Let $V = \mathbb{C}^n$, $A = \Lambda^{\bullet}(\mathbb{C}^n)$ is a graded algebra (with $A_0 =
\mathbb{C}, A_1 = V$, etc).
Consider $A_0$ as a left $A$-module, how do we compute the graded ring $\text{Ext}^{\bullet}_A(A_0, A_0)$? (Doing the $n=3$ example should be enough; then it would be easy to generalize.)
(I was trying to understand Koszul duality for symmetric/exterior algebras from [BGS]; and this is the first step.)
Best Answer
Consider the Koszul complex $$ \dots \to S^3V\otimes A(-3) \to S^2V\otimes A(-2) \to V\otimes A(-1) \to A \to A_0 \to 0, $$ where $(-i)$ is the shift of grading. This is a free resolution of $A_0$. Using this to compute $Ext$ you obtain $Ext^\bullet(A_0,A_0) = S^\bullet(V^*)$.