The Stacks project

Proposition 96.13.2. The stack $\mathcal{H}_ d$ is equivalent to the quotient stack $[X/G]$ described above. In particular $\mathcal{H}_ d$ is an algebraic stack.

Proof. Note that by Groupoids in Spaces, Definition 77.20.1 the quotient stack $[X/G]$ is the stackification of the category fibred in groupoids associated to the “presheaf in groupoids” which associates to a scheme $T$ the groupoid

\[ (X(T), G(T) \times X(T), s, t, c). \]

Since this “presheaf in groupoids” is isomorphic to $FA_ d$ by Lemma 96.13.1 it suffices to prove that the $\mathcal{H}_ d$ is the stackification of (the category fibred in groupoids associated to the “presheaf in groupoids”) $FA_ d$. To do this we first define a functor

\[ \mathop{\mathrm{Spec}}: FA_ d \longrightarrow \mathcal{H}_ d \]

Recall that the fibre category of $\mathcal{H}_ d$ over a scheme $T$ is the category of finite locally free morphisms $Z \to T$ of degree $d$. Thus given a scheme $T$ and a free $d$-dimensional $\Gamma (T, \mathcal{O}_ T)$-algebra $m$ we may assign to this the object

\[ Z = \underline{\mathop{\mathrm{Spec}}}_ T(\mathcal{A}) \]

of $\mathcal{H}_{d, T}$ where $\mathcal{A} = \mathcal{O}_ T^{\oplus d}$ endowed with a $\mathcal{O}_ T$-algebra structure via $m$. Moreover, if $m'$ is a second such free $d$-dimensional $\Gamma (T, \mathcal{O}_ T)$-algebra and if $\varphi : m \to m'$ is an isomorphism of these, then the induced $\mathcal{O}_ T$-linear map $\varphi : \mathcal{O}_ T^{\oplus d} \to \mathcal{O}_ T^{\oplus d}$ induces an isomorphism

\[ \varphi : \mathcal{A}' \longrightarrow \mathcal{A} \]

of quasi-coherent $\mathcal{O}_ T$-algebras. Hence

\[ \underline{\mathop{\mathrm{Spec}}}_ T(\varphi ) : \underline{\mathop{\mathrm{Spec}}}_ T(\mathcal{A}) \longrightarrow \underline{\mathop{\mathrm{Spec}}}_ T(\mathcal{A}') \]

is a morphism in the fibre category $\mathcal{H}_{d, T}$. We omit the verification that this construction is compatible with base change so we get indeed a functor $\mathop{\mathrm{Spec}}: FA_ d \to \mathcal{H}_ d$ as claimed above.

To show that $\mathop{\mathrm{Spec}}: FA_ d \to \mathcal{H}_ d$ induces an equivalence between the stackification of $FA_ d$ and $\mathcal{H}_ d$ it suffices to check that

  1. $\mathit{Isom}(m, m') = \mathit{Isom}(\mathop{\mathrm{Spec}}(m), \mathop{\mathrm{Spec}}(m'))$ for any $m, m' \in FA_ d(T)$.

  2. for any scheme $T$ and any object $Z \to T$ of $\mathcal{H}_{d, T}$ there exists a covering $\{ T_ i \to T\} $ such that $Z|_{T_ i}$ is isomorphic to $\mathop{\mathrm{Spec}}(m)$ for some $m \in FA_ d(T_ i)$, and

see Stacks, Lemma 8.9.1. The first statement follows from the observation that any isomorphism

\[ \underline{\mathop{\mathrm{Spec}}}_ T(\mathcal{A}) \longrightarrow \underline{\mathop{\mathrm{Spec}}}_ T(\mathcal{A}') \]

is necessarily given by a global invertible matrix $g$ when $\mathcal{A} = \mathcal{A}' = \mathcal{O}_ T^{\oplus d}$ as modules. To prove the second statement let $\pi : Z \to T$ be a finite locally free morphism of degree $d$. Then $\mathcal{A}$ is a locally free sheaf $\mathcal{O}_ T$-modules of rank $d$. Consider the element $1 \in \Gamma (T, \mathcal{A})$. This element is nonzero in $\mathcal{A} \otimes _{\mathcal{O}_{T, t}} \kappa (t)$ for every $t \in T$ since the scheme $Z_ t = \mathop{\mathrm{Spec}}(\mathcal{A} \otimes _{\mathcal{O}_{T, t}} \kappa (t))$ is nonempty being of degree $d > 0$ over $\kappa (t)$. Thus $1 : \mathcal{O}_ T \to \mathcal{A}$ can locally be used as the first basis element (for example you can use Algebra, Lemma 10.79.4 parts (1) and (2) to see this). Thus, after localizing on $T$ we may assume that there exists an isomorphism $\varphi : \mathcal{A} \to \mathcal{O}_ T^{\oplus d}$ such that $1 \in \Gamma (\mathcal{A})$ corresponds to the first basis element. In this situation the multiplication map $\mathcal{A} \otimes _{\mathcal{O}_ T} \mathcal{A} \to \mathcal{A}$ translates via $\varphi $ into a free $d$-dimensional algebra $m$ over $\Gamma (T, \mathcal{O}_ T)$. This finishes the proof. $\square$


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