim n 1C n0 0 \to \underset{\longleftarrow}{\lim}_n A_n \to \underset{\longleftarrow}{\lim}_n B_n \to \underset{\longleftarrow}{\lim}_n C_n \to \underset{\longleftarrow}{\lim}_n^1 A_n \to \underset{\longleftarrow}{\lim}_n^1 B_n \to \underset{\longleftarrow}{\lim}_n^1 C_n \to 0

    is a long exact sequence of abelian groups;

  2. if A A_\bullet is a tower such that all maps are surjections, then lim n 1A n0\underset{\longleftarrow}{\lim}^1_n A_n \simeq 0.

(e.g. Bousfield-Kan 72, ch IX, prop. 2.3 and 2.4, Switzer 75, prop. 7.63, Goerss-Jardine 96, section VI. lemma 2.11)

Proof

For the first property: Given A A_\bullet a tower of abelian groups, write

L (A )[0nA ndeg0nA ndeg10] L^\bullet(A_\bullet) \coloneqq \left[ 0 \to \underset{deg \, 0}{\underbrace{\underset{n}{\prod} A_n}} \overset{\partial}{\longrightarrow} \underset{deg\, 1}{\underbrace{\underset{n}{\prod} A_n}} \to 0 \right]

for the homomorphism from def. 1 regarded as the single non-trivial differential in a cochain complex of abelian groups. Then by remark 1 and def. 2 we have H 0(L(A ))limA H^0(L(A_\bullet)) \simeq \underset{\longleftarrow}{\lim} A_\bullet and H 1(L(A ))lim 1A H^1(L(A_\bullet)) \simeq \underset{\longleftarrow}{\lim}^1 A_\bullet.

With this, then for a short exact sequence of towers 0A B C 00 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 the long exact sequence in question is the long exact sequence in homology of the corresponding short exact sequence of complexes

0L (A )L (B )L (C )0. 0 \to L^\bullet(A_\bullet) \longrightarrow L^\bullet(B_\bullet) \longrightarrow L^\bullet(C_\bullet) \to 0 \,.

For the second statement: If all the f kf_k are surjective, then inspection shows that the homomorphism \partial in def. 1 is surjective. Hence its cokernel vanishes.

Lemma

The category Ab (,)Ab^{(\mathbb{N}, \geq)} of towers of abelian groups has enough injectives.

Proof

The functor () n:Ab (,)Ab(-)_n \colon Ab^{(\mathbb{N}, \geq)} \to Ab that picks the nn-th component of the tower has a right adjoint r nr_n, which sends an abelian group AA to the tower

r n[idAidA=(r n) n+1idA=(r n) nid0=(r n) n1000]. r_n \coloneqq \left[ \cdots \overset{id}{\to} A \overset{id}{\to} \underset{= (r_n)_{n+1}}{\underbrace{A}} \overset{id}{\to} \underset{= (r_n)_n}{\underbrace{A}} \overset{id}{\to} \underset{= (r_n)_{n-1}}{\underbrace{0}} \to 0 \to \cdots \to 0 \to 0 \right] \,.

Since () n(-)_n itself is evidently an exact functor, its right adjoint preserves injective objects (prop.).

So with A Ab (,)A_\bullet \in Ab^{(\mathbb{N}, \geq)}, let A nA˜ nA_n \hookrightarrow \tilde A_n be an injective resolution of the abelian group A nA_n, for each nn \in \mathbb{N}. Then

A (η n) nnr nA nnr nA˜ n A_\bullet \overset{(\eta_n)_{n \in \mathbb{N}}}{\longrightarrow} \underset{n \in \mathbb{R}}{\prod} r_n A_n \hookrightarrow \underset{n \in \mathbb{N}}{\prod} r_n \tilde A_n

is an injective resolution for A A_\bullet.

