## Syllabus

The syllabus can be found at the syllabus tab.

## Notes

Course notes can be found in the tab to the left.

## Previous semester

This course is a continuation of the Fall 2021 course Derived Categories I.

## Me

If you are interested in a bit about who I am and what I do visit my website.

# Quasi-coherent sheaves on projective varieties

Let $R$ be a $N$-graded commutative ring.

We constructed a functor $(−) :GrModR→ModX$ where $X=ProjR$ and a functor in the other direction $Γ (X,−):ModXF →GrModR↦i∈Z⨁ Γ(X,F(i)) $

Let's think a little more about their relationsip. For any $m∈M_{l}$, we get a map $m:Rr →M(l)↦rm $ Applying $(−) $, we get a map $O_{X}→M(l)$ and hence an element of $Γ(X,M(l))$. This gives a natural transformation $η_{M}:M→Γ (X,M)$

Similarly, for any $O_{X}$-module $F$, there is a natural transformation $ϵ_{F}:Γ (F) →F$ which we describe on the principal opens $U_{x}$ for $x∈R_{i}$ with $i>0$. We have $Γ (F) (U_{x})=(Γ (F)_{x})_{0}$ Each element of $(Γ (F)_x)_0$ can be written as $x_{s}f $ where $f∈Γ(X,F(si))$ and $x∈R_{i}$. Restricting to $U_{x}$ we get an element $f∣_{U_{x}}∈Γ(U_{x},F(si))$.

On $U_{x}$, we have an isomorphism $1/x_{s}:F(is)→F$. Thus, we get an element $x_{s}1 f∣_{U_{x}}∈Γ(U_{x},F)$ giving an element of $F(U_{x})$.

**Proposition**. The natural transformations $η$ and $ϵ$
induce an adjunction $(−) ⊣Γ $.

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

Using a [theorem](https://738.f21.matthewrobertballard.com/notes/2021-08-31/) from last semester, we have to check that $M→η_{M} Γ (M) →ϵ_{M}M$ and $Γ (F)→η_{Γ(F)}Γ (Γ (F) )→Γ (ϵ_{F})Γ (F)$ are both identity maps.

For the $ϵ_{M}∘η_{M} $, we check the map is the identity for sections over each principal open $U_{x}$. Given $m/x_{i}∈M(U_{x})$, we have $η_{M} (m/x_{i})=x_{i}η_{M}(m) $ restricting $η_{M}(m)$ back to $U_{x}$ gives $m∈Γ(U_{x},M(∣m∣))$.

For $Γ (ϵ_{F})∘η_{Γ(F)}$, if we take $f∈Γ(X,F(l))$, then $η_{Γ(F)}(f):O→Γ (F) (l)$ corresponds to global section $Γ (F) (l)$ of which is $f$ on each $U_{x}$. Applying $Γ (ϵ_{F})$, we see we get $f$ back.

▢Under some mild assumptions, we see that $(−) $ hits all quasi-coherent sheaves on $X$.

**Lemma**. If $R$ is finitely-generated in degree $1$
as a graded $k$-algebra and $F$ is quasi-coherent,
then $ϵ_{F}$ is an isomorphism.

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

From the above, we see that $η_{F}(f/x_{s})=0$ if and only if $f∣_{U_{x}}=0$. Thus, $f=0$ away from $Z(x)$.

Let $x_{0},…,x_{n}$ denote our generators in degree $1$. We
have a affine cover $U_{x_{j}}$ and
$Z(x)∩U_{x_{j}}=Z(x/x_{j})$
is a hypersurface in each $U_{j}$. Then $M:=Γ(U_{x_{j}},F(is))$
is a $R:=(R_{x_{j}})0$* module and $f∈M$. We know that
$f=0$ in $M_{x/x_{j}}$ so there is some $n_{j}$ with
$x_{j}x_{n_{j}} f=0$
on $U_{x_{j}}$. Taking the max of the $n_{j}$, we see that there is
some $x$ with $x_{n}f=0$ on all of $X$. Thus, $f=0$ in
$Γ (F)$*x already and $ϵ_F$
is injective.

