§4 The generalised Eilenberg-Zilber theorem

We give a proof of the generalised Eilenberg-Zilber theorem of Dold,PuppeandCartier[9]. The arguments used here are adapted from the articles [12], [13] of EilenbergandMac Lane.

Throughout this section, we suppose given a bisimplicial objectAin an abelian category A.

(3.26) Definition (shuffle). We letn∈N0 andp∈ bn,0c. A(p, n−p)-shuffle is a permutationµ∈S_[0,n−1]

such that µ|_[0,p−1] and µ|_[p,n−1] are strictly monotonically increasing maps, where we write S_[0,n−1] for the symmetric group on[0, n−1]. The set of all(p, n−p)-shuffles is denoted bySh_p,n−p.

(3.27) Definition. We let n∈N0, p∈ bn,0c.

(a) The morphism A_n,n ^AW

A p,n−p

−−−−−→A_p,n−p

is defined by

AWp,n−p :=AW^A_p,n−p:= d^h_bn,p+1cd^v_bp−1,0c. (b) Further, we let

A_p,n−p−−−−−→^∇^p,n−p An,n

be given by

∇p,n−p:=∇^A_p,n−p:= X

µ∈Shp,n−p

(sgnµ)s^v_d0,p−1eµs^h_dp,n−1eµ.

Our first aim is to show that these morphisms yield complex morphisms betweenC DiagAandTot C⁽²⁾A.

(3.28) Remark. Defining C_nDiagA−−→^AWⁿ Tot_nC⁽²⁾A

byAWnpr_p,n−p:=AW_p,n−pfor allp∈ bn,0c,n∈N⁰, we obtain a complex morphism C DiagA−→^AW Tot C⁽²⁾A.

Proof. We have

AWn∂pr_p,n−1−p =AWp+1,n∂^h+ (−1)^pAWp,n−p∂^v

= d^h_bn,p+2cd^v_bp,0c( X

i∈[0,p+1]

(−1)ⁱd^h_i) + (−1)^pd^h_bn,p+1cd^v_bp−1,0c( X

j∈[0,n−p]

(−1)^jd^v_j)

= X

i∈[0,p+1]

(−1)ⁱd^h_bn,p+2cd^h_id^v_bp,0c+ X

j∈[0,n−p]

(−1)^j+pd^h_bn,p+1cd^v_j+pd^v_bp−1,0c

= X

i∈[0,p+1]

(−1)ⁱd^h_bn,p+2cd^h_id^v_bp,0c+ X

j∈[p,n]

(−1)^jd^h_bn,p+1cd^v_jd^v_bp−1,0c

= X

(3.29) Definition (Alexander-Whitney morphism). The complex morphism C DiagA−→^AW Tot C⁽²⁾A

(3.30) Proposition (recursive characterisation of the shuffle morphism via path simplicial objects). We have

∇^A_0,0= id_A_0,0 and

we can conclude

∇^A_p,n−p= s^v_n−p X

µ∈Shp,n−p

(p−1)µ=n−1

(sgnµ)s^v_d0,p−2eµs^h_dp,n−1eµ+ s^h_p X

µ∈Shp,n−p

(n−1)µ=n−1

(sgnµ)s^v_d0,p−1eµs^h_dp,n−2eµ

= (−1)^n−ps^v_n−p X

µ∈Shp−1,n−p+1

(n−1)µ=n−1

(sgnµ)s^v_d0,p−2eµs^h_{dp−1,n−2eµ}+ s^h_p X

µ∈Shp,n−p

(n−1)µ=n−1

(sgnµ)s^v_d0,p−1eµs^h_dp,n−2eµ

= (−1)^n−ps^v_n−p X

µ∈Shp−1,n−p

(sgnµ)s^v_d0,p−2eµs^h_{dp−1,n−2eµ}+ s^h_p X

µ∈Shp,n−1−p

(sgnµ)s^v_d0,p−1eµs^h_dp,n−2eµ

= (−1)^n−ps^v_n−p∇^P_p−1,n−p⁽²⁾^A + s^h_p∇^P_p,n−1−p⁽²⁾^A

The proof forp=norp= 0is easier since in this case the sum in the shuffle morphism does not split into two sums and since the only(n,0)-shuffle resp.(0, n)-shuffle is the identity: We have

∇^A_n,0= s^v_d0,n−1e= s^v₀s^v_d0,n−2e= s^v₀∇^P_n−1,0⁽²⁾^A and

∇^A_0,n= s^h_d0,n−1e= s^h₀s^h_d0,n−2e= s^h₀∇^P_0,n−1⁽²⁾^A. (3.31) Lemma. We have

∇^A_p,n−ps^Diag_n ^A= s^v_n−ps^h_p∇^P_p,n−p⁽²⁾^A and

∇^P_p,n−p⁽²⁾^Ad^Diag_n+1^A= d^v_n+1−pd^h_p+1∇^A_p,n−p for allp∈ bn,0c,n∈N0.

Proof. We compute

∇^A_p,n−ps^Diag_n ^A= X

µ∈Shp,n−p

(sgnµ)s^v_d0,p−1eµs^h_dp,n−1eµs^v_ns^h_n = s^v_n−ps^h_p X

µ∈Shp,n−p

(sgnµ)s^v_d0,p−1eµs^h_dp,n−1eµ

= s^v_n−ps^h_p∇^P_p,n−p⁽²⁾^A and

∇^P_p,n−p⁽²⁾^Ad^Diag_n+1^A= X

µ∈Shp,n−p

(sgnµ)s^v_d0,p−1eµs^h_dp,n−1eµd^v_n+1d^h_n+1

= d^v_n+1−pd^h_p+1 X

µ∈Shp,n−p

(sgnµ)s^v_d0,p−1eµs^h_dp,n−1eµ= d^v_n+1−pd^h_p+1∇^A_p,n−p for allp∈ bn,0c,n∈N0.