Proposition

The functor lim 1:Ab (,)Ab\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab (def. 2) is the first right derived functor of the limit functor lim:Ab (,)Ab\underset{\longleftarrow}{\lim} \colon Ab^{(\mathbb{N},\geq)} \longrightarrow Ab.

(Bousfield-Kan 72, chapter IX.2, remark 2.6)

Proof

By lemma 1 there are enough injectives in Ab (,)Ab^{(\mathbb{N}, \geq)}. So for A Ab (,)A_\bullet \in Ab^{(\mathbb{N}, \geq)} the given tower of abelian groups, let

0A j 0J 0j 1J 1j 2J 2 0 \to A_\bullet \overset{j^0}{\longrightarrow} J^0_\bullet \overset{j^1}{\longrightarrow} J^1_\bullet \overset{j^2}{\longrightarrow} J^2_\bullet \overset{}{\longrightarrow} \cdots

be an injective resolution. We need to show that

lim 1A ker(lim(j 2))/im(lim(j 1)). \underset{\longleftarrow}{\lim}^1 A_\bullet \simeq ker(\underset{\longleftarrow}{\lim}(j^2))/im(\underset{\longleftarrow}{\lim}(j^1)) \,.

Since limits preserve kernels, this is equivalently

lim 1A (lim(ker(j 2) ))/im(lim(j 1)) \underset{\longleftarrow}{\lim}^1 A_\bullet \simeq (\underset{\longleftarrow}{\lim}(ker(j^2)_\bullet))/im(\underset{\longleftarrow}{\lim}(j^1))

Now observe that each injective J qJ^q_\bullet is a tower of epimorphism. This follows by the defining right lifting property applied against the monomorphisms of towers of the following form

0 0 0 id id id id id incl id id id 0 0 id id id id \array{ \cdots &\to & 0 &\to& 0 &\longrightarrow& 0 &\longrightarrow& \mathbb{Z} &\overset{id}{\longrightarrow}& \cdots &\overset{id}{\longrightarrow}& \mathbb{Z} &\overset{id}{\longrightarrow}& \mathbb{Z} \\ \cdots && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{id}} && && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} \\ \cdots &\to& 0 &\to& 0 &\to & \mathbb{Z} &\underset{id}{\longrightarrow}& \mathbb{Z} &\underset{id}{\longrightarrow}& \cdots &\underset{id}{\longrightarrow}& \mathbb{Z} &\underset{id}{\longrightarrow}& \mathbb{Z} }

Therefore by the second item of prop. 1 the long exact sequence from the first item of prop. 1 applied to the short exact sequence

0A j 0J 0j 1ker(j 2) 0 0 \to A_\bullet \overset{j^0}{\longrightarrow} J^0_\bullet \overset{j^1}{\longrightarrow} ker(j^2)_\bullet \to 0

becomes

0limA limj 0limJ 0limj 1lim(ker(j 2) )lim 1A 0. 0 \to \underset{\longleftarrow}{\lim} A_\bullet \overset{\underset{\longleftarrow}{\lim} j^0}{\longrightarrow} \underset{\longleftarrow}{\lim} J^0_\bullet \overset{\underset{\longleftarrow}{\lim}j^1}{\longrightarrow} \underset{\longleftarrow}{\lim}(ker(j^2)_\bullet) \longrightarrow \underset{\longleftarrow}{\lim}^1 A_\bullet \longrightarrow 0 \,.

Exactness of this sequence gives the desired identification lim 1A (lim(ker(j 2) ))/im(lim(j 1)). \underset{\longleftarrow}{\lim}^1 A_\bullet \simeq (\underset{\longleftarrow}{\lim}(ker(j^2)_\bullet))/im(\underset{\longleftarrow}{\lim}(j^1)) \,.

Proposition

The functor lim 1:Ab (,)Ab\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab (def. 2) is in fact the unique functor, up to natural isomorphism, satisfying the conditions in prop. 3.