A similar argument shows that for any $s∈Γ(U_{x},F)$ there is some $n>0$ so that $x_{n}s$ lifts to $Γ(X,F(ni)$. This shows that $ϵ_{F}$ is surjective.

▢Recall that for a graded $R$-module $M$, we have its
*torsion* submodule
$τ(M):={m∈M∣∃n_{0}>0,(R_{+})_{n}m=0∀n≥n_{0}}$
and we say a $M$ is *torsion* if
$τ(M)=M$
The module is *torsion-free* if
$τ(M)=0$

There is a short exact sequence $0→τ(M)→M→M/τ(M)→0$ Since $τ(M) =0$ there is a natural map $α_{M}:M/τ(M)→Γ (M)$

**Lemma**. Assume $R$ is finitely-generated over $k$ in degree $1$
by $x0n$ and that each $x$* is not a zero-divisor. Then
$α$*M is injective

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

The kernel of $M→Γ (M)$ consists of $m∈M$ such that $(m)=0$ in $(M_x)_0$ for all homogeneous $x∈R_{+}$.

Let $x0n$ denote the generators of $R$ over $k$ in degree $1$. Let $m∈kerα_{M}$ be an element of degree $l$. Then $m/x_{i}=0$ for each $i$ and $x_{i}m=0$ for some $n_{i}$. Thus, for large enough $n$ we have $(R_{+})_{n}m=0$ and so $m∈Tors(M)$.

▢We can describe $τ(M)$ as $τ(M)≅colim_{s}Hom (R/(R_{+})_{s},M)$ where $Hom (M,N):=i∈Z⨁ Hom(M,N(i))$ is the internal Hom for graded modules.

There are also the exact sequences $0→(R_{+})_{s}→R→R/(R_{+})_{s}$ which gives an exact sequence $0→colim_{s}Hom (R/(R_{+})_{s},M)→M→colim_{s}Hom ((R_{+})_{s},M)$ similar to the sequence $0→τ(M)→M→Γ (M)$ which leads us to compare $Γ (M)$ and $colim+)_{s},M)$

Since $⋯→(R_{+})_{s+1} →(R_{+})_{s} →⋯→R_{+} →R$ are all isomorphisms, we see that any $ϕ∈colimHom((R_{+})_{s},M(l))$ induces a well-defined $ϕ :O→M(l)$ and hence an element of $Γ (M)$.

Let's denote this map by $β_{M}:colim_{s}Hom ((R_{+})_{s},M)→Γ (M)$

**Lemma**. Assume that $R$ is finitely-generated over $k$ by
elements of degree $1$. Then
$β_{M}$ is an isomorphism for all $M$.

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

An element of $Γ(X,M(l))$ is a collection of homogeneous elements $m_{f}∈M$ for each homogeneous $f∈R_{+}$ so that - there is $s_{f}$ such that $m_{f}/f_{s_{f}}$ has degree $l$ and - for any $f,g$ $f_{s_{f}}m_{f} =g_{s_{g}}m_{g} $ in $M_{fg}$.

Let $x_{1},…,x_{n}$ denote a set of homogeneous generators for $R$ over $k$. We claim that the function $ϕ$ which assigns $ϕ(x_{i})=m_{x_{i}}$ extends to a well-defined module map $(R_{+})_{s}→M(l)$ for some $s$. This gives the inverse to $β_{M}$.

Let's check that this construction yields a well-defined homomorphism. Up to replacing $m_{i}$ with $x_{i}$ and $s_{i}$ with $s_{i}+l_{i}$, we can assume that $m_{i}=x_{j}x_{i} m_{j}$ in $M_{x_{j}}$. If we have a relation of the form $a_{1}x_{1}+⋯+a_{n}x_{n}=0$ then in $M_{x_{i}}$ we have $a_{1}m_{1}+⋯a_{n}m_{n} =a_{1}x_{i}x_{1} m_{i}+⋯+a_{n}x_{i}x_{1} m_{i}=x_{i}m_{i} (a_{1}x_{1}+⋯+a_{n}x_{n})=0 $ Thus, $a_{1}m_{1}+⋯+a_{n}m_{n}$ is annihilated by some power of each $x_{i}$. Passing to a sufficient high power of $R_{+}$ forces this to vanish.

