Loading web-font TeX/Main/Regular

The Stacks project

Lemma 78.25.2. Notation and assumption as in Lemma 78.21.1. The morphism of quotient stacks

[f] : [U/R] \longrightarrow [U'/R']

is an equivalence if and only if

  1. (U, R, s, t, c) is the restriction of (U', R', s', t', c') via f : U \to U', and

  2. the map

    \xymatrix{ U \times _{f, U', t'} R' \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^ h & R' \ar[r]_{s'} & U' }

    is a surjection of sheaves.

Part (2) holds for example if \{ h : U \times _{f, U', t'} R' \to U'\} is an fppf covering, or if f : U \to U' is a surjection of sheaves, or if \{ f : U \to U'\} is an fppf covering.

Proof. We already know that part (1) is equivalent to fully faithfulness by Lemma 78.25.1. Hence we may assume that (1) holds and that [f] is fully faithful. Our goal is to show, under these assumptions, that [f] is an equivalence if and only if (2) holds. We may use Stacks, Lemma 8.4.8 which characterizes equivalences.

Assume (2). We will use Stacks, Lemma 8.4.8 to prove [f] is an equivalence. Suppose that T is a scheme and x' \in \mathop{\mathrm{Ob}}\nolimits ([U'/R']_ T). There exists a covering \{ g_ i : T_ i \to T\} such that g_ i^*x' is the image of some element a'_ i \in U'(T_ i), see Lemma 78.24.1. Hence we may assume that x' is the image of a' \in U'(T). By the assumption that h is a surjection of sheaves, we can find an fppf covering \{ \varphi _ i : T_ i \to T\} and morphisms b_ i : T_ i \to U \times _{g, U', t'} R' such that a' \circ \varphi _ i = h \circ b_ i. Denote a_ i = \text{pr}_0 \circ b_ i : T_ i \to U. Then we see that a_ i \in U(T_ i) maps to f \circ a_ i \in U'(T_ i) and that f \circ a_ i \cong _{T_ i} h \circ b_ i = a' \circ \varphi _ i, where \cong _{T_ i} denotes isomorphism in the fibre category [U'/R']_{T_ i}. Namely, the element of R'(T_ i) giving the isomorphism is \text{pr}_1 \circ b_ i. This means that the restriction of x to T_ i is in the essential image of the functor [U/R]_{T_ i} \to [U'/R']_{T_ i} as desired.

Assume [f] is an equivalence. Let \xi ' \in [U'/R']_{U'} denote the object corresponding to the identity morphism of U'. Applying Stacks, Lemma 8.4.8 we see there exists an fppf covering \mathcal{U}' = \{ g'_ i : U'_ i \to U'\} such that (g'_ i)^*\xi ' \cong [f](\xi _ i) for some \xi _ i in [U/R]_{U'_ i}. After refining the covering \mathcal{U}' (using Lemma 78.24.1) we may assume \xi _ i comes from a morphism a_ i : U'_ i \to U. The fact that [f](\xi _ i) \cong (g'_ i)^*\xi ' means that, after possibly refining the covering \mathcal{U}' once more, there exist morphisms r'_ i : U'_ i \to R' with t' \circ r'_ i = f \circ a_ i and s' \circ r'_ i = \text{id}_{U'} \circ g'_ i. Picture

\xymatrix{ U \ar[d]^ f & & U'_ i \ar[ll]^{a_ i} \ar[ld]_{r'_ i} \ar[d]^{g'_ i} \\ U' & R' \ar[l]_{t'} \ar[r]^{s'} & U' }

Thus (a_ i, r'_ i) : U'_ i \to U \times _{g, U', t'} R' are morphisms such that h \circ (a_ i, r'_ i) = g'_ i and we conclude that \{ h : U \times _{g, U', t'} R' \to U'\} can be refined by the fppf covering \mathcal{U}' which means that h induces a surjection of sheaves, see Topologies on Spaces, Lemma 73.7.5.

If \{ h\} is an fppf covering, then it induces a surjection of sheaves, see Topologies on Spaces, Lemma 73.7.5. If U' \to U is surjective, then also h is surjective as s has a section (namely the neutral element e of the groupoid in algebraic spaces). \square


Comments (0)


Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.