Proof

The proof of prop. 2 only used the conditions from prop. 1, hence any functor satisfying these conditions is the first right derived functor of lim\underset{\longleftarrow}{\lim}, up to natural isomorphism.

Vanishing of lim 1\lim^1

Definition

A tower A A_\bullet of abelian groups

A 3A 2A 1A 0 \cdots \to A_3 \to A_2 \to A_1 \to A_0

is said to satisfy the Mittag-Leffler condition if for all kk there exists iki \geq k such that for all jikj \geq i \geq k the image of the homomorphism A iA kA_i \to A_k equals that of A jA kA_j \to A_k

im(A iA k)im(A jA k). im(A_i \to A_k) \simeq im(A_j \to A_k) \,.

(e.g. Switzer 75, def. 7.74)

Example

The Mittag-Leffler condition, def. 3, is satisfied in particular when all morphisms A i+1A iA_{i+1}\to A_i are epimorphisms (hence surjections of the underlying sets).

Proposition

If a tower A A_\bullet satisfies the Mittag-Leffler condition, def. 3, then its lim 1\underset{\leftarrow}{\lim}^1 vanishes:

lim 1A =0. \underset{\longleftarrow}{\lim}^1 A_\bullet = 0 \,.

e.g. (Switzer 75, theorem 7.75, Kochmann 96, prop. 4.2.3, Weibel 94, prop. 3.5.7)

Proof idea

One needs to show that with the Mittag-Leffler condition, then the cokernel of \partial in def. 1 vanishes, hence that \partial is an epimorphism in this case, hence that every (a n) nnA n(a_n)_{n \in \mathbb{N}} \in \underset{n}{\prod} A_n has a preimage under \partial. So use the Mittag-Leffler condition to find pre-images of a na_n by induction over nn.

Relation to ExtExt-groups

Lemma

Given a cotower

A =(A 0f 0A 1f 1A 2) A_\bullet = (A_0 \overset{f_0}{\to} A _1 \overset{f_1}{\to} A_2 \to \cdots)

of abelian groups, then for every abelian group BAbB \in Ab there is a short exact sequence of the form

0lim n 1Hom(A n,B)Ext 1(lim nA n,B)lim nExt 1(A n,B)0, 0 \to \underset{\longleftarrow}{\lim}^1_n Hom(A_n, B) \longrightarrow Ext^1( \underset{\longrightarrow}{\lim}_n A_n, B ) \longrightarrow \underset{\longleftarrow}{\lim}_n Ext^1( A_n, B) \to 0 \,,

where Hom(,)Hom(-,-) denotes the hom-group, Ext 1(,)Ext^1(-,-) denotes the first Ext-group (and so Hom(,)=Ext 0(,)Hom(-,-) = Ext^0(-,-)).

Proof

Consider the homomorphism

˜:nA nnA n \tilde \partial \;\colon\; \underset{n}{\oplus} A_n \longrightarrow \underset{n}{\oplus} A_n

which sends a nA na_n \in A_n to a nf n(a n)a_n - f_n(a_n). Its cokernel is the colimit over the cotower, but its kernel is trivial (in contrast to the otherwise formally dual situation in remark 1). Hence (as opposed to the long exact sequence in def. 2) there is a short exact sequence of the form

0nA n˜nA nlim nA n0. 0 \to \underset{n}{\oplus} A_n \overset{\tilde \partial}{\longrightarrow} \underset{n}{\oplus} A_n \overset{}{\longrightarrow} \underset{\longrightarrow}{lim}_n A_n \to 0 \,.