▢As a corollary, we get short exact sequences.

**Corollary**. Assume that $R$ is finitely-generated by
elements of degree $1$ over $k$. Then, for each injective
graded $R$-module $I$ we have a short exact sequence
$0→τ(I)→I→Γ (I)→0$

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

Since $I$ is injective, we get short exact sequences $0→Hom(R/(R_{+})_{s},I(l))→I_{l}→Hom((R_{+})_{s},I(l))→0$ taking direct sums and passing to the colimit gives the result.

▢Note the map $M→Γ (M)$ is not always onto.

**Example**. Take $k[x]$ with $∣x∣=1$. Then,
$Γ (k[x] )=k[x,x_{−1}]$

As consequence, we get a semi-orthogonal decomposition relating graded $R$-modules and quasi-coherent sheaves on $ProjR$.

**Proposition**. Assume that $R$ is finitely-generated by
elements of degree $1$ over $k$. Then there are
semi-orthogonal decompositions
$D_{†}(GrModR)=⟨D_{†}(GrMod_{Tors}R),D_{†}(QcohProjR)⟩$
where $†∈{b,+,−,∅}$.

Here $GrMod_{Tors}R$ denotes the abelian subcategory of torsion graded $R$-modules.

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

For each complex of injectives $I$, we have a short exact sequence of complexes $0→τ(I)→I→Γ (I_{∙})→0$ and thus a triangle $τ(I)→I→Γ (I_{∙})→τ(I)[1]$ Each component of $τ(I)$ is torsion.

Thanks to adjunction we have $Hom(τ(I),Γ (J))≅Hom(τ(I) ,J)=0$ so we have semi-orthogonality.

It remains to check that $Γ $ is fully-faithful and the derived functors of $τ$ and $Γ $ preserve cohomological boundedness.

Using adjunction, we have $Hom(Γ (I),Γ (J))≅Hom(I,Γ (J) )$ And $Hom(I,Γ (J) )≅Hom(I,J)$ since $Γ (J) →J$ is an isomorphism.

Finally, from the triangle we see that $R_{i}τ=0$ for $i≫0$ if and only if $R_{i}Γ =0$ for $i≫0$. We will see [next]({% link notes/2022_01_13.md %}) that $R_{i}Γ =0$ for $i>N$ where $N$ is the number of generators of $R$ over $k$.

▢**Remark**.

- One can remove the condition that the finite set
of generators of $R$ are of degree $1$ if we work with
the
*quotient stack*$[U/G_m]$ with the quasi-affine $U=SpecR∖{R_{+}}$ instead of $ProjR$. - One would like to replace $GrMod$ and $Qcoh$ with their finite versions $grmod$ and $coh$. This can be done but one needs to truncate the grading.

# 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.

# Computing sheaf cohomology on $P_{n}$

Last time we saw that computing $H_{i}(P_{k},O(j)):=R_{i}Γ(P_{k},O(j))$ could be achieved by computing the cohomology of the (augmented) Čech complex $0→k[x_{0},…,x_{n}]→i=0⨁n k[x_{0},…,x_{n}]_{x_{i}}→⋯→∣I∣=n⨁ k[x_{0},…,x_{n}]_{x_{I}}→k[x_{0},…,x_{n}]_{x_{0}⋯x_{n}}→0$

Let's look at the case $n=0$ to get a foothold. This reduces to $0→k[x_{0}]→k[x_{0},x_{0}]→0$ Since $x_{0}$ is regular, there is no $H_{0}$. We have $H_{1}=k[x_{0},x_{0}]/k[x_{0}]$ which as a $k[x_{0}]$-module has a filtration $0⊂Ann(x_{0})⊂⋯⊂Ann(x_{0})⊂Ann(x_{0})⊂⋯$ whose associated graded pieces are $Ann(x_{0})Ann(x_{0}) ≅x_{0}k $ In particular, $H_{1}$ is a free $k$-module. It is also graded with $∣x_{0}∣=−1$.

