Quasi-coherent sheaves on projective varieties

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

We constructed a functor $(-) : GrMod R \to Mod X$ where $X = Proj R$ and a functor in the other direction $\underline{Γ} (X, -) : Mod X F \to GrMod R \mapsto i \in Z ⨁ Γ (X, F (i))$

Let's think a little more about their relationsip. For any $m \in M_{l}$ , we get a map $m : R r \to M (l) \mapsto r m$ Applying $(-)$ , we get a map $O_{X} \to M (l)$ and hence an element of $Γ (X, M (l))$ . This gives a natural transformation $η_{M} : M \to \underline{Γ} (X, M)$

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

On $U_{x}$ , we have an isomorphism $1/ x^{s} : F (i s) \to F$ . Thus, we get an element $\frac{1}{x ^{s}} f ∣_{U_{x}} \in Γ (U_{x}, F)$ giving an element of $F (U_{x})$ .

Proposition. The natural transformations $η$ and $ϵ$ induce an adjunction $(-) ⊣ \underline{Γ}$ .

Proof. (Expand to view)

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

For the $ϵ_{M} \circ η_{M}$ , we check the map is the identity for sections over each principal open $U_{x}$ . Given $m / x^{i} \in M (U_{x})$ , we have $η_{M} (m / x^{i}) = \frac{η _{M} ( m )}{x ^{i}}$ restricting $η_{M} (m)$ back to $U_{x}$ gives $m \in Γ (U_{x}, M (∣ m ∣))$ .

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

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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 $x_{0}, \dots, x_{n}$ denote our generators in degree $1$ . We have a affine cover $U_{x_{j}}$ and $Z (x) \cap U_{x_{j}} = Z (x / x_{j}^{i})$ is a hypersurface in each $U_{j}$ . Then $M := Γ (U_{x_{j}}, F (i s))$ is a $R := (R_{x_{j}}) 0$ module and $f \in M$ . We know that $f = 0$ in $M_{x / x_{j}^{i}}$ so there is some $n_{j}$ with $\frac{x ^{n_{j}}}{x _{j}^{n_{j} i}} f = 0$ 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 $\underline{Γ} (F)$ x already and $ϵ_F$ is injective.

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

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Recall that for a graded $R$ -module $M$ , we have its torsion submodule $τ (M) := {m \in M ∣ \exists n_{0} > 0, (R_{+})^{n} m = 0 \forall n \geq n_{0}}$ 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 $0 \to τ (M) \to M \to M / τ (M) \to 0$ Since $τ (M) = 0$ there is a natural map $α_{M} : M / τ (M) \to \underline{Γ} (M)$

Lemma. Assume $R$ is finitely-generated over $k$ in degree $1$ by $x 0, \dots, x n$ and that each $x i$ is not a zero-divisor. Then $α$ M is injective

Proof. (Expand to view)

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

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

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

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

Since $\dots \to (R_{+})^{s + 1} \to (R_{+})^{s} \to \dots \to R_{+} \to R$ are all isomorphisms, we see that any $ϕ \in colim Hom ((R_{+})^{s}, M (l))$ induces a well-defined $ϕ : O \to M (l)$ and hence an element of $\underline{Γ} (M)$ .

Let's denote this map by $β_{M} : colim_{s} \underline{Hom} ((R_{+})^{s}, M) \to \underline{Γ} (M)$

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 $m_{f} \in M$ for each homogeneous $f \in R_{+}$ so that - there is $s_{f}$ such that $m_{f} / f^{s_{f}}$ has degree $l$ and - for any $f, g$ $\frac{m _{f}}{f ^{s_{f}}} = \frac{m _{g}}{g ^{s_{g}}}$ in $M_{f g}$ .

Let $x_{1}, \dots, x_{n}$ denote a set of homogeneous generators for $R$ over $k$ . We claim that the function $ϕ$ which assigns $ϕ (x_{i}^{s_{i}}) = m_{x_{i}}$ extends to a well-defined module map $(R_{+})^{s} \to M (l)$ 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 $x_{i}^{l_{i}}$ and $s_{i}$ with $s_{i} + l_{i}$ , we can assume that $m_{i} = \frac{x _{i}^{s_{i}}}{x _{j}^{s_{j}}} m_{j}$ in $M_{x_{j}}$ . If we have a relation of the form $a_{1} x_{1}^{s_{1}} + \dots + a_{n} x_{n}^{s_{n}} = 0$ then in $M_{x_{i}}$ we have $a_{1} m_{1} + \dots a_{n} m_{n} = a_{1} \frac{x _{1}^{s_{1}}}{x _{i}^{s_{i}}} m_{i} + \dots + a_{n} \frac{x _{1}^{s_{1}}}{x _{i}^{s_{i}}} m_{i} = \frac{m _{i}}{x _{i}^{s_{i}}} (a_{1} x_{1}^{s_{1}} + \dots + a_{n} x_{n}^{s_{n}}) = 0$ Thus, $a_{1} m_{1} + \dots + a_{n} m_{n}$ is annihilated by some power of each $x_{i}$ . Passing to a sufficient high power of $R_{+}$ forces this to vanish.

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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 $0 \to τ (I) \to I \to \underline{Γ} (I) \to 0$

Proof. (Expand to view)

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

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Note the map $M \to \underline{Γ} (M)$ is not always onto.

Example. Take $k [x]$ with $∣ x ∣ = 1$ . Then, $\underline{Γ} (k [x]) = k [x, x^{- 1}]$

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

Proposition. Assume that $R$ is finitely-generated by elements of degree $1$ over $k$ . Then there are semi-orthogonal decompositions $D^{†} (GrMod R) = ⟨ D^{†} (GrMod_{Tors} R), D^{†} (Qcoh Proj R)⟩$ where $† \in {b, +, -, \emptyset}$ .

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 $0 \to τ (I) \to I \to \underline{Γ} (I^{∙}) \to 0$ and thus a triangle $τ (I) \to I \to \underline{Γ} (I^{∙}) \to τ (I) [1]$ Each component of $τ (I)$ is torsion.

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

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

Using adjunction, we have $Hom (\underline{Γ} (I), \underline{Γ} (J)) ≅ Hom (I, \underline{Γ} (J))$ And $Hom (I, \underline{Γ} (J)) ≅ Hom (I, J)$ since $\underline{Γ} (J) \to J$ is an isomorphism.

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

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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 $U = Spec R ∖ {R_{+}}$ instead of $Proj R$ .
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.

Derived Categories II

Quasi-coherent sheaves on projective varieties