Every short exact sequence gives rise to a long exact sequence of derived functors (prop.) which in the present case starts out as

0Hom(lim nA n,B)nHom(A n,B)nHom(A n,B)Ext 1(lim nA n,B)nExt 1(A n,B)nExt 1(A n,B) 0 \to Hom(\underset{\longrightarrow}{\lim}_n A_n,B) \longrightarrow \underset{n}{\prod} Hom( A_n, B ) \overset{\partial}{\longrightarrow} \underset{n}{\prod} Hom( A_n, B ) \longrightarrow Ext^1(\underset{\longrightarrow}{\lim}_n A_n,B) \longrightarrow \underset{n}{\prod} Ext^1( A_n, B ) \overset{\partial}{\longrightarrow} \underset{n}{\prod} Ext^1( A_n, B ) \longrightarrow \cdots

where we used that direct sum is the coproduct in abelian groups, so that homs out of it yield a product, and where the morphism \partial is the one from def. 1 corresponding to the tower

Hom(A ,B)=(Hom(A 2,B)Hom(A 1,B)Hom(A 0,B)). Hom(A_\bullet,B) = ( \cdots \to Hom(A_2,B) \to Hom(A_1,B) \to Hom(A_0,B) ) \,.

Hence truncating this long sequence by forming kernel and cokernel of \partial, respectively, it becomes the short exact sequence in question.

Milnor exact sequences

For homotopy groups

Proposition

(Milnor exact sequence for homotopy groups)

Let

X 3p 2X 2p 1X 1p 0X 0 \cdots \to X_3 \overset{p_2}{\longrightarrow} X_2 \overset{p_1}{\longrightarrow} X_1 \overset{p_0}{\longrightarrow} X_0

be a tower of fibrations, for instance a tower of simplicial sets with each map a Kan fibration (and X 0X_0, hence each X nX_n a Kan complex), or a tower of topological spaces with each map a Serre fibration. Then for each qq \in \mathbb{N} there is a short exact sequence

0lim i 1π q+1(X i)π q(lim iX i)lim iπ q(X i)0, 0 \to \underset{\longleftarrow}{\lim}^1_i \pi_{q+1}(X_i) \longrightarrow \pi_q(\underset{\longleftarrow}{\lim}_i X_i) \longrightarrow \underset{\longleftarrow}{\lim}_i \pi_q(X_i) \to 0 \,,

for π \pi_\bullet the homotopy group-functor (exact as pointed sets for i=0i = 0, as groups for i1i \geq 1) which says that

  1. the failure of the limit over the homotopy groups of the stages of the tower to equal the homotopy groups of the limit of the tower is at most in the kernel of the canonical comparison map;

  2. that kernel is the lim 1\underset{\longleftarrow}{\lim}^1 (def. 2) of the homotopy groups of the stages.

e.g. (Bousfield-Kan 72, chapter IX, theorem 3.1, Goerss-Jardine 96, section VI. prop. 2.15)

Proof

With respect to the classical model structure on simplicial sets or the classical model structure on topological spaces, a tower of fibrations as stated is a fibrant object in the injective model structure on functors [(,),sSet] inj[(\mathbb{N},\geq), sSet]_{inj} ([(,),Top] inj[(\mathbb{N},\geq), Top]_{inj}) (prop). Hence the plain limit over this diagram represents the homotopy limit. By the discussion there, up to weak equivalence that homotopy limit is also the pullback in

holimX nPath(X n) (pb) nX n (id,p n) n nX n×X n, \array{ holim X_\bullet &\longrightarrow& \underset{n}{\prod} Path(X_n) \\ \downarrow &(pb)& \downarrow \\ \underset{n}{\prod} X_n &\underset{(id,p_n)_n}{\longrightarrow}& \underset{n}{\prod} X_ n \times X_n } \,,

where on the right we have the product over all the canonical fibrations out of the path space objects. Hence also the left vertical morphism is a fibration, and so by taking its fiber over a basepoint, the pasting law gives a homotopy fiber sequence

nΩX nholimX nX n. \underset{n}{\prod} \Omega X_n \longrightarrow holim X_\bullet \longrightarrow \underset{n}{\prod} X_n \,.