The general complex is isomorphic to the tensor product of complexes $i=0⨂n (k[x_{i}]→k[x_{i}]_{x_{i}})$ over $k$. Since the homologies of the complexes are flat over $k$, we have $H_{s}(i=0⨂n (k[x_{i}]→k[x_{i}]_{x_{i}}))≅i_{0}+⋯+i_{n}=s⨁ H_{i_{j}}(k[x_{j}]→k[x_{j}]_{x_{j}})$ From our computation above we see that $H_{s}=⎩⎨⎧ 0∑_{∣I∣=n}k[x_{0},…,x_{n}]_{x_{I}}k[x_{0},…,x_{n}]_{x_{0}⋯x_{n}} s=ns=n $

**Theorem**. We have
$Ext_{P_{k}}(O(i),O(j))≅H_{l}(P_{k},O(j−i))≅⎩⎨⎧ k[x_{0},…,x_{n}]_{j−i}k[x_{0},…,x_{n}]_{i−j−n−1}0 l=0l=notherwise $

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

We have an isomorphism of functors $Hom(O(i),F)≅Γ(P_{k},F(−i))$ which gives the isomorphims of derived functors $Ext_{l}(O(i),F)≅R_{i}Γ(P_{k},F(−i)).$

Now we use the computation above. Recall that
$H_{s}(P_{k},O(d))≅H_{s}(0→i=0⨁n R_{x_{i}}→⋯→R_{x_{0}⋯x_{n}}→0)_{d}$
where $R=k[x_{0},…,x_{n}]$. Thus, we have
$H_{0}(P_{k},O(j−i))≅R_{j−i}$
and we get the vanishing for $0<s<n$. For $s=n$, from the discussion above,

$x_{0}⋯x_{n}1 ,a_{0},…,a_{n}≥1$
forms a $k$-basis for $H_{n}$. Rewriting this as
$x_{0}⋯x_{n}1 =x_{0}⋯x_{n}1 x_{0}⋯x_{n}1 $
we get a bijection between $H_{d}$ and $R_{−d−n−1}$.

Fixing the basis vector $1/x_{0}⋯x_{n}$ gives an isomorphism $H_{n}(P_{k},O(−n−1))≅k$ From our computations, we see that composition $Ext_{i}(O(i),O(j))⊗_{k}Ext_{n−i}(O(j),O(i−n−1))→Ext_{n}(O(i),O(i−n−1))≅H_{n}(P_{k},O(−n−1))$ is given by multiplication $f,g1 ↦gf $ up to isomorphism.

**Corollary**. Composition
$Ext_{i}(O(i),O(j))⊗_{k}Ext_{n−i}(O(j),O(i−n−1))→H_{n}(P_{k},O(−n−1))$
is a perfect pairing. In other words, this induces an isomorphism
$Ext_{i}(O(i),O(j))_{∨}≅Ext_{n−i}(O(j),O(i−n−1))$

For a scheme $X→Speck$, we can ask if the functor $RHom_{k}(RHom_{X}(E,F),k)$ is quasi-isomorphic to $RHom_{X}(F,S(E))$ for some $k$-linear auto-equivalence $S$. In the case $k$ is simply a field, then this is just a Serre functor. In this statement, it is an enhanced version of a Serre functor.

**Proposition**. For any objects $E,F∈D_{b}(cohP_{k})$ there
are natural quasi-isomorphisms
$RHom_{k}(RHom(E,F),k)≅RHom(F,S(E))$
where $S(E)=E(−n−1)[n]$.

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

Some care is required for a general $k$ which we won't take. From composition we have $RHom(E,F)⊗L _{k}RHom(F,S(E))≅RHom(RHom(E,E),O(−n−1)[n])$ Using the natural map $O→RHom(E,E)$ lands us in \mathbf{R}\operatorame{Hom}(\mathcal O, \mathcal O(-n-1)[n]) which is quasi-isomorphic to $H_{n}(P_{k},O(−n−1))≅k$.

