## 17.21 Symmetric and exterior powers

Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. We define the tensor algebra of $\mathcal{F}$ to be the sheaf of noncommutative $\mathcal{O}_ X$-algebras

$\text{T}(\mathcal{F}) = \text{T}_{\mathcal{O}_ X}(\mathcal{F}) = \bigoplus \nolimits _{n \geq 0} \text{T}^ n(\mathcal{F}).$

Here $\text{T}^0(\mathcal{F}) = \mathcal{O}_ X$, $\text{T}^1(\mathcal{F}) = \mathcal{F}$ and for $n \geq 2$ we have

$\text{T}^ n(\mathcal{F}) = \mathcal{F} \otimes _{\mathcal{O}_ X} \ldots \otimes _{\mathcal{O}_ X} \mathcal{F} \ \ (n\text{ factors})$

We define $\wedge (\mathcal{F})$ to be the quotient of $\text{T}(\mathcal{F})$ by the two sided ideal generated by local sections $s \otimes s$ of $\text{T}^2(\mathcal{F})$ where $s$ is a local section of $\mathcal{F}$. This is called the exterior algebra of $\mathcal{F}$. Similarly, we define $\text{Sym}(\mathcal{F})$ to be the quotient of $\text{T}(\mathcal{F})$ by the two sided ideal generated by local sections of the form $s \otimes t - t \otimes s$ of $\text{T}^2(\mathcal{F})$.

Both $\wedge (\mathcal{F})$ and $\text{Sym}(\mathcal{F})$ are graded $\mathcal{O}_ X$-algebras, with grading inherited from $\text{T}(\mathcal{F})$. Moreover $\text{Sym}(\mathcal{F})$ is commutative, and $\wedge (\mathcal{F})$ is graded commutative.

Lemma 17.21.1. In the situation described above. The sheaf $\wedge ^ n\mathcal{F}$ is the sheafification of the presheaf

$U \longmapsto \wedge ^ n_{\mathcal{O}_ X(U)}(\mathcal{F}(U)).$

See Algebra, Section 10.13. Similarly, the sheaf $\text{Sym}^ n\mathcal{F}$ is the sheafification of the presheaf

$U \longmapsto \text{Sym}^ n_{\mathcal{O}_ X(U)}(\mathcal{F}(U)).$

Proof. Omitted. It may be more efficient to define $\text{Sym}(\mathcal{F})$ and $\wedge (\mathcal{F})$ in this way instead of the method given above. $\square$

Lemma 17.21.2. In the situation described above. Let $x \in X$. There are canonical isomorphisms of $\mathcal{O}_{X, x}$-modules $\text{T}(\mathcal{F})_ x = \text{T}(\mathcal{F}_ x)$, $\text{Sym}(\mathcal{F})_ x = \text{Sym}(\mathcal{F}_ x)$, and $\wedge (\mathcal{F})_ x = \wedge (\mathcal{F}_ x)$.

Proof. Clear from Lemma 17.21.1 above, and Algebra, Lemma 10.13.5. $\square$

Lemma 17.21.3. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ Y$-modules. Then $f^*\text{T}(\mathcal{F}) = \text{T}(f^*\mathcal{F})$, and similarly for the exterior and symmetric algebras associated to $\mathcal{F}$.

Proof. Omitted. $\square$

Lemma 17.21.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F} \to 0$ be an exact sequence of sheaves of $\mathcal{O}_ X$-modules. For each $n \geq 1$ there is an exact sequence

$\mathcal{F}_2 \otimes _{\mathcal{O}_ X} \text{Sym}^{n - 1}(\mathcal{F}_1) \to \text{Sym}^ n(\mathcal{F}_1) \to \text{Sym}^ n(\mathcal{F}) \to 0$

and similarly an exact sequence

$\mathcal{F}_2 \otimes _{\mathcal{O}_ X} \wedge ^{n - 1}(\mathcal{F}_1) \to \wedge ^ n(\mathcal{F}_1) \to \wedge ^ n(\mathcal{F}) \to 0$

Proof. See Algebra, Lemma 10.13.2. $\square$

Lemma 17.21.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules.