The long exact sequence of homotopy groups of this fiber sequence goes

nπ q+1(X n)nπ q+1(X n)π q(limX )nπ q(X n)nπ q(X n). \cdots \to \underset{n}{\prod} \pi_{q+1}(X_n) \longrightarrow \underset{n}{\prod} \pi_{q+1}(X_n) \longrightarrow \pi_q (\underset{\longleftarrow}{\lim} X_\bullet) \longrightarrow \underset{n}{\prod} \pi_q(X_n) \longrightarrow \underset{n}{\prod} \pi_q(X_n) \to \cdots \,.

Chopping that off by forming kernel and cokernel yields the claim for positive qq. For q=0q = 0 it follows by inspection.

For chain homology

Example

Let

C 3C 2C 1C 0 \cdots \to C_3 \to C_2 \to C_1 \to C_0

be a tower of chain complexes (of abelian groups) such that it satisfies degree-wise the Mittag-Leffler condition, def. 3, and write

Clim nC n C \coloneqq \underset{\longleftarrow}{\lim}_n C_n

for its limit. Then for each qq \in \mathbb{Z} the chain homology H q()H_q(-) of the limit sits in a short exact sequence with the ordinary lim\underset{\longleftarrow}{\lim} and the lim 1\underset{\longleftarrow}{\lim}^1 of the chain homologies:

0lim i 1H q+1(C i)H q(C)lim iH q(C i)0. 0 \to \underset{\longleftarrow}{\lim}^1_i H_{q+1}(C_i) \longrightarrow H_q(C) \longrightarrow \underset{\longleftarrow}{\lim}_i H_q(C_i) \to 0 \,.

(e.g. Weibel 94, prop. 3.5.8)

For generalized cohomology groups

Of spaces

Proposition

(Milnor exact sequence for generalized cohomology)

Let XX be a pointed CW-complex, X=lim nX nX = \underset{\longrightarrow}{\lim}_n X_n and let E˜ \tilde E^\bullet an additive reduced cohomology theory.

Then the canonical morphisms make a short exact sequence

0lim n 1E˜ 1(X n)E˜ (X)lim nE˜ (X n)0, 0 \to \underset{\longleftarrow}{\lim}^1_n \tilde E^{\bullet-1}(X_n) \longrightarrow \tilde E^{\bullet}(X) \longrightarrow \underset{\longleftarrow}{\lim}_n \tilde E^{\bullet}(X_n) \to 0 \,,

saying that

  1. the failure of the canonical comparison map E˜ (X)limE˜ (X n)\tilde E^\bullet(X) \to \underset{\longleftarrow}{\lim} \tilde E^\bullet(X_n) to the limit of the cohomology groups on the finite stages to be an isomorphism is at most in a non-vanishing kernel;

  2. this kernel is precisely the lim 1\lim^1 (def. 2) of the cohomology groups at the finite stages in one degree lower.

e.g. (Switzer 75, prop. 7.66, Kochmann 96, prop. 4.2.2)

Proof

For

X =(X 0i 0X 1i 1X 2i 1) X_\bullet = \left( X_0 \overset{i_0}{\hookrightarrow} X_1 \overset{i_1}{\hookrightarrow} X_2 \overset{i_1}{\hookrightarrow} \cdots \right)

the sequence of stages of the (pointed) CW-complex X=lim nX nX = \underset{\longleftarrow}{\lim}_n X_n, write

A X nX 2n×[2n,2n+1]; B X nX (2n+1)×[2n+1,2n+2]. \begin{aligned} A_X &\coloneqq \underset{n \in \mathbb{N}}{\sqcup} X_{2n} \times [2n,{2n}+1]; \\ B_X &\coloneqq \underset{n \in \mathbb{N}}{\sqcup} X_{(2n+1)} \times [2n+1,{2n}+2]. \end{aligned}

for the disjoint unions of the cylinders over all the stages in even and all those in odd degree, respectively.

