Zoran Skoda
tensorF

[(Δ Rid)F](F1)ϕ(1(r)1) = (F (1) 1 R(F (2) 1F 1r)F2 = β(F (2) 1(F 2r))F (1) 1F 2 = β(F (2) 1(F 2r))F (1) 1F 1 RF 2 = F 1β(F 2r)F 1 RF 2\array{ [ (\Delta \otimes_R id) F ] (F\otimes 1) \phi(1\otimes (\triangleright r) \otimes 1) & = & (F^1_{(1)}\otimes_R (F^1_{(2)}F'^1\triangleright r)\otimes F'2 \\ &=& \beta(F^1_{(2)}\triangleright(F'^2\triangleright r))F^1_{(1)} F'^2 \\ &=& \beta(F^1_{(2)}\triangleright (F'^2\triangleright r))F^1_{(1)}F'^1\otimes_R F^2 \\ &=& F^1\beta(F'^2\triangleright r)F'^1\otimes_R F^2 }

where we have used gβ(b)=β(g (2)b)g (1)g\beta(b) = \beta(g_{(2)}\triangleright b) g_{(1)} for g=F 1g = F^1. The identity

[(Δ Rid)F](F1)=[(id RΔ)F](1F)ϕ [ (\Delta \otimes_R id) F ] (F\otimes 1) = [ (id \otimes_R \Delta) F ] (1\otimes F) \phi

shows that the same can be calculated as

[(id RΔ)F](1F)ϕ(1(r)1) = F 1 R(F (1) 2F 1 RF (2) 2F 2)ϕ(1(r)1) = F 1ϕ 1 Rα(F (1) 2(F 1ϕ 2a))F (2) 2F 2 = F 1ϕ 1F 2α(F 1ϕ 2r)F 2ϕ 3\array{ [ (id \otimes_R \Delta) F ] (1\otimes F) \phi(1\otimes (\triangleright r) \otimes 1) & = & F^1 \otimes_R (F^2_{(1)} F'^{1}\otimes_R F^2_{(2)} F'^2) \phi (1 \otimes (\triangleright r)\otimes 1) \\ &=& F^1 \phi^1 \otimes_R \alpha (F^2_{(1)}\triangleright (F'^1 \phi^2 \triangleright a)) F^2_{(2)} F'^2 \\ &=& F^1\phi^1\otimes F^2\alpha(F'^1\phi^2\triangleright r)F'_2\phi^3 }

where we have used gα(b)=α(g (1)b)g (2)g\alpha(b) = \alpha(g_{(1)}\triangleright b)g_{(2)} for g=F 2g = F^2.

We conclude that

F 1β(F 2r)F 1 RF 2=F 1ϕ 1 RF 2α(F 1ϕ 2r)F 2ϕ 3 F^1\beta(F'^2\triangleright r) F'^1\otimes_R F^2 = F^1 \phi^1\otimes_R F^2\alpha(F'^1\phi^2\triangleright r)F'^2\phi^3

for all rr. In different terms,

FI FI F I_F \subseteq I

where I=Ker(HHH RH)I = Ker(H\otimes H\to H\otimes_R H) and I F=Ker(HHH FH)I_F = Ker(H\otimes H\to H\otimes_F H) where H FHH\otimes_F H is the coequalizer of

(HR)Hϕ˜H(RH)HαHH(HH)Hμ HHH (H\otimes R)\otimes H \stackrel{\tilde\phi}\longrightarrow H\otimes(R\otimes H) \stackrel{H\otimes\alpha\otimes H}\longrightarrow H\otimes(H\otimes H)\stackrel{H\otimes \mu_H} \longrightarrow H\otimes H

and

(HR)Hτ R,HH(RH)H(βH)H(HH)Hμ HHHH (H\otimes R)\otimes H \stackrel{\tau_{R,H}\otimes H}\longrightarrow (R\otimes H)\otimes H \stackrel{(\beta\otimes H)\otimes H}\longrightarrow (H\otimes H)\otimes H\stackrel{\mu_H\otimes H} \longrightarrow H\otimes H

Created on October 20, 2019 at 15:58:53. See the history of this page for a list of all contributions to it.