1. If $\mathcal{F}$ is locally generated by sections, then so is each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$.

2. If $\mathcal{F}$ is of finite type, then so is each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$.

3. If $\mathcal{F}$ is of finite presentation, then so is each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$.

4. If $\mathcal{F}$ is coherent, then for $n > 0$ each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$ is coherent.

5. If $\mathcal{F}$ is quasi-coherent, then so is each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$.

6. If $\mathcal{F}$ is locally free, then so is each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$.

Proof. These statements for $\text{T}^ n(\mathcal{F})$ follow from Lemma 17.16.6.

Statements (1) and (2) follow from the fact that $\wedge ^ n(\mathcal{F})$ and $\text{Sym}^ n(\mathcal{F})$ are quotients of $\text{T}^ n(\mathcal{F})$.

Statement (6) follows from Algebra, Lemma 10.13.1.

For (3) and (5) we will use Lemma 17.21.4 above. By locally choosing a presentation $\mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F} \to 0$ with $\mathcal{F}_ i$ free, or finite free and applying the lemma we see that $\text{Sym}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$ has a similar presentation; here we use (6) and Lemma 17.16.6.

To prove (4) we will use Algebra, Lemma 10.13.3. We may localize on $X$ and assume that $\mathcal{F}$ is generated by a finite set $(s_ i)_{i \in I}$ of global sections. The lemma mentioned above combined with Lemma 17.21.1 above implies that for $n \geq 2$ there exists an exact sequence

$\bigoplus \nolimits _{j \in J} \text{T}^{n - 2}(\mathcal{F}) \to \text{T}^ n(\mathcal{F}) \to \text{Sym}^ n(\mathcal{F}) \to 0$

where the index set $J$ is finite. Now we know that $\text{T}^{n - 2}(\mathcal{F})$ is finitely generated and hence the image of the first arrow is a coherent subsheaf of $\text{T}^ n(\mathcal{F})$, see Lemma 17.12.4. By that same lemma we conclude that $\text{Sym}^ n(\mathcal{F})$ is coherent. $\square$

Lemma 17.21.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules.

1. If $\mathcal{F}$ is quasi-coherent, then so is each $\text{T}(\mathcal{F})$, $\wedge (\mathcal{F})$, and $\text{Sym}(\mathcal{F})$.

2. If $\mathcal{F}$ is locally free, then so is each $\text{T}(\mathcal{F})$, $\wedge (\mathcal{F})$, and $\text{Sym}(\mathcal{F})$.

Proof. It is not true that an infinite direct sum $\bigoplus \mathcal{G}_ i$ of locally free modules is locally free, or that an infinite direct sum of quasi-coherent modules is quasi-coherent. The problem is that given a point $x \in X$ the open neighbourhoods $U_ i$ of $x$ on which $\mathcal{G}_ i$ becomes free (resp. has a suitable presentation) may have an intersection which is not an open neighbourhood of $x$. However, in the proof of Lemma 17.21.5 we saw that once a suitable open neighbourhood for $\mathcal{F}$ has been chosen, then this open neighbourhood works for each of the sheaves $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$ and $\text{Sym}^ n(\mathcal{F})$. The lemma follows. $\square$

Comment #1807 by OS on

The second sentence of this section should be "Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules." (or maybe just "an $\mathcal{O}_X$-module") and not "Let $\mathcal{F}$ be an $\mathcal{O}_X$-algebra."

Comment #4180 by udhav on

Do you have a proof for Lemma 17.19.4 ?

Comment #4376 by on

@#4180: As said in the proof, this lemma follows from Lemma 10.13.2 perhaps combined with Lemma 17.21.1. Proving these kinds of lemmas is an excellent way to get used to this material.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).