(3.32) Remark. We have

∇_p,n−pd^Diag_n ^A=







d^h_n∇_n−1,0 ifp=n,

(−1)^n−pd^h_p∇p−1,n−p+ d^v_n−p∇p,n−1−p ifp∈ bn−1,1c,

d^v_n∇_0,n−1 ifp= 0

for allp∈ bn,0c,n∈N.

Proof. According to proposition (3.30) and lemma (3.31), we have

∇^A_p,n−pd^Diag_n ^A= ((−1)^n−ps^v_n−p∇^P_p−1,n−p⁽²⁾^A + s^h_p∇^P_p,n−1−p⁽²⁾^A )d^Diag_n ^A

= (−1)^n−ps^v_n−p∇^P_p−1,n−p⁽²⁾^A d^Diag_n ^A+ s^h_p∇^P_p,n−1−p⁽²⁾^A d^Diag_n ^A

= (−1)^n−ps^v_n−pd^v_n+1−pd^h_p∇^A_p−1,n−p+ s^h_pd^v_n−pd^h_p+1∇^A_p,n−1−p

= (−1)^n−pd^h_p∇^A_p−1,n−p+ d^v_n−p∇^A_p,n−1−p

forp∈ bn−1,1c. The computation forp=nandp= 0 is analogous: We have

∇^A_n,0d^Diag_n ^A= (s^v₀∇^P_n−1,0⁽²⁾^A)d^Diag_n ^A= s^v₀d^v₁d^h_n∇^A_n−1,0= d^h_n∇^A_n−1,0 and

∇^A_0,nd^Diag_n ^A= (s^h₀∇^P_0,n−1⁽²⁾^A)d^Diag_n ^A= s^h₀d^v_nd^h₁∇^A_0,n−1= d^v_n∇^A_0,n−1. (3.33) Remark. Defining

TotnC⁽²⁾A−−→^∇ⁿ CnDiagA

byemb_p,n−p∇n:=∇_p,n−p for allp∈ bn,0c,n∈N0, we obtain a complex morphism Tot C⁽²⁾A−→^∇ C DiagA.

Proof. By the definition of the differential morphisms inTot C⁽²⁾A, we have to show that

∇^A_p,n−p∂^{C Diag}^A=







∂^C⁽²⁾^A,h∇^A_n−1,0 ifp=n,

∂^C⁽²⁾^A,h∇^A_p−1,n−p+ (−1)^p∂^C⁽²⁾^A,v∇^A_p,n−1−p ifp∈ bn−1,1c,

∂^C⁽²⁾^A,v∇^A_0,n−1 ifp= 0

for allp∈ bn,0c,n∈N0. First, we consider the boundary cases: We have

∇^A_n,0∂^{C Diag}^A= s^v_d0,n−1e X

k∈[0,n]

(−1)^kd^Diag_k ^A= s^v_d0,n−1e X

k∈[0,n]

(−1)^kd^v_kd^h_k

= X

k∈[0,n]

(−1)^ks^v_d0,n−1ed^v_kd^h_k = X

k∈[0,n]

(−1)^kd^h_ks^v_d0,n−2e=∂^C⁽²⁾^A,h∇^A_n−1,0

and analogously

∇^A_0,n∂^{C Diag}^A= s^h_d0,n−1e X

k∈[0,n]

(−1)^kd^Diag_k ^A= s^h_d0,n−1e X

k∈[0,n]

(−1)^kd^v_kd^h_k

= X

k∈[0,n]

(−1)^kd^v_ks^h_d0,n−1ed^h_k = X

k∈[0,n]

(−1)^kd^v_ks^h_d0,n−2e=∂^C⁽²⁾^A,v∇^A_0,n−1

for alln∈N. It remains to prove

∇^A_p,n−p∂^{C Diag}^A=∂^C⁽²⁾^A,h∇^A_p−1,n−p+ (−1)^p∂^C⁽²⁾^A,v∇^A_p,n−1−p forp∈ bn−1,1c,n∈N,n≥2. Thereto, it suffices to show that

∇^A_p,n−p∂^{C Diag P}⁽²⁾^A=∂^C⁽²⁾^P⁽²⁾^A,h∇^A_p−1,n−p+ (−1)^p∂^C⁽²⁾^P⁽²⁾^A,v∇^A_p,n−1−p because according to remark (3.32) this implies

∇^A_p,n−p∂^{C Diag}^A=∇^A_p,n−p(∂^{C Diag P}⁽²⁾^A+ (−1)ⁿd^Diag_n ^A) =∇^A_p,n−p∂^{C Diag P}⁽²⁾^A+ (−1)ⁿ∇^A_p,n−pd^Diag_n ^A

=∂^C⁽²⁾^P⁽²⁾^A,h∇^A_p−1,n−p+ (−1)^p∂^C⁽²⁾^P⁽²⁾^A,v∇^A_p,n−1−p+ (−1)ⁿ((−1)^n−pd^h_p∇^A_p−1,n−p+ d^v_n−p∇^A_p,n−1−p)

=∂^C⁽²⁾^P⁽²⁾^A,h∇^A_p−1,n−p+ (−1)^p∂^C⁽²⁾^P⁽²⁾^A,v∇^A_p,n−1−p+ (−1)^pd^h_p∇^A_p−1,n−p+ (−1)ⁿd^v_n−p∇^A_p,n−1−p

=∂^C⁽²⁾^A,h∇^A_p−1,n−p+ (−1)^p∂^C⁽²⁾^A,v∇^A_p,n−1−p.

