Manuscript IX-C@1 · IX-C.tex · section · 03ce8e6b

Actual-functional descent of the second stage

Actual-functional descent of the second stage · lines 801–824

IX-C@1#eq:leftpresent, IX-C@1#eq:rightpresent

All tensor products in this section are algebraic tensor products over $C$ . No completion or quotient topology is silently introduced.

Let

E_{L} = T \otimes T, E_{R} = T \otimes T .

Define presentation maps

p_{L} (f \otimes g) p_{R} (g \otimes f^{'}) = B_{f} W_{g}, = W_{g} B_{f^{'}}, q_{L} (f \otimes g) q_{R} (g \otimes f^{'}) = Q_{f, g}, = Q^{R}_{g, f^{'}} .

Their images are the corresponding actual-functional spaces. Their kernels contain all finite linear relations between the displayed presentations.

Displayed formulas in this passage

IX-C@1#eq:leftpresent · line 813
$p_{L} (f \otimes g) p_{R} (g \otimes f^{'}) = B_{f} W_{g}, = W_{g} B_{f^{'}}, q_{L} (f \otimes g) q_{R} (g \otimes f^{'}) = Q_{f, g}, = Q^{R}_{g, f^{'}} .$
IX-C@1#eq:rightpresent · line 813
$p_{L} (f \otimes g) p_{R} (g \otimes f^{'}) = B_{f} W_{g}, = W_{g} B_{f^{'}}, q_{L} (f \otimes g) q_{R} (g \otimes f^{'}) = Q_{f, g}, = Q^{R}_{g, f^{'}} .$
unlabeled · line 808
$E_{L} = T \otimes T, E_{R} = T \otimes T .$