The Blueprint of the Empire State Building
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?
iP?
P?
P?
F?
F?
Q?
iQ?
Q?
A
iA
A
A
A
F?
Q?
f
B
f
iB
B
f
B
B
Shilin Yu
The Blueprint of the Empire State Building
psulogo
The Empire State Building made of projective resolutions
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P1
F1
d
Q1
P0
d i P0 F0
Q0
A
iA
f
B
P1
F1
P1
F1
Q1
d d
P0
F0
Q1
d
d
P0
F0
Q0
Q0
A
A
A
iB f
B
B f
B
Shilin Yu
The Blueprint of the Empire State Building
psulogo
Construct the 'beam' F0
P0
iP0
P0
P0
F0
F0
Q0
iQ0
Q0
A
iA
A
A
A
F0
Q0
f
B
f
iB
B
f
B
B
Want the red square in the middle to be commutive, i.e., F0 - f = 0.
psulogo
Shilin Yu
The Blueprint of the Empire State Building
Check on the subobject P0 P0
P0
iP0
P0
P0
F0
F0
Q0
iQ0
Q0
A
iA
A
A
A
F0
Q0
f
B
f
iB
B
f
B
B
Check on the subobject P0 P0, i.e., ( F0 - f ) iP0 = 0.
Shilin Yu
The Blueprint of the Empire State Building
psulogo
Check on the subobject P0 P0
P0
iP0
P0
P0
F0
F0
Q0
iQ0
Q0
A
iA
A
A
A
F0
Q0
f
B
f
iB
B
f
B
B
F0 iP0 .
Shilin Yu
The Blueprint of the Empire State Building
psulogo
Check on the subobject P0 P0
P0
iP0
P0
P0
F0
F0
Q0
iQ0
Q0
A
iA
A
A
A
F0
Q0
f
B
f
iB
B
f
B
B
F0 iP0 = iQ0 F0.
Shilin Yu
The Blueprint of the Empire State Building
psulogo
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