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17.19 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 17.19.1. 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 10.12. 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 17.19.2. 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 17.19.1 above, and Algebra, Lemma 10.12.4. $\square$

Lemma 17.19.3. 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 17.19.4. 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 10.12.2. $\square$

Lemma 17.19.5. 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 17.15.5.

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 10.12.1.

For (3) and (5) we will use Lemma 17.19.4 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 17.15.5.

To prove (4) we will use Algebra, Lemma 10.12.3. 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 17.19.1 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 17.19.6. 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 17.19.5 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. $\square$


Comments (1)

Comment #1807 by OS on

The second sentence of this section should be "Let be a sheaf of -modules." (or maybe just "an -module") and not "Let be an -algebra."


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