On the derived categories of coherent sheaves on a quadric

Question

I am reading On the derived categories of coherent sheaves on some homogeneous spaces - Kapranov for the proof for quadrics. Consider a vector space $V=\mathbb{C}^{n+2}$ with the standard quadratic form $\langle-,-\rangle$, we define a graded Clifford algebra $A=\bigoplus A_i$ generated by $v\in V$ with $\deg(v)=1$ and a letter $h$ with $\deg(h)=2$ subjecting the relations

• $vw+wv=2\langle v,w\rangle h$

• $vh=hv$

for any $v,w\in V$. Now pick a quadric $Q\subset\mathbb{P}(V)\cong\mathbb{P}^{n+1}$ and denote $B:=\bigoplus H^0(Q,\mathcal{O}_Q(i))$. He claimed that

$A$ and $B$ are quadric Koszul algebras such that $A\cong\text{Ext}_B^{\bullet}(\mathbb{C},\mathbb{C})$ and $B\cong\text{Ext}_A^{\bullet}(\mathbb{C},\mathbb{C})$

Moreover, he claimed that

the generalised Koszul complex $$\cdots\rightarrow A_2^*\otimes B\rightarrow A_1^*\otimes B\rightarrow A_0^*\otimes B$$ coincides with the Tate resolution of $B$-module $\mathbb{C}$.

It seems that they are known results. Could anyone provide a reference?

Answer 1

(Answered for me by M. Kapranov)

These statements are not stated as that before in literatures (he found them independently), but they are not so hard to prove. A reference for Koszul algebra can be Koszul Algebras - Ralf Fröberg.

It is clear that both $A$ and $B$ are quadratic algebras: generators in degree $1$, relations in degree $2$. Next, we see by a direct computation that the spaces of generators of $A$ and $B $ are naturally dual to each other and the spaces of relations are the orthogonals of each other. Hence, they are Priddy dual to each other.

Fianlly, $B$ is Koszul because it is a complete intersection algebra (in which case, the Tate resolution is the Koszul complex for it).