This gives a natural map $RHom(F,S(E))→RHom_{k}(RHom(E,F),k)$ which we see is a quasi-isomorphism for a collection of generators $O(j)$. Therefore, it is for all objects of $D_{b}(cohP_{k})$.

▢# Compact generation and Grothendieck duality

We saw last semester that compact generation of a triangulated category was a very useful property. Here we establish that $D(QcohX)$ is compactly generated for a quasi-projective schemes $X$ over an affine base. In addition, we provide some useful replacements for deriving common functors.

**Lemma**. Let $R$ be a $Z$-graded ring. Then $R(i)$ for $i∈Z$ is a set of compact generators for $D(GrModR)$.

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

One checks that $Hom(R(i)[j],C)≅H_{−j}(C)_{i}$ Thus, if $C$ is right orthogonal to all $R(i)$ it must be acyclic.

▢**Proposition**. If $R$ is finitely-generated in degree $1$ over $R_{0}$,
then $D(QcohProjR)$ is compactly generated.

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

Let $X=ProjR$. Recall that we [have](2022_01_11.md) a SOD $D(GrModR)=⟨D(TorsR),D(QcohX)⟩$ Given by $(−) $ and $Γ $. Thus, $Hom(O(i)[j],F)≅Hom(R(i)[j],Gamma (F))$ This vanishes for all $i,j∈Z$ if and only if $Γ (F)$ is acyclic. As $Γ (F) ≅F$ we see that $F$ is acyclic also.

▢**Lemma**. If $D(QcohX)$ is compactly-generated and $U$ is an
open subscheme, then so is $U$.

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

Let $l:U→X$ denote the inclusion. Then the counit $l_{∗}Rl_{∗}F→F$ is an isomorphism. Thus, if $C_{i}$ is a set of compact generators for $X$, then $l_{∗}C_{i}$ is for $U$.

▢**Corollary**. Let $X$ be quasi-projective over an affine base. Then any compact
object is quasi-isomorphic to a bounded complex of locally-free sheaves.

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

We have seen that $O_{X}(i)$ form a set of compact generators. As we saw [last semester](https://738.f21.matthewrobertballard.com/notes/2021_10_05/), the compact objects in $D(QcohX)$ are summands of iterated cones over maps to objects from the set of compact generators. Such a complex is bounded and has locally-free components.

▢We saw last time that $E↦E(−n−1)[n]$ is the Serre
functor on $P_{k}$. In fact, this is a consequence of a more general
fact known as *Grothendieck duality*. One formulation of Grothendieck duality
is that for a map $f:X→Y$ of schemes possesses a right adjoint
$f_{!}:D(QcohY→QcohX)$
to $Rf_{∗}$. This is not automatic in full generality. But it is
true with mild conditions on $X$ and $Y$.

For the projection \pi : \mathbb{P}^n_k \to \operatorname{Spec} k} we have $π_{!}O_{Speck}≅O(−n−1)[n]$

Originally proof of the existence of a right adjoint was arduous. However, [Neeman](https://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00174- 9/S0894-0347-96-00174-9.pdf) demonstrated how Brown Representability can be used to give a clean proof.

Thanks to Brown Representability, we know that for $Rf_{∗}$ to have a right adjoint is equivalent to the natural map $i∈I⨁ Rf_{∗}E_{i}→Rf_{∗}(⨁E_{i})$ being an isomorphism for all sums. This question is local on $Y$ so we reduce to asking -- when does taking cohomology commute with direct sums?

One useful fact: formation of the Čech complex for an affine cover where all intersections are affine commutes with direct sums.

**Proposition**. For any map $f:X→Y$ between quasi-projective schemes over
an affine base, there is a right adjoint $f_{!}$ to $Rf_{∗}$.

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

From the discussion above, we reduce to the checking the case where $Y$ is affine. Here we are computing global sections over a quasi-affine scheme which can be done using a Čech complex.

▢**Remark**. Neeman proves the existence of $f_{!}$ whenver $X$ and $Y$ are
quasi-compact and quasi-separated.

## Deriving tensor and sheaf Hom

We have implicitly used both the tensor product $E⊗_{O_{X}}F$ and sheaf Hom $Hom_{O_{X}}(E,F)$ of $O_{X}$-modules.

