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The Blueprint of the Empire State Building
Shilin Yu
Department of Mathematics
University of Pennsylvania
MATH 628 Homological Algebra
Oct 1, 2014
Shilin Yu The Blueprint of the Empire State Building
psulogo
The Empire State Building in New York
Shilin Yu The Blueprint of the Empire State Building
psulogo
The Empire State Building in Homological Algebra
P′• P• P′′•
Q′• Q• Q′′•
A′ A A′′
B′ B B′′
iP′•
F′•
ε′ ε
F•
ε′′
F′′•
iQ′•
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
Shilin Yu The Blueprint of the Empire State Building
psulogo
The Empire State Building made of projective resolutions
· · · · · · · · ·
· · · · · · · · ·
P′1 P1 P′′1
Q′1 Q1 Q′′1
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
F′1
d′
d
F1
d′′
F′′1
d′′iP′0
F′0
ε′ ε
F0
ε′′
F′′0
d′ d
η′′iA
f ′ f
πA
f ′′iB
η′
πB
η
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct the ’beam’ F0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
ε′′
F′′0
iQ′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
Want the red square in the middle to be commutive, i.e., η ◦ F0 − f ◦ε = 0.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
ε′′
F′′0
iQ′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
Check on the subobject P′0 ⊂ P0, i.e., (η ◦ F0 − f ◦ε) ◦ iP′0= 0.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
ε′′
F′′0
iQ′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ F0 ◦ iP′0.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
ε′′
F′′0
iQ′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ F0 ◦ iP′0= η ◦ iQ′0
◦ F′0.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
ε′′
F′′0
iQ′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ F0 ◦ iP′0= η ◦ iQ′0
◦ F′0 = iB ◦ η′ ◦ F′0.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
ε′′
F′′0
iQ′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ F0 ◦ iP′0= η ◦ iQ′0
◦ F′0 = iB ◦ η′ ◦ F′0 = iB ◦ f ′ ◦ε′.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
ε′′
F′′0
iQ′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ F0 ◦ iP′0= η ◦ iQ′0
◦ F′0 = iB ◦ η′ ◦ F′0 = iB ◦ f ′ ◦ε′ = f ◦ iA ◦ε′.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
ε′′
F′′0
iQ′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ F0 ◦ iP′0= η ◦ iQ′0
◦ F′0 = iB ◦ η′ ◦ F′0 = iB ◦ f ′ ◦ε′ = f ◦ iA ◦ε′ = f ◦ε ◦ iP′0.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
ε′′
F′′0
iQ′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ F0 ◦ iP′0= η ◦ iQ′0
◦ F′0 = iB ◦ η′ ◦ F′0 = iB ◦ f ′ ◦ε′ = f ◦ iA ◦ε′ = f ◦ε ◦ iP′0.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
ε′′
F′′0
iQ′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ F0 ◦ iP′0= η ◦ iQ′0
◦ F′0 = iB ◦ η′ ◦ F′0 = iB ◦ f ′ ◦ε′ = f ◦ iA ◦ε′ = f ◦ε ◦ iP′0.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
ε′′
F′′0
iQ′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ F0 ◦ iP′0= η ◦ iQ′0
◦ F′0 = iB ◦ η′ ◦ F′0 = iB ◦ f ′ ◦ε′ = f ◦ iA ◦ε′ = f ◦ε ◦ iP′0.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
iP′′0
ε′′
F′′0
iQ′0iQ′′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
Claim: f ◦ε ◦ i′′P0− η ◦ iQ′′0
◦ F′′0 factors through iB.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
iP′′0
ε′′
F′′0
iQ′0iQ′′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
Check: f ◦ε ◦ i′′P0− η ◦ iQ′′0
◦ F′′0 factors through iB, i.e., its composition with
πB is zero.
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
iP′′0
ε′′
F′′0
iQ′0iQ′′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ iQ′′0◦ F′′0
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
iP′′0
ε′′
F′′0
iQ′0iQ′′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ iQ′′0◦ F′′0 = η′′ ◦ πQ0 ◦ iQ′′0
◦ F′′0
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
iP′′0
ε′′
F′′0
iQ′0iQ′′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
