Let \(S\) be a scheme and \(\Sch_{/S}\) the category of \(S\)-schemes with the étale topology (D-ETFPPF).
(Stack)
A stack over \(S\) is a category \(\mathcal{X}\) with a functor \(p \colon \mathcal{X} \to \Sch_{/S}\) such that
\(\mathcal{X}\) is fibred in groupoids: for every morphism \(T' \to T\) and object \(\xi\) over \(T\) there is a pullback \(\xi|_{T'}\) over \(T'\), unique up to unique isomorphism, and every morphism lying over an identity is an isomorphism;
for all objects \(\xi, \eta\) over \(T\), the presheaf \(\operatorname{Isom}(\xi, \eta) \colon (T' \to T) \mapsto \{\text{isomorphisms } \xi|_{T'} \to \eta|_{T'} \text{ over } T'\}\) is a sheaf on \((\Sch_{/T})_{\mathrm{et}}\);
every descent datum is effective: for an étale covering \(\{T_i \to T\}\), objects \(\xi_i\) over \(T_i\) with isomorphisms \(\varphi_{ij} \colon \xi_j|_{T_{ij}} \to \xi_i|_{T_{ij}}\) satisfying \(\varphi_{ij} \circ \varphi_{jk} = \varphi_{ik}\) on \(T_{ijk}\) come from an object \(\xi\) over \(T\).
A \(1\)-morphism \(\mathcal{Y} \to \mathcal{X}\) of stacks is representable if for every \(S\)-scheme \(T\) and \(T \to \mathcal{X}\), the \(2\)-fibre product \(\mathcal{Y} \times_{\mathcal{X}} T\) is an algebraic space (D-ALGSPACE). For a class \(P\) of morphisms of algebraic spaces that is stable under base change and étale local on the target (D-MORLOCAL), a representable \(\mathcal{Y} \to \mathcal{X}\) has \(P\) if every such base change \(\mathcal{Y} \times_{\mathcal{X}} T \to T\) has \(P\).
(Algebraic and Deligne--Mumford stacks)
A stack \(\mathcal{X}\) over \(S\) is algebraic if its diagonal \(\mathcal{X} \to \mathcal{X} \times_S \mathcal{X}\) is representable and there are a scheme \(U\) and a representable, smooth, surjective morphism \(U \to \mathcal{X}\), a smooth atlas. It is Deligne–Mumford if moreover it has an atlas \(U \to \mathcal{X}\) that is étale.
(Substacks)
A substack of a stack \(\mathcal{X}\) is a full subcategory \(\mathcal{Z} \subseteq \mathcal{X}\) closed under isomorphisms and pullbacks. It is an open, closed or locally closed substack if the inclusion \(\mathcal{Z} \to \mathcal{X}\) is representable by open immersions, closed immersions or immersions: for every \(S\)-scheme \(T\) and object of \(\mathcal{X}\) over \(T\), the base change \(\mathcal{Z} \times_{\mathcal{X}} T \to T\) is an open, closed or locally closed immersion of schemes.
(Smooth stack)
An algebraic stack \(\mathcal{X}\) over \(S\) is smooth over \(S\) if for some smooth atlas \(U \to \mathcal{X}\) the composite \(U \to S\) is smooth.
Smoothness of \(U \to S\) does not depend on the choice of smooth atlas.
(Orbifold)
An orbifold over \(\CC\) is a smooth, separated Deligne–Mumford stack of finite type over \(\CC\) whose stabilizer group at a general point is trivial.
The weighted projective line \(\PP(1,2)\) is the quotient stack \([(\AA^2 \setminus \{0\}) / \GG_m]\) for the action \(\lambda \cdot (x, y) = (\lambda x, \lambda^2 y)\) over \(\CC\). The atlas \(\AA^2 \setminus \{0\} \to \PP(1,2)\) is smooth, and the stabilizers are finite and reduced: a point \((x, y)\) with \(x \neq 0\) has trivial stabilizer, while \((0, 1)\) has stabilizer \(\{\lambda : \lambda^2 = 1\} = \mu_2\). So \(\PP(1,2)\) is a smooth separated Deligne–Mumford stack with trivial generic stabilizer, an orbifold with one point of stabilizer \(\mu_2\). The moduli stacks \(\mathcal{M}_g\) of smooth curves of genus \(g \geq 2\) are smooth Deligne–Mumford stacks (T-MGSMOOTH).
Dependencies
D-MORLOCALClasses of morphisms stable under base change and local on the source or targetusesD-ALGSPACEÉtale equivalence relations and algebraic spacesuses
Backlinks
D-ALGSPACEÉtale equivalence relations and algebraic spacesrelated-to