Recall that $E$ is *locally-free* if each $p∈X$ has a neighborhood $U$ with
$E∣_{U}≅O_{U}$
Since exactness can be checked locally, $E⊗_{O_{X}}−$ preserves
exactness of complexes of $O_{X}$-modules if $E$ is locally-free. Thus, if we
can replace any complex with one whose components are locally-free, we can use
these adapted objects to derive $⊗_{O_{X}}$.

**Proposition**. For any quasi-projective schemes $X$ and any chain complex $E$
of quasi-coherent sheaves, there is complex $P$ with locally-free components
and a quasi-isomorphism $P→E$.

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

It suffices to prove this for projective $X$. Any complex of quasi-coherent sheaves on $X$ is of the form $C$ for some complex of graded modules $C$. There are enough K-projectives in K(\operatarname{GrMod} R). Indeed, the twists of $R$ suffice. Thus, we get a $K$-projective $P$ and a quasi-isomorphism $P→C$. Applying $(−) $ gives the result.

▢Note that despite the notation we cannot find enough K-projective complexes in $D(QcohX)$.

Using this result, we take $E⊗L −:=P⊗−$ for some such $P$.

Similarly, we can derive $Hom(E,F)$ by replacing $E$ by a complex of locally-frees or $F$ by a K-injective. The choices are naturally quasi-isomorphic so we denote either by $RHom(E,F).$

# The projection formula and standard adjunctions

The main functors in play for sheaves of modules are

- $f_{∗}$ for a map $f:X→Y$
- $f_{∗}$ for $f:X→Y$
- $⊗_{O_{X}}$ and
- $Hom_{O_{X}}$

We will also use the notation $E_{∨}:=Hom_{O_{X}}(E,O_{X})$ for both the underived and derived duals over $O_{X}$.

We record some back facts about them and we omit the details for the most part.

**Lemma**. We have an adjunction $f_{∗}⊣f_{∗}$.

**Lemma**. We also have an adjunction
$⊗_{O_{X}}E⊣Hom_{O_{X}}(E,−)$

**Lemma**. Tensor products commute with pullbacks
$f_{∗}(E⊗_{O_{X}}F)≅f_{∗}E⊗_{O_{X}}f_{∗}F$

From these, we can derive some less immediate ones.

**Lemma**. We have an adjunction
$f_{∗}(−⊗_{O_{Y}}F)⊣Hom_{O_{Y}}(F,f_{∗}(−))$

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

We have natural isomorphisms $Hom_{X}(f_{∗}(E⊗F),G) ≅Hom_{Y}(E⊗F,f_{∗}G)≅Hom_{Y}(E,Hom(F,f_{∗}G)) $

▢We also have a sheafy version of pull-push adjunction.

**Lemma**. There is a natural isomorphism
$Hom_{O_{Y}}(E,f_{∗}F)≅f_{∗}Hom_{O_{X}}(f_{∗}E,F)$

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

We have $Hom_{X}(f_{∗}E⊗f_{∗}F,G) ≅Hom_{X}(f_{∗}E,Hom(f_{∗}F,G)≅Hom_{Y}(E,f_{∗}Hom_{O_{X}}(f_{∗}F,G))) $ From the previous lemma, we have two right adjoints. Thus, they must be isomorphic.

▢One of the most useful relationships between these functors is one that is not true.
Thanks to the map
$f_{∗}(f_{∗}E⊗F)≅f_{∗}f_{∗}E⊗f_{∗}F→E⊗f_{∗}F$
coming from the counit and adjunction we have a natural map
$f_{∗}E⊗F→f_{∗}(E⊗f_{∗}F)$
However, this is *not* an isomorphism of quasi-coherent sheaves in general.

For what $E$ and $F$ is the map $f_{∗}E⊗F→f_{∗}(E⊗f_{∗}F)$ an isomorphism? Taking $F=O_{Y}$ works. Since isomorphism can be detected locally, it suffices that $F$ is locally isomorphic to $O_{Y}$.

