# 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.