This is a follow-up to Computing Ext in Exterior algebra (related to Koszul duality) .

Let $V = \mathbb{C} x$, $A = \Lambda^{\bullet}(V) = A_0 \oplus A_1$ is graded (with $A_0 = \mathbb{C}, A_1 = V$). Consider $A_0$ as a left $A$-module, how do we compute the multiplication in the algebra $\text{Ext}^{\bullet}_A(A_0, A_0)$? I've heard the word "Yoneda product" mentioned, but I don't know how it works.

We can compute $\text{Ext}^{k}_A(A_0, A_0)$ as follows (see the answer in my above post for more details). We have a Koszul complex $\cdots \rightarrow S^2 V \otimes \Lambda^{\bullet}V[-2] \rightarrow V \otimes \Lambda^{\bullet}V[-1] \rightarrow \Lambda^{\bullet}V \rightarrow A_0$; and we have to compute the cohomology of the complex $\text{Hom}_A(A_0, A_0) \rightarrow \text{Hom}_A(A, A_0) \rightarrow \text{Hom}_A(V \otimes A[-1], A_0) \rightarrow \cdots$. It's easy to check that $\text{Hom}_A(S^k V \otimes A[-k], A_0)=(S^k V)^*[-k]$, and that all maps in this chain complex are $0$ (except for the first); this means $\text{Ext}^k_A(A_0, A_0)=(S^k V)^*[-k]$.

I'm also interested in multiplication for the general case (where $\text{dim } V>1$), but I'm guessing it follows from using the exact same method.

Homology of Noetherian rings and local rings, for computing the product structures on Ext and Tor in these sorts of situations. The basic idea is to build the projective resolution in such a way that you can further equip it with the structure of a DGA, which then induces the product structure you want. The word "Koszul" does not appear, but nonetheless it's fun and easy and you can learn it in an afternoon. projecteuclid.org/euclid.ijm/1255378502 – Eric Peterson Aug 7 '13 at 16:13