Exceptional collections and Čech cohomology

We have developed most of the background necessary to describe the derived category $D^b(\mathrm{coh}{\mathbb{P}}_k^n)$ but we require one more important notion.

Definition. Let $\mathcal{T}$ be a $k$-linear triangulated category. An object $E$ is ($k$-)exceptional if $$\mathrm{Hom}(E,E[l])=\{\begin{array}{ll}0& l\ne 0\\ k\cdot 1_E& l=0\end{array}$$

A sequence of objects $E_1,\dots ,E_s$ is called an exceptional collection if

• each $E_i$ is exceptional
• and $$\mathrm{Hom}(E_i,E_j[l])=0$$ for all $l$ and $j<i$.

We say an exceptional colletion is strong if $$\mathrm{Hom}(E_i,E_j[l])=0$$ for all $l\ne 0$.

An exceptional collection is full if it generates $\mathcal{T}$.

Assume we have an abelian category $\mathcal{A}$ and a full exceptional collection for $D^{†}(\mathcal{A})$.

Since full exceptional collections generate, we can apply the results from last semester to get an equivalence $$D(\mathcal{A})\cong \mathrm{perf}A$$ where $$A=\mathbf{R}\mathrm{End}\left(⨁E_i\right)$$

If, in addition, the collection is strong then we can take $$A=\mathrm{End}\left(⨁E_i\right)$$ and $D^{†}(\mathcal{A})$ is equivalent to the category of perfect complexes of modules over an honest $k$-algebra. This algebra is usually non-commutative.

We can now state a fundamental result of Beilinson.

Theorem. For any $i\in \mathbb{Z}$, the sheaves $$\mathcal{O}(i-n),\dots ,\mathcal{O}(i-1),\mathcal{O}(i)$$ form a full strong exceptional collection for $D^b(\mathrm{coh}{\mathbb{P}}_k^n)$.

Showing that something is a full (strong) exceptional collection breaks up into two parts:

• a computation to check that vanishing of the appropriate Hom spaces and
• a generation check.

We handle the generation statement now and the cohomology computation after introducing Čech cohomology.

Lemma. We have $$\mathcal{O}(j)\in ⟨\mathcal{O}(i-n),\dots ,\mathcal{O}(i-1),\mathcal{O}(i){⟩}_{\infty }$$

Lemma. We have $$\mathcal{F}\in ⟨\mathcal{O}(i-n),\dots ,\mathcal{O}(i-1),\mathcal{O}(i){⟩}_{\infty }$$ for any coherent sheaf $\mathcal{F}$.

Lemma. The collection $$\mathcal{O}(i-n),\dots ,\mathcal{O}(i-1),\mathcal{O}(i)$$ generates $D^b(\mathrm{coh}{\mathbb{P}}_k^n)$.

To finish the proof of Beilinson's result, we need to compute $${\mathrm{Ext}}^l(\mathcal{O}(i),\mathcal{O}(j))\cong H^l({\mathbb{P}}_k^n,\mathcal{O}(j-i))$$ We will do this next.

Čech cohomology

The universal way to "compute" $H^i(X,\mathcal{F})$ is to

• replace $\mathcal{F}$ with an injective resolution,
• apply $\Gamma (X,-)$, and
• compute cohomology of the resulting complex of $k$-modules.

Unfortunately, projective resolutions are hard enough to handle and we have some very explicit projective modules. Injective objects are generally more complicated. Having enough control over their explicit structure to compute cohomology of a sequence is a big ask.

However, there is a bit more flexibility in computing derived using adapated objects

If we can construct a resolution $$0\to \mathcal{F}\to {\mathcal{A}}^0\to {\mathcal{A}}^1\to \cdots$$ with $$H^i(X,{\mathcal{A}}^j)=0,\forall i>0$$ then $$H^i(X,\mathcal{F})\cong H^i(\Gamma (X,{\mathcal{A}}^{\bullet }))$$

Recall that $\Gamma (X,-)$ is yields an equivalence between $\mathrm{Qcoh}X$ and $\mathrm{Mod}R$ if $X=\mathrm{Spec}R$. Consequently, if $\mathcal{F}$ is a quasi-coherent sheaf supported on affine patch, then $$H^i(X,\mathcal{F})=0,\forall i>0$$ for general $\mathcal{F}$.

For a general $\mathcal{F}$ and open cover $U_1,\dots ,U_n$ by affines, each $\mathcal{F}{|}_{U_i}$ is adapted and we have the obvious map $$\mathcal{F}\to \underset{i=1}{\overset{n}{⨁}}\mathcal{F}{|}_{U_i}$$ Since $\mathcal{F}$ is a sheaf, this is injective but it is rarely surjective.

To fix it and make a resolution, let $${\check{C}}^s(\mathcal{F}):=\underset{i_0<\cdots <i_s}{⨁}\mathcal{F}{|}_{U_{i_0}\cap \cdots \cap U_{i_s}}$$ and $${\delta }^s:{\check{C}}^s(\mathcal{F})\to {\check{C}}^{s+1}(\mathcal{F})$$ where the map $${\delta }^s:F{|}_{U_{i_0}\cap \cdots \widehat{U_{i_j}}\cdots \cap U_{i_s}}\to F{|}_{U_{i_0}\cap \cdots \cap U_{i_s}}$$ is restricting scaled by $(-1{)}^j$.

Lemma. ${\check{C}}^{\bullet }(\mathcal{F})$ is a resolution.

Lemma. If each $U_{i_0}\cap \cdots \cap U_{i_s}$ is affine, then $$H^i(X,\mathcal{F})\cong H^i(\Gamma (X,{\check{C}}^{\bullet }(\mathcal{F})))$$

It would be nice to know that given two affines $U,V$ in $X$ the intersection $U\cap V$ is also affine. Such a scheme is called semi-separated.

A separated scheme is one where the diagonal morphism is a closed immersion. This implies semi-separated.

We will sidestep the generalities here because we will use very specific affines: principal opens in $\mathrm{Proj}R$. Here the intersection is another principal open so the conditions of the previous lemma are immediate.

In our example of ${\mathbb{P}}_k^n$ we use the principal opens $U_{x_i}$. Then, $\Gamma (X,\mathcal{O}(l))=$ $$0\to ⨁k[x_0,\dots ,x_n,x_i^{-1}{]}_l\to \cdots \to k[x_0,\dots ,x_n,(x_0\dots x_n{)}^{-1}{]}_l\to 0$$

It makes sense to sum over $l$ and consider the complex $$0\to ⨁k[x_0,\dots ,x_n,x_i^{-1}]\to \cdots \to k[x_0,\dots ,x_n,(x_0\dots x_n{)}^{-1}]\to 0$$ of graded $k[x_0,\dots ,x_n]$ modules all at once.