Moreover, if $f_{∗}E⊗F_{a}→f_{∗}(E⊗f_{∗}F_{a})$ for some $a∈A$, then $f_{∗}E⊗a∈A⨁ F_{a}→f_{∗}(E⊗f_{∗}(⨁F_{a}))$ will also be an isomorphism if $f_{∗}$ commutes with coproducts which is true for quasi-projective schemes (and more generally).

**Proposition**. Let $f:X→Y$ be a map of quasi-projective schemes. Then for
any $E∈D(QcohX)$ and $F∈D(QcohY)$, the natural map
$Rf_{∗}E⊗L F→Rf_{∗}(E⊗L Lf_{∗}F)$
is a quasi-isomorphism.

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

We can replace $E$ with a $K$-injective complex and $F$ with complexe of locally-free sheaves. Then, from the discussion, above we know that the induced map on the components is an isomorphism.

▢The previous result is often called the *projection formula*.

# Base change

Recall in commutative algebras given a diagram

$B↓⏐ S R$
we can form a pushout (or fiber coproduct)
$B↓⏐ S R↓⏐ S⊗_{B}R $
using the tensor product $S⊗_{B}R$.

Consequently, in the category of affine schemes, we have fiber products (or pullbacks) $X×_{Z}Y↓⏐ X Y↓⏐ Z $ where, if $X=SpecR$, $Y=SpecS$, and $Z=SpecB$, we have $X×_{Z}Y=Spec(S⊗_{B}R).$

For schemes generally, we can use this to build fiber products. Given $X f Y↓⏐ gZ $ $X×_{Z}Y$ exists. Given affine $U⊂X,V⊂Y$, and $W⊂Z$ with $f(U),g(V)⊂W$, we have an affine chart $U×_{W}V$ and these can be glued together appropriately to give the fiber product as a scheme.

A square
$W↓⏐ X Y↓⏐ Z $
isomorphic to a fiber product square is commonly called *Cartesian*.

Given a Cartesian square $W↓⏐ g~ X f~ f Y↓⏐ gZ $ and a $E∈D(QcohX)$ can we compare $Lg_{∗}Rf_{∗}EandRf~ _{∗}Lg~ _{∗}E$ In general, they will not be quasi-isomorphic. However, under some assumptions, this is the case.

**Theorem**. Assume that
$Tor_{i}(O_{X},O_{Y})=0$
for $i=0$. Then, there is a natural isomorphism
$Lg_{∗}Rf_{∗}E≅Rf~ _{∗}Lg~ _{∗}E$

In particular, if either $O_{X}$ or $O_{Y}$ is flat over $O_{Z}$, then we can exchange pushing and pulling.

The previous result is often called *(flat) base change*. We won't provide more
details on this result. Lipman has
some very thorough notes on Grothendieck duality which include exposition on Tor-independent
base change.

Along with the projection formula, base change will serve as regular tool in our future investigations of derived categories.

# Kernels and integral transforms

We saw last semester that for two rings $S$ and $R$ the category of $(S,R)$-bimodules was an imporant source of functors between $D(ModS)$ and $D(ModR)$.

Here we globalize this idea. Note that an $(S,R)$-bimodule is the same data as a $S⊗_{Z}R$-module. Geometrically, $S⊗R$ corresponds to the fiber product. Therefore, it is natural to look to objects on the (fiber) product for a source of functors.

For $X←π_{X}X×_{Z}Y→π_{Y}Y$ and $Q∈D(QcohX×_{Z}Y)$, we get a functor $Φ_{Q}:D(QcohX)E →D(QcohY)↦Rπ_{Y∗}(Lπ_{X}E⊗L Q) $

We call $Φ_{Q}$ an *integral transform* and $Q$ its *kernel*. If $Φ_{Q}$ is an equivalence, we
say $Φ_{Q}$ and $Q$ are *Fourier-Mukai* and $X$ and $Y$ are *Fourier-Mukai* partners.

The language is inspired by the analogy with integral transformations in analysis. $f(x)↦∫_{X}Q(x,y)f(x)dx.$

Each integral transform comes with a right adjoint thanks to Grothendieck duality.

