10_cohomology.tex

Paper X: Universal Residual Seeds and Conditional Occupied-System Completion

10_cohomology.tex

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]
2 \path[use as bounding box] (0,-.6) rectangle (14,4.8);
3 \node[font=\small,align=center,font=\small] at (2.06,4.65) {$d_0:C^0\to C^1$};
4 \path[draw=black!30,fill=black!8] (1.4,3.29) rectangle (2.06,3.95);
5 \node[font=\small,align=center,font=\small] at (1.73,3.62) {$1$};
6 \path[draw=black!30,fill=white] (2.06,3.29) rectangle (2.72,3.95);
7 \node[font=\small,align=center,font=\small] at (2.39,3.62) {$0$};
8 \path[draw=black!30,fill=white] (1.4,2.63) rectangle (2.06,3.29);
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19 \node[font=\small,align=center,font=\small] at (2.39,1.64) {$0$};
20 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (1.25,3.62) {$u_0$};
21 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (1.25,2.96) {$u_1$};
22 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (1.25,2.3) {$u_2$};
23 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (1.25,1.64) {$u_3$};
24 \node[font=\small,align=center,font=\scriptsize] at (1.73,4.18) {$e_0$};
25 \node[font=\small,align=center,font=\scriptsize] at (2.39,4.18) {$e_1$};
26 \node[font=\small,align=center,font=\small] at (6.48,4) {$d_1:C^1\to C^2$};
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43 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (4.95,2.955) {$f_0$};
44 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (4.95,2.265) {$f_1$};
45 \node[font=\small,align=center,font=\scriptsize] at (5.445,3.53) {$u_0$};
46 \node[font=\small,align=center,font=\scriptsize] at (6.135,3.53) {$u_1$};
47 \node[font=\small,align=center,font=\scriptsize] at (6.825,3.53) {$u_2$};
48 \node[font=\small,align=center,font=\scriptsize] at (7.515,3.53) {$u_3$};
49 \node[font=\small,align=center] at (11,3.7) {$d_1d_0=0$};
50 \node[font=\small,align=center] at (11,2.65) {$H^0=\langle[e_1]\rangle$};
51 \node[font=\small,align=center] at (11,1.85) {$H^1=\langle[u_2],[u_3]\rangle$};
52 \node[font=\small,align=center] at (11,1.05) {$H^2=\langle[f_1]\rangle$};
53 \node[font=\small,align=center,font=\footnotesize] at (6.8,0.2) {Packet A is exact. Packet B has zero differentials and carries all cohomology.};
54 \end{tikzpicture}

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