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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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"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?2012-05-08
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    Yes, all the time. For a cheap example, abelian groups are $\mathbb{Z}$-modules!2012-05-08
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    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.2012-05-08
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    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.2012-05-11
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    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.)2012-05-11