We proceed by induction on n∈N, n≥2, to show the second identity involving the path bisimplicial object P⁽²⁾A and use the recursive characterisation of proposition (3.30). However, by the induction hypothesis, we may also use the first identity involvingAduring our calculations since this is implied by the second as already shown.

First, forn= 2andp= 1we compute

∇^A_1,1∂^{C Diag P}⁽²⁾^A= (s^h₁∇^P_1,0⁽²⁾^A−s^v₁∇^P_0,1⁽²⁾^A)∂^{C Diag P}⁽²⁾^A= s^h₁∇^P_1,0⁽²⁾^A∂^{C Diag P}⁽²⁾^A−s^v₁∇^P_0,1⁽²⁾^A∂^{C Diag P}⁽²⁾^A

= s^h₁∂^C⁽²⁾^P⁽²⁾^A,h∇^P_0,0⁽²⁾^A−s^v₁∂^C⁽²⁾^P⁽²⁾^A,v∇^P_0,0⁽²⁾^A= s^h₁∂^C⁽²⁾^P⁽²⁾^A,h−s^v₁∂^C⁽²⁾^P⁽²⁾^A,v

= s^h₁(d^h₀−d^h₁)−s^v₁(d^v₀−d^v₁) = d^h₀s^h₀−idA_1,1−d^v₀s^v₀+ idA_1,1 = d^h₀s^h₀−d^v₀s^v₀

=∂^C⁽²⁾^P⁽²⁾^A,h∇^A_0,1−∂^C⁽²⁾^P⁽²⁾^A,v∇^A_1,0. Next, we show the asserted formula forp=n−1,n∈N,n≥3:

∇^A_n−1,1∂^{C Diag P}⁽²⁾^A= (−s^v₁∇^P_n−2,1⁽²⁾^A+ s^h_n−1∇^P_n−1,0⁽²⁾^A)∂^{C Diag P}⁽²⁾^A

=−s^v₁∇^P_n−2,1⁽²⁾^A∂^{C Diag P}⁽²⁾^A+ s^h_n−1∇^P_n−1,0⁽²⁾^A∂^{C Diag P}⁽²⁾^A

=−s^v₁(∂^C⁽²⁾^P⁽²⁾^A,h∇^P_n−3,1⁽²⁾^A + (−1)ⁿ⁻²∂^C⁽²⁾^P⁽²⁾^A,v∇^P_n−2,0⁽²⁾^A) + s^h_n−1∂^C⁽²⁾^P⁽²⁾^A,h∇^P_n−2,0⁽²⁾^A

=−s^v₁∂^C⁽²⁾^P⁽²⁾^A,h∇^P_n−3,1⁽²⁾^A + (−1)ⁿ⁻¹s^v₁∂^C⁽²⁾^P⁽²⁾^A,v∇^P_n−2,0⁽²⁾^A + s^h_n−1∂^C⁽²⁾^P⁽²⁾^A,h∇^P_n−2,0⁽²⁾^A

=−∂^C⁽²⁾^P⁽²⁾^A,hs^v₁∇^P_n−3,1⁽²⁾^A + (−1)ⁿ⁻¹(∂^C⁽²⁾^P⁽²⁾^A,vs^v₀−id_A_n−1,1)∇^P_n−2,0⁽²⁾^A + (∂^C⁽²⁾^P⁽²⁾^A,hs^h_n−2+ (−1)ⁿ⁻¹id_A_n−1,1)∇^P_n−2,0⁽²⁾^A

=−∂^C⁽²⁾^P⁽²⁾^A,hs^v₁∇^P_n−3,1⁽²⁾^A + (−1)ⁿ⁻¹∂^C⁽²⁾^P⁽²⁾^A,vs^v₀∇^P_n−2,0⁽²⁾^A +∂^C⁽²⁾^P⁽²⁾^A,hs^h_n−2∇^P_n−2,0⁽²⁾^A

=∂^C⁽²⁾^P⁽²⁾^A,h(s^h_n−2∇^P_n−2,0⁽²⁾^A−s^v₁∇^P_n−3,1⁽²⁾^A) + (−1)ⁿ⁻¹∂^C⁽²⁾^P⁽²⁾^A,vs^v₀∇^P_n−2,0⁽²⁾^A

=∂^C⁽²⁾^P⁽²⁾^A,h∇^A_n−2,1+ (−1)ⁿ⁻¹∂^C⁽²⁾^P⁽²⁾^A,v∇^A_n−1,0. Analogously, forp= 1, n∈N,n≥3, we have

∇^A_1,n−1∂^{C Diag P}⁽²⁾^A= ((−1)ⁿ⁻¹s^v_n−1∇^P_0,n−1⁽²⁾^A+ s^h₁∇^P_1,n−2⁽²⁾^A)∂^{C Diag P}⁽²⁾^A

= (−1)ⁿ⁻¹s^v_n−1∇^P_0,n−1⁽²⁾^A∂^{C Diag P}⁽²⁾^A+ s^h₁∇^P_1,n−2⁽²⁾^A∂^{C Diag P}⁽²⁾^A

