The Cotangent Stack

1.2. We refer to [LMB] for the basic background on stacks. Let us quickly recall: a stack over a scheme S is a sheaf of groupoids on the faithfully flat topology of S-schemes, where the presheaf requirement that the composition of the restriction maps is the restriction map of the composition is understood in the weak sense. Stacks form a 2-category with fiber products and there is a clear embedding of the category of schemes as a full subcategory. A morphism X −→ Y is representable if for any map to Y from a scheme the pull-back of X along this map is an algebraic space. For representable morphisms, it makes sense to say it has a local property like flatness or smoothness, and we will call such morphisms flat or smooth withholding the word representable where convenient. An Artin (or algebraic) stack is a stack X for which there exists a smooth presentation, i.e., a smooth, surjective map X −→X from a scheme X, and such that the diagonal map X −→X ×X is representable and quasicompact. Note that to say the diagonal map is representable is equivalent to requiring that any morphism from a scheme to X is representable. By this condition, for any point x : S −→X from a scheme to a stack, one can make sense of the automorphisms of this point Aut(x) = S ×sX S, and if this stack is an Artin stack then this is naturally given the structure of a group scheme over S. An accessible example of an algebraic stack is given by taking the quotient stack X/G where G is an algebraic group acting on X. The S-points of this stack are given by the groupoid of G-torsors on S equipped with an equivariant map to X. An Artin stack X is smooth if for every scheme U mapping smoothly to X , the scheme U is smooth. One can check that a stack X is smooth if and only if there is a smooth scheme U equipped with a smooth epimorphism U −→X . In general, it makes sense to talk about properties of X which are local properties for the smooth topology. We denote by Xsm the smooth topology of X , i.e., it is the site for which the underlying category is that of schemes U equipped with a smooth map U πU −→X and for which morphisms are “smooth 2-morphisms” and for which covering maps are