These come with canonical inclusion maps into the mapping telescope Tel(X )Tel(X_\bullet) (def.), which we denote by

A X B X ι A x ι B x Tel(X ). \array{ A_X && && B_X \\ & {}_{\mathllap{\iota_{A_x}}}\searrow && \swarrow_{\mathrlap{\iota_{B_x}}} \\ && Tel(X_\bullet) } \,.

Observe that

  1. A XB XTel(X )A_X \cup B_X \simeq Tel(X_\bullet);

  2. A XB XnX nA_X \cap B_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_n;

and that there are homotopy equivalences

  1. A XnX 2n+1A_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n+1}

  2. B XnX 2nB_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n}

  3. Tel(X )XTel(X_\bullet) \simeq X.

The first two are obvious, the third is this proposition.

This implies that the Mayer-Vietoris sequence (prop.) for E˜ \tilde E^\bullet on the cover ABXA \sqcup B \to X is isomorphic to the bottom horizontal sequence in the following diagram:

E˜ 1(A X)E˜ 1(B X) E˜ 1(A XB X) E˜ (X) (ι A x) *(ι B x) * E˜ (A X)E˜ (B X) E˜ (A XB X) = (id,id) nE˜ 1(X n) nE˜ 1(X n) E˜ (X) (i n *) n nE˜ (X n) nE˜ (X n), \array{ \tilde E^{\bullet-1}(A_X)\oplus \tilde E^{\bullet-1}(B_X) &\longrightarrow& \tilde E^{\bullet-1}(A_X \cap B_X) &\longrightarrow& \tilde E^\bullet(X) &\overset{(\iota_{A_x})^\ast - (\iota_{B_x})^\ast}{\longrightarrow}& \tilde E^\bullet(A_X)\oplus \tilde E^\bullet(B_X) &\overset{}{\longrightarrow}& \tilde E^\bullet(A_X \cap B_X) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} && {}^{\mathllap{(id, -id)}}\downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \underset{n}{\prod}\tilde E^{\bullet-1}(X_n) &\underset{\partial}{\longrightarrow}& \underset{n}{\prod}\tilde E^{\bullet-1}(X_n) &\longrightarrow& \tilde E^\bullet(X) &\overset{(i_n^\ast)_{n}}{\longrightarrow}& \underset{n}{\prod}\tilde E^\bullet(X_n) &\underset{\partial}{\longrightarrow}& \underset{n}{\prod}\tilde E^\bullet(X_n) } \,,

hence that the bottom sequence is also a long exact sequence.

To identify the morphism \partial, notice that it comes from pulling back EE-cohomology classes along the inclusions ABAA \cap B \to A and ABBA\cap B \to B. Comonentwise these are the inclusions of each X nX_n into the left and the right end of its cylinder inside the mapping telescope, respectively. By the construction of the mapping telescope, one of these ends is embedded via i n:X nX n+1i_n \colon X_n \hookrightarrow X_{n+1} into the cylinder over X n+1X_{n+1}. In conclusion, \partial acts by

:(a n) n(a ni n *(a n+1)). \partial \;\colon\; (a_n)_{n \in \mathbb{N}} \mapsto ( a_n - i_n^\ast(a_{n+1}) ) \,.

(The relative sign is the one in (ι A x) *(ι B x) *(\iota_{A_x})^\ast - (\iota_{B_x})^\ast originating in the definition of the Mayer-Vietoris sequence and properly propagated to the bottom sequence while ensuring that E˜ (X) nE˜ (X n)\tilde E^\bullet(X)\to \prod_n \tilde E^\bullet(X_n) is really (i n *) n(i_n^\ast)_n and not (1) n(i n *) n(-1)^n(i_n^\ast)_n, as needed for the statement to be proven.)

This is the morphism from def. 1 for the sequence

E˜ (X n+1)i n *E˜ (X n)i n *E˜ (X n1). \cdots \to \tilde E^\bullet(X_{n+1}) \overset{i_n^\ast}{\longrightarrow} \tilde E^\bullet(X_n) \overset{i_n^\ast}{\longrightarrow} \tilde E^{\bullet}(X_{n-1}) \to \cdots \,.