= (−1)ⁿ⁻¹s^v_n−1∂^C⁽²⁾^P⁽²⁾^A,v∇^P_0,n−2⁽²⁾^A+ s^h₁(∂^C⁽²⁾^P⁽²⁾^A,h∇^P_0,n−2⁽²⁾^A−∂^C⁽²⁾^P⁽²⁾^A,v∇^P_1,n−3⁽²⁾^A)

= (−1)ⁿ⁻¹s^v_n−1∂^C⁽²⁾^P⁽²⁾^A,v∇^P_0,n−2⁽²⁾^A+ s^h₁∂^C⁽²⁾^P⁽²⁾^A,h∇^P_0,n−2⁽²⁾^A−s^h₁∂^C⁽²⁾^P⁽²⁾^A,v∇^P_1,n−3⁽²⁾^A

= (−1)ⁿ⁻¹(∂^C⁽²⁾^P⁽²⁾^A,vs^v_n−2+ (−1)ⁿ⁻¹idA1,n−1)∇^P_0,n−2⁽²⁾^A

+ (∂^C⁽²⁾^P⁽²⁾^A,hs^h₀−idA1,n−1)∇^P_0,n−2⁽²⁾^A −∂^C⁽²⁾^P⁽²⁾^A,vs^h₁∇^P_1,n−3⁽²⁾^A

= (−1)ⁿ⁻¹∂^C⁽²⁾^P⁽²⁾^A,vs^v_n−2∇^P_0,n−2⁽²⁾^A+∂^C⁽²⁾^P⁽²⁾^A,hs^h₀∇^P_0,n−2⁽²⁾^A−∂^C⁽²⁾^P⁽²⁾^A,vs^h₁∇^P_1,n−3⁽²⁾^A

=∂^C⁽²⁾^P⁽²⁾^A,hs^h₀∇^P_0,n−2⁽²⁾^A−((−1)ⁿ⁻²∂^C⁽²⁾^P⁽²⁾^A,vs^v_n−2∇^P_0,n−2⁽²⁾^A+∂^C⁽²⁾^P⁽²⁾^A,vs^h₁∇^P_1,n−3⁽²⁾^A)

=∂^C⁽²⁾^P⁽²⁾^A,hs^h₀∇^P_0,n−2⁽²⁾^A−∂^C⁽²⁾^P⁽²⁾^A,v((−1)ⁿ⁻²s^v_n−2∇^P_0,n−2⁽²⁾^A+ s^h₁∇^P_1,n−3⁽²⁾^A)

=∂^C⁽²⁾^P⁽²⁾^A,h∇^A_0,n−1−∂^C⁽²⁾^P⁽²⁾^A,v∇^A_1,n−2. Finally, we letp∈[n−2,2],n∈N, n≥4. Then we get

∇^A_p,n−p∂^{C Diag P}⁽²⁾^A= ((−1)^n−ps^v_n−p∇^P_p−1,n−p⁽²⁾^A + s^h_p∇^P_p,n−1−p⁽²⁾^A )∂^{C Diag P}⁽²⁾^A

= (−1)^n−ps^v_n−p∇^P_p−1,n−p⁽²⁾^A ∂^{C Diag P}⁽²⁾^A+ s^h_p∇^P_p,n−1−p⁽²⁾^A ∂^{C Diag P}⁽²⁾^A

= (−1)^n−ps^v_n−p(∂^C⁽²⁾^P⁽²⁾^A,h∇^P_p−2,n−p⁽²⁾^A + (−1)^p−1∂^C⁽²⁾^P⁽²⁾^A,v∇^P_{p−1,n−1−p}⁽²⁾^A ) + s^h_p(∂^C⁽²⁾^P⁽²⁾^A,h∇^P_{p−1,n−1−p}⁽²⁾^A + (−1)^p∂^C⁽²⁾^P⁽²⁾^A,v∇^P_p,n−2−p⁽²⁾^A )

= (−1)^n−ps^v_n−p∂^C⁽²⁾^P⁽²⁾^A,h∇^P_p−2,n−p⁽²⁾^A + (−1)ⁿ⁻¹s^v_n−p∂^C⁽²⁾^P⁽²⁾^A,v∇^P_{p−1,n−1−p}⁽²⁾^A + s^h_p∂^C⁽²⁾^P⁽²⁾^A,h∇^P_{p−1,n−1−p}⁽²⁾^A + (−1)^ps^h_p∂^C⁽²⁾^P⁽²⁾^A,v∇^P_p,n−2−p⁽²⁾^A

= (−1)^n−p∂^C⁽²⁾^P⁽²⁾^A,hs^v_n−p∇^P_p−2,n−p⁽²⁾^A

+ (−1)ⁿ⁻¹(∂^C⁽²⁾^P⁽²⁾^A,vs^v_n−1−p+ (−1)^n−pid_A_p,n−p)∇^P_{p−1,n−1−p}⁽²⁾^A

+ (∂^C⁽²⁾^P⁽²⁾^A,hs^h_p−1+ (−1)^pid_A_p,n−p)∇^P_{p−1,n−1−p}⁽²⁾^A + (−1)^p∂^C⁽²⁾^P⁽²⁾^A,vs^h_p∇^P_p,n−2−p⁽²⁾^A

