A relative notion of algebraic Lie group and applications to n-stacks

Let X be the big etale site of schemes over k = C. If S is a scheme of finite type over k, let X/S denote the big etale site of schemes over S. The goal of this paper is to introduce a full subcategory of the category of sheaves of groups on X/S, which we will call the category of presentable group sheaves (§2), with the following properties. 1. The category of presentable group sheaves contains those group sheaves which are representable by group schemes of finite type over S (Corollary 2.6). 2. The category of presentable group sheaves is closed under kernel, quotient (by a normal subgroup sheaf which is presentable), and extension (Theorem 1.13). 3. If S′ → S is a morphism then pullback takes presentable group sheaves on S to presentable group sheaves on S′ (Lemma 3.2). 4. If S′ → S is a finite morphism then direct image takes presentable group sheaves on S′ to presentable group sheaves on S (Lemma 3.3). 5. If S = Spec(k) then presentable group sheaves are just group schemes of finite type over Spec(k) (Theorem 6.4). In particular if G is a presentable group sheaf over any S then the pullback to each point Spec(k) → S is an algebraic group. 6. There is a notion of connectedness extending the usual notion over Spec(k) and compatible with quotients, extensions, pullbacks and finite direct images; and a presentable group sheaf G has a largest connected presentable subsheaf G0 ⊂ G which we call the connected component (Theorem 7.2). 7. A presentable group sheaf G has a Lie algebra object Lie(G) (Theorem 9.1) which is a vector sheaf with bracket operation (see below for a discussion of the notion of vector sheaf—in the case S = Spec(k) it is the same thing as a finite dimensional k-vector space). 8. If G is a connected presentable group sheaf then G/Z(G) is determined up to isomorphism by the Lie algebra sheaf Lie(G) (where Z(G) denotes the center of G). This is Theorem 9.6 below.