I have been reading the appendix in Serre's Local fields, to do with explicit computations of cup products (pg 176), but I'm stuck on one bit of lemma 4. It goes as follows Let B …

One way to define the Steenrod Operations is to use the cup-i product, as in Mosher and Tangora's book. It basically says, given the chain complex from mod-2 homology $C_*$, defin …

I want to make a Kunneth product of sorts on Ext. In particular, letting $C_*$ be a $R$-free resolution for $k$ over a $k$-hopf algebra $R$, elements in $Ext_R(k,k)$ are represent …

Let $X$ be a smooth projective variety over $\mathbb{C}$, and let $\alpha \in H^{2k}(X)$ be an algebraic cohomology class. Let us fix an $l$ such that $H^l(X) \neq 0$ and $H^{l+2k} …

Suppose that we are given a CW complex X in terms of the cells and the gluing maps. My understanding is that computing the cup product of the singular cohomology ring from this inf …

I was discussing with a student today the nature of the non-commutativity of cup-product at the level of cochains. In trying to explain what happens, I came up with the following …

I am wondering if there is a way to see the cup product, in some cases, without using cochain complexes. The situation I am interested in is the following: Let $G=F/R$ be a finite …