= (−1)^n−p∂^C⁽²⁾^P⁽²⁾^A,hs^v_n−p∇^P_p−2,n−p⁽²⁾^A + (−1)ⁿ⁻¹∂^C⁽²⁾^P⁽²⁾^A,vs^v_n−1−p∇^P_{p−1,n−1−p}⁽²⁾^A +∂^C⁽²⁾^P⁽²⁾^A,hs^h_p−1∇^P_{p−1,n−1−p}⁽²⁾^A + (−1)^p∂^C⁽²⁾^P⁽²⁾^A,vs^h_p∇^P_p,n−2−p⁽²⁾^A

=∂^C⁽²⁾^P⁽²⁾^A,h((−1)^n−ps^v_n−p∇^P_p−2,n−p⁽²⁾^A + s^h_p−1∇^P_{p−1,n−1−p}⁽²⁾^A )

+ (−1)^p∂^C⁽²⁾^P⁽²⁾^A,v((−1)^n−p−1s^v_n−1−p∇^P_{p−1,n−1−p}⁽²⁾^A + s^h_p∇^P_p,n−2−p⁽²⁾^A )

=∂^C⁽²⁾^P⁽²⁾^A,h∇^A_p−1,n−p+ (−1)^p∂^C⁽²⁾^P⁽²⁾^A,v∇^A_p,n−1−p

By induction, we have shown that the morphisms∇n forn∈N0 yield a complex morphism Tot C⁽²⁾A−→^∇ C DiagA.

(3.34) Definition (shuffle morphism). The complex morphism Tot C⁽²⁾A−→^∇ C DiagA

given as in remark (3.33) byembp,n−p∇n=∇p,n−pfor allp∈ bn,0c, that is,

∇n=







∇n,0

...

∇_0,n







as a morphism fromTotnC⁽²⁾A=L

p∈bn,0cAp,n−ptoCnDiagA=An,nfor alln∈N0, is called (Eilenberg-Mac Lane)shuffle morphism.

At next, we will show that the Alexander-Whitney morphism and the Eilenberg-Mac Lane shuffle morphism restrict to well-defined morphisms onM DiagAresp.Tot M⁽²⁾A.

(3.35) Proposition.

(a) We have a morphism of split short exact sequences D DiagA //

C DiagA //

M DiagA

Tot D⁽²⁾A //Tot C⁽²⁾A //Tot M⁽²⁾A

By abuse of notation, the induced morphism M DiagA−→Tot M⁽²⁾Ais also denoted byAW:=AW^A. (b) We have a morphism of split short exact sequences

Tot D⁽²⁾A //

Tot C⁽²⁾A //

∇

Tot M⁽²⁾A D DiagA //C DiagA //M DiagA

By abuse of notation, the induced morphism Tot M⁽²⁾A−→M DiagAis also denoted by∇:=∇^A. Proof.

(a) We have

s^Diag_k ^AAWp,n−p= s^h_ks^v_kd^h_bn,p+1cd^v_bp−1,0c= s^h_kd^h_bn,p+1cs^v_kd^v_bp−1,0c=

((s^h_kd^h_bn,p+1cd^v_bp−1,0c)s^v_k−p ifp≤k, (d^h_bn−1,pcs^v_kd^v_bp−1,0c)s^h_k ifp > k,

for every k∈[0, n−1], that is,(Im s^Diag_k ^A)AW_p,n−p= Im(s^Diag_k ^AAW_p,n−p)D⁽²⁾_p,n−pAfor allk∈[0, n−1]

and therefore(DnDiagA)AWp,n−pD⁽²⁾_p,n−pAfor allp∈ bn,0c. Hence we have an induced morphism D DiagA−→Tot D⁽²⁾A.

Moreover, M DiagA ∼= C DiagA/D DiagA and Tot M⁽²⁾A ∼= Tot(C⁽²⁾A/D⁽²⁾A) ∼= Tot C⁽²⁾A/Tot D⁽²⁾A by the normalisation theorem (2.28). Hence we have an induced morphism on the cokernels

M DiagA−→^AW Tot M⁽²⁾A.

(b) We will show that

(D⁽²⁾_p,n−pA)∇^A_p,n−pDnDiagA

for allp∈ bn,0c, n∈N0. Thereto, we proceed by induction onn, where for n= 0the assertion is trivial since D⁽²⁾_0,0A ∼= 0and D₀DiagA ∼= 0. So we let a natural number n∈ N with n ≥1 and p ∈ bn,0c be given and we assume that the asserted inclusion holds for all bisimplicial sets up to dimension n−1. By proposition (3.30), we compute

s^h_i∇^A_p,n−p=

(s^h_is^v₀∇^P_n−1,0⁽²⁾^A ifp=n, s^h_i((−1)^n−ps^v_n−p∇^P_p−1,n−p⁽²⁾^A + s^h_p∇^P_p,n−1−p⁽²⁾^A ) ifp∈ bn−1,1c

)

(s^h_is^v₀∇^P_n−1,0⁽²⁾^A ifp=n, (−1)^n−ps^h_is^v_n−p∇^P_p−1,n−p⁽²⁾^A + s^h_is^h_p∇^P_p,n−1−p⁽²⁾^A ifp∈ bn−1,1c

)

(s^v₀s^h_i∇^P_n−1,0⁽²⁾^A ifp=n,

(−1)^n−ps^v_n−ps^h_i∇^P_p−1,n−p⁽²⁾^A + s^h_p−1s^h_i∇^P_p,n−1−p⁽²⁾^A ifp∈ bn−1,1c fori∈[0, p−1],p∈ bn,1c, and therefore

