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How much connection is there between Commutative Algebra and Algebraic Topology?

I am looking for general highlights, not complex details.

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1 Answer 1

up vote 3 down vote accepted

"How much connection is there between the wheel and bike racing ?"

Basically, one is an essential tool for the other : algebraic topology is full of commutative algebra concepts, such as polynomial rings or exact sequences, and uses many commutative algebra methods or results, such as the five-lemma or diagram-chasing arguments.

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Do modules appear in algebraic topology? –  Manos May 8 '12 at 15:46
Yes, all the time. For a cheap example, abelian groups are $\mathbb{Z}$-modules! –  user641 May 8 '12 at 15:49
And for a slightly less cheap example, the cohomology of the total space of a fiber bundle is a module over the cohomology of the base space. –  Samuel T May 8 '12 at 23:14
And for an important and more algebra-heavy example, the mod-p cohomology of any space is a module over the (rather complicated, and in particular non-commutative!) Steenrod algebra. –  Aaron Mazel-Gee May 11 '12 at 14:52
Oops, you were asking about commutative algebra. I suppose I should amend my statement: the mod-p homology of any space is a comodule over the (commutative) dual Steenrod algebra. (The Steenrod algebra is actually a Hopf algebra, and it has non-commutative multiplication and commutative comultiplication, so its dual has commutative multiplication and non-commutative comultiplication.) –  Aaron Mazel-Gee May 11 '12 at 17:54

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