Lemma 17.21.1. In the situation described above. The sheaf \wedge ^ n\mathcal{F} is the sheafification of the presheaf
See Algebra, Section 10.13. Similarly, the sheaf \text{Sym}^ n\mathcal{F} is the sheafification of the presheaf
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
Here \text{T}^0(\mathcal{F}) = \mathcal{O}_ X, \text{T}^1(\mathcal{F}) = \mathcal{F} and for n \geq 2 we have
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
See Algebra, Section 10.13. Similarly, the sheaf \text{Sym}^ n\mathcal{F} is the sheafification of the presheaf
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
and similarly an exact sequence
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.
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}).
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}).
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}).
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.
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}).
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
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.
If \mathcal{F} is quasi-coherent, then so is each \text{T}(\mathcal{F}), \wedge (\mathcal{F}), and \text{Sym}(\mathcal{F}).
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
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