Im(s^h_i∇^A_n,0)Im(s^v₀s^h_i∇^P_n−1,0⁽²⁾^A) fori∈[0, n−1]and

Im(s^h_i∇^A_p,n−p)Im((−1)^n−ps^v_n−ps^h_i∇^P_p−1,n−p⁽²⁾^A + s^h_p−1s^h_i∇^P_p,n−1−p⁽²⁾^A ) Im((−1)^n−ps^v_n−ps^h_i∇^P_p−1,n−p⁽²⁾^A ) + Im(s^h_p−1s^h_i∇^P_p,n−1−p⁽²⁾^A ) Im(s^v_n−ps^h_i∇^P_p−1,n−p⁽²⁾^A ) + Im(s^h_i∇^P_p,n−1−p⁽²⁾^A )

fori∈[0, p−1],p∈ bn−1,1c. Now by the induction hypothesis, we have

Im(s^h_i∇^P_p,n−1−p⁽²⁾^A )(D⁽²⁾_p,n−1−pP⁽²⁾A)∇^P_p,n−1−p⁽²⁾^A Dn−1Diag P⁽²⁾ADnDiagA fori∈[0, p−1],p∈ bn−1,1c, and

Im(s^v_n−ps^h_i∇^P_p−1,n−p⁽²⁾^A )Im(s^h_i∇^P_p−1,n−p⁽²⁾^A )(D⁽²⁾_p−1,n−pP⁽²⁾A)∇^P_p−1,n−p⁽²⁾^A D_n−1Diag P⁽²⁾A DnDiagA

fori∈[0, p−2],p∈ bn,1c. Since additionally, by lemma (3.31),

Im(s^v_n−ps^h_p−1∇^P_p−1,n−p⁽²⁾^A ) = Im(∇^A_p−1,n−ps^Diag_n−1^A)Im(s^Diag_n−1^A)DnDiagA

forp∈ bn,1c, we can conclude that Im(s^h_i∇^A_p,n−p)D_nDiagA fori∈[0, p−1],p∈ bn,1c.

Analogously, we show Im(s^v_j∇^A_p,n−p) DnDiagA for all j ∈ [0, n−1−p], p ∈ bn−1,0c. Indeed, by proposition (3.30), we have

s^v_j∇^A_p,n−p=

(s^v_j((−1)^n−ps^v_n−p∇^P_p−1,n−p⁽²⁾^A + s^h_p∇^P_p,n−1−p⁽²⁾^A ) ifp∈ bn−1,1c, s^v_js^h₀∇^P_0,n−1⁽²⁾^A ifp= 0

)

((−1)^n−ps^v_js^v_n−p∇^P_p−1,n−p⁽²⁾^A + s^v_js^h_p∇^P_p,n−1−p⁽²⁾^A ifp∈ bn−1,1c, s^v_js^h₀∇^P_0,n−1⁽²⁾^A ifp= 0

)

((−1)^n−ps^v_n−1−ps^v_j∇^P_p−1,n−p⁽²⁾^A + s^h_ps^v_j∇^P_p,n−1−p⁽²⁾^A ifp∈ bn−1,1c, s^h₀s^v_j∇^P_0,n−1⁽²⁾^A ifp= 0

forj∈[0, n−1−p], p∈ bn−1,0c, and therefore

Im(s^v_j∇^A_p,n−p)Im((−1)^n−ps^v_n−1−ps^v_j∇^P_p−1,n−p⁽²⁾^A + s^h_ps^v_j∇^P_p,n−1−p⁽²⁾^A ) Im((−1)^n−ps^v_n−1−ps^v_j∇^P_p−1,n−p⁽²⁾^A ) + Im(s^h_ps^v_j∇^P_p,n−1−p⁽²⁾^A ) Im(s^v_j∇^P_p−1,n−p⁽²⁾^A ) + Im(s^h_ps^v_j∇^P_p,n−1−p⁽²⁾^A )

forj∈[0, n−1−p], p∈ bn−1,1c, and Im(s^v_j∇^A_0,n)Im(s^h₀s^v_j∇^P_0,n−1⁽²⁾^A)

forj∈[0, n−1]. With the induction hypothesis, it follows that

Im(s^v_j∇^P_p−1,n−p⁽²⁾^A )(D⁽²⁾_p−1,n−pP⁽²⁾A)∇^P_p−1,n−p⁽²⁾^A D_n−1Diag P⁽²⁾AD_nDiagA forj∈[0, n−1−p], p∈ bn−1,1cand

Im(s^h_ps^v_j∇^P_p,n−1−p⁽²⁾^A )Im(s^v_j∇^P_p,n−1−p⁽²⁾^A )(D⁽²⁾_p,n−1−pP⁽²⁾A)∇^P_p,n−1−p⁽²⁾^A D_n−1Diag P⁽²⁾ADnDiagA forj∈[0, n−2−p], p∈ bn−1,0c. Since additionally, by lemma (3.31),

Im(s^h_ps^v_n−1−p∇^P_p,n−1−p⁽²⁾^A ) = Im(∇^A_p,n−1−ps^Diag_n−1^A)Im(s^Diag_n−1^A)D_nDiagA

forp∈ bn−1,0c. HenceIm(s^v_j∇^A_p,n−p)D_nDiagAforj∈[0, n−1−p],p∈ bn−1,0c.

Therefore

(D⁽²⁾_p,n−pA)∇^A_p,n−pDnDiagA.

