Q:

What is the difference between the cohomology rings of odd and even dimensional spheres?

I am reading Hatcher example 3.16 (3.18 in the old edition).
It says that the cohomology ring of an odd-dimensional sphere is $\Lambda_\mathbb{Z}[\alpha]$ which is the exterior algebra generated by the odd-dimensional $\alpha$; the cohomology ring of an even-dimensional sphere is $\mathbb{Z}[\alpha]/(\alpha^2)$ with even-dimensional $\alpha$.
Why is the cohomology ring of an odd dimensional sphere not $\mathbb{Z}[\alpha]/(\alpha^2)$? And why is the other one not the exterior algebra?
Here is the book: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
Thanks for any help...

A:

The point is that we are considering these rings as graded commutative rings, and elements of even degree and elements of odd degree behave very differently in such rings.  In particular, elements of even degree commute with all others, so they can generate a polynomial ring.  On the other hand, two elements of odd degree anticommute, and in particular this means an element $\alpha$ of odd degree always satisfies $\alpha^2=-\alpha^2$.  So, over a base ring in which $2\neq 0$, you cannot have a polynomial ring generated by an element in odd degree.  In particular, then, this is why Hatcher doesn't write the cohomology ring of an odd-dimensional sphere as $\mathbb{Z}[\alpha]/(\alpha^2)$: there actually is no such thing as $\mathbb{Z}[\alpha]$ as a graded commutative ring, when $\alpha$ has odd degree.
As for why not to call the cohomology ring of an even dimensional sphere an exterior algebra, I'm guessing that Hatcher wants to be able to say that a tensor product of exterior algebras is an exterior algebra.  This isn't true if you allow exterior algebras on generators of even degree, since the different generators will commute instead of anticommuting in the tensor product.  Indeed, just as an element of odd degree cannot generate a polynomial ring (if $2\neq 0$), an element of even degree cannot be a generator of an exterior algebra unless there are no other generators (since it needs to anticommute with all the other generators, but it instead commutes with them since it has even degree).

