Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

An exercise in Hatcher's book asks to prove that whenever $X$ is a space with the homology groups $H_n(X; \mathbb{Z})$ finitely generated free abelian for each $n \geq 0$, then $H^*(X; \mathbb{Z}) \otimes \mathbb{Z}_p$ and $H^*(X;\mathbb{Z}_p)$ are isomorphic as graded rings.

I've only started to learn about the cup product and I have no idea how to proceed. I see that the two objects in question are isomorphic as groups (from the universal coefficients theorem), but that's about it. So the question is: how to prove the statement in the title of the question?

share|improve this question

1 Answer 1

Hint: $H^n(X)\to H^n(X;\mathbb{Z}_p)$ is a ring homomorphism. Putting that in a well-chosen commutative diagram, you ultimately need to show that a map $\prod\mathbb{Z}\to\prod\mathbb{Z}_p$ induces an isomorphism $(\prod\mathbb{Z})\otimes\mathbb{Z}_p\to\prod\mathbb{Z}_p$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.