Hence truncating the above long exact sequence by forming kernel and cokernel of \partial, the result follows via remark 1 and definition 2.

In contrast:

Proposition

Let XX be a pointed CW-complex, X=lim nX nX = \underset{\longleftarrow}{\lim}_n X_n.

For E˜ \tilde E_\bullet an additive reduced generalized homology theory, then

lim nE˜ (X n)E˜ (X) \underset{\longrightarrow}{\lim}_n \tilde E_\bullet(X_n) \overset{\simeq}{\longrightarrow} \tilde E_\bullet(X)

is an isomorphism.

(Switzer 75, prop. 7.53)

Of spectra

For X,EHo(Spectra)X, E \in Ho(Spectra) two spectra, then the EE-generalized cohomology of XX is the graded group of homs in the stable homotopy category (def., exmpl.)

E (X) [X,E] [Σ X,E]. \begin{aligned} E^\bullet(X) & \coloneqq [X,E]_{-\bullet} \\ & \coloneqq [\Sigma^\bullet X, E] \end{aligned} \,.

The stable homotopy category is, in particular, the homotopy category of the stable model structure on orthogonal spectra, in that its localization at the stable weak homotopy equivalences is of the form

γ:OrthSpec(Top cg) stableHo(Spectra). \gamma \;\colon\; OrthSpec(Top_{cg})_{stable} \longrightarrow Ho(Spectra) \,.

In the following when considering an orthogonal spectrum XOrthSpec(Top cg)X \in OrthSpec(Top_{cg}), we use, for brevity, the same symbol for its image under γ\gamma.

Proposition

For X,EOrthSpec(Top cg)X, E \in OrthSpec(Top_{cg}) two orthogonal spectra (or two symmetric spectra such that XX is a semistable symmetric spectrum) then there is a short exact sequence of the form

0lim n 1E +n1(X n)E (X)lim nE +n(X n)0 0 \to \underset{\longleftarrow}{\lim}^1_n E^{\bullet + n -1}(X_{n}) \longrightarrow E^\bullet(X) \longrightarrow \underset{\longleftarrow}{\lim}_n E^{\bullet + n}(X_n) \to 0

where lim 1\underset{\longleftarrow}{\lim}^1 denotes the lim^1, and where this and the limit on the right are taken over the following structure morphisms

E +n+1(X n+1)E +1n+1(Σ n X)E +n+1(X nS 1)E +n(X n). E^{\bullet + n + 1}(X_{n+1}) \overset{E^{\bullet+1n+1}(\Sigma^X_n)}{\longrightarrow} E^{\bullet+n+1}(X_n \wedge S^1) \overset{\simeq}{\longrightarrow} E^{\bullet + n}(X_n) \,.

(Schwede 12, chapter II prop. 6.5 (ii)) (using that symmetric spectra underlying orthogonal spectra are semistable (Schwede 12, p. 40))

Corollary

For X,EHo(Spectra)X,E \in Ho(Spectra) two spectra such that the tower nE n1(X n)n \mapsto E^{n -1}(X_{n}) satisfies the Mittag-Leffler condition (def. 3), then two morphisms of spectra XEX \longrightarrow E are homotopic already if all their morphisms of component spaces X nE nX_n \to E_n are.

Proof

By prop. 4 the assumption implies that the lim 1lim^1-term in prop. 8 vanishes, hence by exactness it follows that in this case there is an isomorphism

[X,E]E 0(X)lim n[X n,E n]. [X,E] \simeq E^0(X) \overset{\simeq}{\longrightarrow} \underset{\longleftarrow}{\lim}_n [X_n, E_n] \,.

References

For generalized cohomology of spectra

Discussion in the context of categories of fibrant objects is in

Discussion in the context of conditional convergence of spectral sequences is in

Last revised on January 30, 2018 at 02:50:24. See the history of this page for a list of all contributions to it.