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

???

???

???

???

???

???

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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