So we have induced morphisms Tot D⁽²⁾A−→D DiagA and

Tot M⁽²⁾A ^∇

−−→M DiagA.

(3.36) Theorem (generalised Eilenberg-Zilber theorem, normalised case). The Alexander-Whitney morphism CMA−→^AW Tot M⁽²⁾A

and the Eilenberg-Mac Lane shuffle morphism Tot M⁽²⁾A−→^∇ M DiagA

are mutually inverse homotopy equivalences. In particular, M DiagA'Tot M⁽²⁾A.

Proof. First, we want to show that ∇^AAW^A = id_{Tot M}(2)A. We let n∈N0 be given. For each p, q ∈ bn,0cwe have

∇^A_p,n−pAW^A_q,n−q= ( X

µ∈Shp,n−p

(sgnµ)s^v_d0,p−1eµs^h_dp,n−1eµ)d^h_bn,q+1cd^v_bq−1,0c

= X

µ∈Shp,n−p

(sgnµ)(s^v_d0,p−1eµd^v_bq−1,0c)(s^h_dp,n−1eµd^h_bn,q+1c).

By applying the simplicial identities, we recognise that each summand ends with a vertical degeneracy if q < p resp. with a horizontal degeneracy if q > p. Since we are in the normalised case, this means that

∇^A_p,n−pAW^A_q,n−q = 0forp6=q. It remains to consider the caseq=p. Then we have

since the only summand that is not trivial because it ends with a degeneracy, is the one whereµ = id_[0,n−1]. Thus

We have to show that these morphisms induce morphisms on the entries of the normalised complexM DiagA.

Thereto, we prove that they restrict to morphisms DnDiagA ^h

For the first summand, we get by induction that Im(s^Diag_k ^Ah^P_n−1⁽²⁾^A)DnDiag P⁽²⁾ADn+1DiagA

ifk ≤n−2. We consider the case k =n−1. By definition and proposition (2.33)(c), h^A_n is a certain linear combination of morphisms, each one being a composite of a horizonal backal and a vertical backal morphism.

Thus, by proposition (2.33)(a) applied vertically and horizontally, we have s^Diag_n−1^Ah^P_n−1⁽²⁾^A = h^A_n−1s^Diag_n ^A and hence

Im(s^Diag_n−1^Ah^P_n−1⁽²⁾^A)Dn+1DiagA.

Altogether,

Im(s^Diag_k ^Ah^A_n)D_n+1DiagA.

It remains to show that(h_n∈_A(M_nDiagA,M_n+1DiagA)|n∈N0)is a complex homotopy from id_{M Diag}_A to f^A, that is,

h^A_n∂^{C Diag}^A+∂^{C Diag}^Ah^A_n−1= id_C_n_Diag_A−f_n^A for alln∈N0

up to sums of morphisms whose images are in the degenerate complex. We proceed by induction on n∈N0. Forn= 0, we have

h^A₀∂^{C Diag}^A= s^Diag₀ ^Af₀^A(d^Diag₀ ^A−d^Diag₁ ^A) = s^Diag₀ ^Ad^Diag₀ ^A−s^Diag₀ ^Ad^Diag₁ ^A= id_C_n_Diag_A−id_C_n_Diag_A

= idC_nDiagA−f_n^A.

Now we assume thatn≥1and that the assumpted relation holds in all lower dimensions. Then we have h^A_n∂^{C Diag P}⁽²⁾^A= (h^P_n−1⁽²⁾^A+ (−1)ⁿs^Diag_n ^Af_n^P⁽²⁾^A)∂^{C Diag P}⁽²⁾^A

=h^P_n−1⁽²⁾^A∂^{C Diag P}⁽²⁾^A+ (−1)ⁿs^Diag_n ^Af_n^P⁽²⁾^A∂^{C Diag P}⁽²⁾^A

=h^P_n−1⁽²⁾^A∂^{C Diag P}⁽²⁾^A+ (−1)ⁿs^Diag_n ^A∂^{C Diag P}⁽²⁾^Af_n−1^P⁽²⁾^A

= (idC_nDiagA−f_n−1^P⁽²⁾^A−∂^{C Diag P}⁽²⁾^Ah^P_n−2⁽²⁾^A) + (−1)ⁿ(∂^{C Diag P}⁽²⁾^As^Diag_n−1^A+ (−1)ⁿid)f_n−1^P⁽²⁾^A

= idC_nDiagA−f_n−1^P⁽²⁾^A−∂^{C Diag P}⁽²⁾^Ah^P_n−2⁽²⁾^A+ (−1)ⁿ∂^{C Diag P}⁽²⁾^As^Diag_n−1^Af_n−1^P⁽²⁾^A+f_n−1^P⁽²⁾^A

= id_C_n_Diag_A−∂^{C Diag P}⁽²⁾^Ah^P_n−2⁽²⁾^A+ (−1)ⁿ∂^{C Diag P}⁽²⁾^As^Diag_n−1^Af_n−1^P⁽²⁾^A

= idC_nDiagA−∂^{C Diag P}⁽²⁾^A(h^P_n−2⁽²⁾^A+ (−1)ⁿ⁻¹s^Diag_n−1^Af_n−1^P⁽²⁾^A) = idC_nDiagA−∂^{C Diag P}⁽²⁾^Ah^A_n−1 as well as, by proposition (2.31)(c),

h^A_nd^Diag_n+1^A= (h^P_n−1⁽²⁾^A+ (−1)ⁿs^Diag_n ^Af_n^P⁽²⁾^A)d^Diag_n+1^A=h^P_n−1⁽²⁾^Ad^Diag_n+1^A+ (−1)ⁿs^Diag_n ^Af_n^P⁽²⁾^Ad^Diag_n+1^A

