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.)



Computing Ext in Exterior algebra (related to Koszul duality)
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 $Ext^\bullet(A\_0,A\_0) = S^\bullet(V^\*)$.


