# Exceptional collections and Čech cohomology

We have developed most of the background necessary to describe the derived category $D_{b}(cohP_{k})$ but we require one more important notion.

**Definition**. Let $T$ be a $k$-linear triangulated
category. An object $E$ is ($k$-)*exceptional* if
$Hom(E,E[l])={0k⋅1_{E} l=0l=0 $

A sequence of objects $E_{1},…,E_{s}$ is called an
*exceptional collection* if

- each $E_{i}$ is exceptional
- and $Hom(E_{i},E_{j}[l])=0$ for all $l$ and $j<i$.

We say an exceptional colletion is *strong* if
$Hom(E_{i},E_{j}[l])=0$
for all $l=0$.

An exceptional collection is *full* if it generates $T$.

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

Since full exceptional collections generate, we can apply the results from last semester to get an equivalence $D(A)≅perfA$ where $A=REnd(⨁E_{i})$

If, in addition, the collection is strong then we can take $A=End(⨁E_{i})$ and $D_{†}(A)$ 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 $i∈Z$, the sheaves
$O(i−n),…,O(i−1),O(i)$
form a full strong exceptional collection for
$D_{b}(cohP_{k})$.

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
$O(j)∈⟨O(i−n),…,O(i−1),O(i)⟩_{∞}$

##
**Proof**. (Expand to view)

The Koszul complex $K(x_{0},…,x_{n})$ yields an exact complex of graded modules $0→R(−n−1)→i=0⨁n R(−n+1)→⋯→i=0⨁n R(−1)→R→R/(x_{0},…,x_{n})→0$ Applying $(−) $ gives the exact sequence $0→O(−n−1)→i=0⨁n O(−n+1)→⋯→i=0⨁n O(−1)→O→0$ Twisting by $O(i)$ we get an exact sequence $0→O(i−n−1)→i=0⨁n O(i−n+1)→⋯→i=0⨁n O(i−1)→O(i)→0$ Consequently, if we have $O(i−n),…,O(i−1),O(i)$ we can generate both $O(i−n−1)$ and $O(i+1)$.

▢**Lemma**. We have
$F∈⟨O(i−n),…,O(i−1),O(i)⟩_{∞}$
for any coherent sheaf $F$.

##
**Proof**. (Expand to view)

We have $F=M$ for a finitely- generated graded module $M$. From the previous lemma and the fact $(−) $ is exact, it is enough to show we can generate $M$ from $O(j)$ for $j∈Z$.

Inductively, we can construct an exact sequence $F_{l}→F_{l+1}→⋯F_{0}→M→0$ with $F_{i}=j=1⨁r R(m_{ij})$ Since $k[x_{0},…,x_{n}]$ is regular and graded local, we have the kernel $F_{−n}→F_{−n+1}$ is free. Thus, for $j>n+1$, the extension of $M$ by $K_{−j}$ $0→K_{−j}→F_{−j}→⋯→F_{0}→M→0$ is $0$ and $F_{−j}→⋯→F_{0}≅M⊕K_{−j}[j]$ in the derived category. So we generate $M$.

▢**Lemma**. The collection
$O(i−n),…,O(i−1),O(i)$
generates $D_{b}(cohP_{k})$.

##
**Proof**. (Expand to view)

Any complex $C$ with $H_{i}(C)=0$ for $i≫0$ and $0≫i$ is an iterated sequence of cones of the form $cone(H_{j}(C)[j+1]→C_{j+1})$ where $C_{m}=H_{m}(C)[m]$ for maximal $m$ with $H_{m}(C)=0$. Thus, we can generate a bounded complex using its cohomology sheaves -- which we generated in the last lemma.

▢To finish the proof of Beilinson's result, we need to compute $Ext_{l}(O(i),O(j))≅H_{l}(P_{k},O(j−i))$ We will do this next.

## Čech cohomology

The universal way to "compute" $H_{i}(X,F)$ is to

- replace $F$ with an injective resolution,
- apply $Γ(X,−)$, 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 $0→F→A_{0}→A_{1}→⋯$ with $H_{i}(X,A_{j})=0,∀i>0$ then $H_{i}(X,F)≅H_{i}(Γ(X,A_{∙}))$

Recall that $Γ(X,−)$ is yields an equivalence between $QcohX$ and $ModR$ if $X=SpecR$. Consequently, if $F$ is a quasi-coherent sheaf supported on affine patch, then $H_{i}(X,F)=0,∀i>0$ for general $F$.

For a general $F$ and open cover $U_{1},…,U_{n}$ by affines, each $F∣_{U_{i}}$ is adapted and we have the obvious map $F→i=1⨁n F∣_{U_{i}}$ Since $F$ is a sheaf, this is injective but it is rarely surjective.

To fix it and make a resolution, let $Cˇ_{s}(F):=i_{0}<⋯<i_{s}⨁ F∣_{U_{i}∩⋯∩U_{i}}$ and $δ_{s}:Cˇ_{s}(F)→Cˇ_{s+1}(F)$ where the map $δ_{s}:F∣_{U_{i}∩⋯U_{i}⋯∩U_{i}}→F∣_{U_{i}∩⋯∩U_{i}}$ is restricting scaled by $(−1)_{j}$.

**Lemma**. $Cˇ_{∙}(F)$ is a resolution.

##
**Proof**. (Expand to view)

It suffices to check exactness on stalks where each restriction map inducts an isomorphisms. Inclusion-exclusion says this is exact.

▢**Lemma**. If each $U_{i_{0}}∩⋯∩U_{i_{s}}$ is affine,
then
$H_{i}(X,F)≅H_{i}(Γ(X,Cˇ_{∙}(F)))$

##
**Proof**. (Expand to view)

As discussed above, we have a resolution by sheaves adapted to $Γ(X,−)$.

▢It would be nice to know that given two affines $U,V$ in $X$
the intersection $U∩V$ 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 $ProjR$. Here the intersection is another principal open so the conditions of the previous lemma are immediate.

In our example of $P_{k}$ we use the principal opens $U_{x_{i}}$. Then, $Γ(X,O(l))=$ $0→⨁k[x_{0},…,x_{n},x_{i}]_{l}→⋯→k[x_{0},…,x_{n},(x_{0}…x_{n})_{−1}]_{l}→0$

It makes sense to sum over $l$ and consider the complex $0→⨁k[x_{0},…,x_{n},x_{i}]→⋯→k[x_{0},…,x_{n},(x_{0}…x_{n})_{−1}]→0$ of graded $k[x_{0},…,x_{n}]$ modules all at once.