= d^Diag_n ^Ah^A_n−1+ (−1)ⁿs^Diag_n ^Ad^Diag_n+1^Af_n^A= d^Diag_n ^Ah^A_n−1+ (−1)ⁿf_n^A. Hence we can conclude

h^A_n∂^{C Diag}^A=h^A_n(∂^{C Diag P}⁽²⁾^A+ (−1)ⁿ⁺¹d^Diag_n+1^A) =h^A_n∂^{C Diag P}⁽²⁾^A+ (−1)ⁿ⁺¹h^A_nd^Diag_n+1^A

= (id_C_n_Diag_A−∂^{C Diag P}⁽²⁾^Ah^A_n−1) + (−1)ⁿ⁺¹(d^Diag_n ^Ah^A_n−1+ (−1)ⁿf_n^A)

= idC_nDiagA−∂^{C Diag P}⁽²⁾^Ah^A_n−1+ (−1)ⁿ⁺¹d^Diag_n ^Ah^A_n−1−f_n^A

=−∂^{C Diag P}⁽²⁾^Ah^A_n−1−(−1)ⁿd^Diag_n ^Ah^A_n−1+ id_C_n_Diag_A−f_n^A

=−(∂^{C Diag P}⁽²⁾^A+ (−1)ⁿd^Diag_n ^A)h^A_n−1+ idC_nDiagA−f_n^A

=−∂^{C Diag}^Ah^A_n−1+ idC_nDiagA−f_n^A, that is,h^A_n∂^{C Diag}^A+∂^{C Diag}^Ah^A_n−1= id_C_n_Diag_A−f_n^A.

(3.37) Theorem (generalised Eilenberg-Zilber theorem ofDold,PuppeandCartier, cf. [9, Satz 2.9]). We have

C DiagA'Tot C⁽²⁾A.

Proof. By theorem (3.36), we have M DiagA'Tot M⁽²⁾A.

Since the normalisation theorem states a homotopy equivalence between the associated (double) complexes and the Moore (double) complexes, cf. theorem (2.28) and theorem (3.24), and since the total complex functor preserves homotopy equivalences due to proposition (3.19), this implies by theorem (3.36) that

C DiagA'M DiagA'Tot M⁽²⁾A'Tot C⁽²⁾A.

Quillenmentions the following corollary in [28] as well-known.

(3.38) Corollary. There exists a spectral sequence E with E_p,n−p¹ ∼= Hn−p(CA_p,−) that converges to the homology groupHn(C DiagA), wherep∈[0, n],n∈N0.

Proof. By the generalised Eilenberg-Zilber theorem (3.37), we haveC DiagA'Tot C⁽²⁾A and hence Hn(C DiagA)∼= Hn(Tot C⁽²⁾A)

for alln∈N0. The spectral sequenceE of the “columnwise” filtered double complexC⁽²⁾Ahas the entries E_p,n−p¹ ∼= Hn−p(C⁽²⁾_p,−A) = Hn−p(CAp,−)

forp∈[0, n],n∈N0.

(3.39) Corollary. We suppose given a bisimplicial setX, a commutative ringRand anR-moduleM. (a) There exists a spectral sequence E with E_p,n−p¹ ∼= Hn−p(X_p,−, M;R) that converges to the homology

group Hn(DiagX, M;R), wherep∈[0, n],n∈N⁰.

(b) There exists a spectral sequence E with E₁^p,n−p ∼= H^n−p(X_p,−, M;R)that converges to the cohomology group Hⁿ(DiagX, M;R), wherep∈[0, n],n∈N0.

Proof.

(a) We apply corollary (3.38) toRX⊗RM. Then we obtain

Hn(C Diag(RX⊗RM)) = Hn(C((DiagRX)⊗RM)) = Hn((C DiagRX)⊗RM)

= Hn((CRDiagX)⊗RM) = Hn(C(DiagX;R)⊗RM)

= H_n(DiagX, M;R) forn∈N0, and

H_n−p(C(RX⊗_RM)_p,−) = H_n−p(C(RX_p,−⊗_RM) = H_n−p((CRX_p,−)⊗_RM)

= Hn−p(C(Xp,−;R)⊗RM) = Hn−p(Xp,−, M;R) forp∈[0, n], n∈N0.

(b) By the generalised Eilenberg-Zilber theorem (3.37), we have

C(DiagX;R) = CRDiagX = C DiagRX'Tot C⁽²⁾RX= Tot C⁽²⁾(X;R) and hence

Hⁿ(DiagX, M;R) = Hⁿ(R(C(DiagX;R), M))∼= Hⁿ(R(Tot C⁽²⁾(X;R), M))

= Hⁿ(Tot_R(C⁽²⁾(X;R), M))

forn∈N0. The spectral sequence of the “columnwise” filtered double complexR(C⁽²⁾(X;R), M)has E₁^p,n−p∼= H^n−p(_R(C⁽²⁾_p,−(X;R), M)) = H^n−p(_R(C(X_p,−;R), M)) = H^n−p(X_p,−, M;R)

forp∈[0, n], n∈N0. (¹)

1The seeming non-duality in the proofs of (a) and (b) is due to the fact that cohomology ofcosimplicial objects has not been defined.

Chapter IV

Im Dokument (Co)homology of crossed modules (Seite 60-71)