**Lemma**. We have
$Φ_{Q}⊣Ψ_{Q}:=Rπ_{X∗}RHom_{X×Y}(Q,π_{Y}(−))$

# Identities, projections, and convolutions

Let's expand out analogy with integral transforms in analysis a bit more. There are three nice properties of such linear transformations:

**Identity kernels**. Let $Δ={(x,x)∈X×X}$ be the diagonal. Then, $∫_{X}δ_{Δ}(x,y)f(x)dx=f(y)$ for the $δ$-function along $Δ$.**Projection kernels**. For a kernel of the form $K(x,y)=e(x)h(y)$, we have $∫_{X}K(x,y)f(x)dx=h(y)∫_{X}e(x)f(x)dx$ is a type of projection onto $h(y)$.**Composition as convolution**. For $K(x,y)$ and $Q(y,z)$ the composition of $f(x)↦∫_{X}K(x,y)f(x)dx$ and $f(y)↦∫_{Y}Q(y,z)f(y)dy$ is also an integral transformation with kernel $Q⋆K:=∫_{Y}K(x,y)Q(y,z)dy$

These three properties still hold integral transforms of derived categories.

## Identity Kernels

Given a $k$-algebra $R$, we have a canonical exact sequence of $R⊗_{k}R$ -modules $0→I→R⊗_{k}R→R→0$ This embeds $SpecR$ as a closed subscheme of $SpecR×_{Speck}SpecR$ cut out by the sheaf of ideals $I$.

For a morphism of schemes $X→Y$, we have a corresponding map

$Δ:X→X×_{Y}X$
as the diagram
$X↓⏐ 1_{X}X 1_{X} X↓⏐ Y $
commutes.

We set $O_{Δ}:=Δ_{∗}O_{X}.$

**Proposition**. $Φ_{O_{Δ}}$ is isomorphic to the identity functor.

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

Suppressing the derived decoration, we have $Φ_{O_{Δ}}(E):=π_{X∗}(π_{X}E⊗Δ_{∗}O_{X})$ Using the projective formula, we get $π_{X∗}(π_{X}E⊗Δ_{∗}O_{X})≅π_{X∗}Δ_{∗}(O_{X}⊗Δ_{∗}π_{X}E)$ From the universal property of the fiber product, we have $π_{X}∘Δ=1_{X}$ so $π_{X∗}Δ_{∗}≅Id$ and $Δ_{∗}π_{∗}≅Id.$

▢## Projections Kernels

Given $E∈D(QcohX)$ and $F∈D(QcohY)$,
we set
$E⊠_{Z}F:=π_{X}E⊗π_{Y}F$
Such an object is called an *external product* of $E$ and $F$. We will omit $Z$
when the context is clear.

**Proposition**. Suppose $Z=Speck$ and $X$ and $Y$ are
Tor-independent over $k$. If $E$ is compact, we have a natural isomorphism
$Φ_{E_{∨}⊠F}(E_{′})≅RHom(E,E_{′})⊗L _{k}F$

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

We have $Φ_{E_{∨}⊠F}(E_{′}):=π_{Y∗}(π_{X}E_{′}⊗π_{X}E_{∨}⊗π_{Y}F)$ Using the projection formula gives $π_{Y∗}π_{X}(E_{∨}⊗E_{′})⊗F$ Since $E$ is compact, the natural map $E_{∨}⊗E_{′}→Hom(E,E_{′})$ is a quasi-isomorphism.

From our assumptions, the Cartesian square $X×_{k}Y↓⏐ p_{Y}X p_{X} Y↓⏐ Speck $ is Tor-independent. Thus, $π_{Y∗}π_{X}Hom(E,E_{′})⊗F≅p_{X}p_{Y∗}Hom(E,E_{′})⊗F$ And $p_{Y∗}Hom(E,E_{′})≅Hom(E,E_{′})$ giving the result.

▢In our analogy here with analysis, the Hom-spaces furnish the analog of a non-degenerate inner product.

## Convolution of kernels

Finally, given $Q∈D(QcohX×Y)$ and $K∈D(Qcoh$