η ◦ iQ′′0◦ F′′0 = η′′ ◦ πQ0 ◦ iQ′′0
◦ F′′0 = η′′ ◦ IdQ′′0◦F′′0 = η′′ ◦ F′′0
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
iP′′0
ε′′
F′′0
iQ′0iQ′′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
πB ◦ f ◦ε ◦ i′′P0
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
iP′′0
ε′′
F′′0
iQ′0iQ′′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
πB ◦ f ◦ε ◦ i′′P0= f ′′ ◦ πA ◦ε ◦ i′′P0
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
iP′′0
ε′′
F′′0
iQ′0iQ′′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
πB ◦ f ◦ε ◦ i′′P0= f ′′ ◦ πA ◦ε ◦ i′′P0
= f ′′ ◦ε′′ ◦ πP0 ◦ πiP′′0
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
iP′′0
ε′′
F′′0
iQ′0iQ′′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
πB ◦ f ◦ε ◦ i′′P0= f ′′ ◦ πA ◦ε ◦ i′′P0
= f ′′ ◦ε′′ ◦ πP0 ◦ πiP′′0= f ′′ ◦ε′′ ◦ IdP′′0
=
f ′′ ◦ε′′
Shilin Yu The Blueprint of the Empire State Building
psulogo
Check on the subobject P′′0 ⊂ P0
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
iP′0
F′0
ε′ ε
F0
iP′′0
ε′′
F′′0
iQ′0iQ′′0
η′′
iA
f ′ f
πA
f ′′
iB
η′
πB
η
πB ◦ f ◦ε ◦ i′′P0= f ′′ ◦ε′′ = πB ◦ η ◦ iQ′′0
◦ F′′0 = η′′ ◦ F′′0
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct higher beams Fn
· · · · · · · · ·
· · · · · · · · ·
P′1 P1 P′′1
Q′1 Q1 Q′′1
P′0 P0 P′′0
Q′0 Q0 Q′′0
A′ A A′′
B′ B B′′
F′1
d′
d
F1
d′′
F′′1
d′′iP′0
F′0
ε′ ε
F0
ε′′
F′′0
d′ d
η′′iA
f ′ f
πA
f ′′iB
η′
πB
η
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct Fn
P′n Pn P′′n
Q′n Qn Q′′n
P′n−1 Pn−1 P′′n−1
Q′n−1 Qn−1 Q′′n−1
P′n−2 Pn−2 P′′n−2
Q′n−2 Qn−2 Q′′n−2
F′n
d′d′′
λn
F′′n
d′′
F′n−1
d′λn−1
d′′F′′n−1d′
λn
d′′
F′n−2 F′′n−2
d′λn−1
πB
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct Fn
P′n Pn P′′n
Q′n Qn Q′′n
P′n−1 Pn−1 P′′n−1
Q′n−1 Qn−1 Q′′n−1
P′n−2 Pn−2 P′′n−2
Q′n−2 Qn−2 Q′′n−2
F′n
d′d′′
λn
F′′n
d′′
F′n−1
d′
d′′F′′n−1
γn−1d′
λn
d′′
F′n−2 F′′n−2
d′
πB
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct Fn
P′n Pn P′′n
Q′n Qn Q′′n
P′n−1 Pn−1 P′′n−1
Q′n−1 Qn−1 Q′′n−1
P′n−2 Pn−2 P′′n−2
Q′n−2 Qn−2 Q′′n−2
F′n
d′d′′
F′′n
d′′
F′n−1
d′
d′′F′′n−1
λn−1
d′
d′′
F′n−2 F′′n−2γn−2
d′ λn−1
πB
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct Fn
P′n Pn P′′n
Q′n Qn Q′′n
P′n−1 Pn−1 P′′n−1
Q′n−1 Qn−1 Q′′n−1
P′n−2 Pn−2 P′′n−2
Q′n−2 Qn−2 Q′′n−2
F′n
d′d′′
F′′n
d′′
F′n−1
d′
d′′F′′n−1
λn−1
d′
d′′
F′n−2 F′′n−2
d′
πB
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct Fn
P′n Pn P′′n
Q′n Qn Q′′n
P′n−1 Pn−1 P′′n−1
Q′n−1 Qn−1 Q′′n−1
P′n−2 Pn−2 P′′n−2
Q′n−2 Qn−2 Q′′n−2
F′n
d′d′′
λn
F′′n
d′′
F′n−1
d′
d′′F′′n−1d′
d′′
F′n−2 F′′n−2
d′
πB
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct Fn
P′n Pn P′′n
Q′n Qn Q′′n
P′n−1 Pn−1 P′′n−1
Q′n−1 Qn−1 Q′′n−1
P′n−2 Pn−2 P′′n−2
Q′n−2 Qn−2 Q′′n−2
F′n
d′d′′
λn
F′′n
d′′
F′n−1
d′
d′′F′′n−1d′
d′′
F′n−2 F′′n−2
d′
πB
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct Fn
P′n Pn P′′n
Q′n Qn Q′′n
P′n−1 Pn−1 P′′n−1
Q′n−1 Qn−1 Q′′n−1
P′n−2 Pn−2 P′′n−2
Q′n−2 Qn−2 Q′′n−2
F′n
d′d′′
λn
F′′n
d′′
F′n−1
d′
d′′F′′n−1d′
d′′
F′n−2 F′′n−2
d′
πB
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct Fn
P′n Pn P′′n
Q′n Qn Q′′n
P′n−1 Pn−1 P′′n−1
Q′n−1 Qn−1 Q′′n−1
P′n−2 Pn−2 P′′n−2
Q′n−2 Qn−2 Q′′n−2
F′n
d′d′′
F′′n
d′′
F′n−1
d′
d′′F′′n−1d′
d′′
F′n−2 F′′n−2
d′ λn−1
πB
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct Fn
P′n Pn P′′n
Q′n Qn Q′′n
P′n−1 Pn−1 P′′n−1
Q′n−1 Qn−1 Q′′n−1
P′n−2 Pn−2 P′′n−2
Q′n−2 Qn−2 Q′′n−2
F′n
d′d′′
F′′n
d′′
F′n−1
d′
d′′F′′n−1d′
d′′
F′n−2 F′′n−2
d′ λn−1
πB
Shilin Yu The Blueprint of the Empire State Building
psulogo
Construct Fn
P′n Pn P′′n
Q′n Qn Q′′n
P′n−1 Pn−1 P′′n−1
Q′n−1 Qn−1 Q′′n−1
P′n−2 Pn−2 P′′n−2
Q′n−2 Qn−2 Q′′n−2
F′n
d′d′′
F′′n
d′′
F′n−1
d′
d′′F′′n−1d′
λn
d′′
F′n−2 F′′n−2
d′
πB
Shilin Yu The Blueprint of the Empire State Building
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