Dec 29, 2019
In a short exact sequence of vector fields 0 → A → B → C→ 0,B must be the direct sum B=A⊕C of the subgroup A and the quotient group C=B/A.
Allen Hatcher
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In a short exact sequence of vector fields 0 → A → B → C→ 0,B must be the direct sum B=A⊕C of the subgroup A and the quotient group C=B/A.
Allen Hatcher
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I’m currently surrounded by two groups of people: One of them is trying to figure out the possible eigenvalues for nilpotent matrices. The other one is trying to solve their problem via spectral sequences.
The first page of a spectral sequence can be ignored, like the prefaces to many books.
Allen Hatcher
Spectral sequences are a powerful book-keeping tool for proving things involving complicated commutative diagrams. They were introduced by Leray in the 1940’s at the same time as he introduced sheaves. They have a reputation for being abstruse and difficult. It has been suggested that the name ‘spectral’ was given because, like spectres, spectral sequences are terrifying, evil, and dangerous. I have heard no one disagree with this interpretation, which is perhaps not surprising since I just made it up.
Ravi Vakil, Foundations of Algebraic Geometry notes