Commutative algebraic groups up to isogeny
Let \(k\) be an arbitrary field and let \(\mathcal{G}_k\) be the category with objects the commutative group schemes of finite type over \(k\) and with morphisms the homomorphisms of \(k\)-group schemes. By a result of Grothendieck \(\mathcal{G}_k\) is an abelian category. Moreover, by results of Serre and Oort, \(\mathcal{G}_k\) has homological dimension \(1\) (resp. \(2\)) if \(k\) is algebraically closed of characteristic \(0\) (resp. positive). However, Milne proved that the homological dimension of \(\mathcal{G}_k\) can be arbitrary large if \(k\) is non-algebraically closed. In this article, the author studies the quotient category \(\mathcal{G}_k/\mathcal{F}_k\), where \(\mathcal{F}_k\) is the Serre subcategory of commutative finite group schemes. Then \(\mathcal{G}_k/\mathcal{F}_k\) is obtained from \(\mathcal{G}_k\) by inverting all isogenies (i.e. all morphisms with finite kernels and cokernels). The full subcategory \({\mathcal{C}_k}\) of \(\mathcal{G}_k/\mathcal{F}_k\) with objects the smooth connected commutative group schemes of finite type over \(k\) turns out to be equivalent to \(\mathcal{G}_k/\mathcal{F}_k\) (Lemma 3.1). Therefore the author works mainly with \({\mathcal{C}}_k\) instead of \(\mathcal{G}_k/\mathcal{F}_k\) as the morphisms in \({\mathcal{C}}_k\) admit a simpler description. Let \({\mathcal{A}}_k\) / \({\mathcal{T}}_k\) / \({\mathcal{U}}_k\) / \({\mathcal{L}}_k\) / \({\mathcal{S}}_k\) denote the full subcategories of \({\mathcal{C}}_k\) with objects abelian varieties / tori / unipotent algebraic groups / linear algebraic groups / semi-abelian varieties. The main results of the article are the following: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] The homological dimension of \({\mathcal{C}}_k\) is \(1\) (\S 4.3). \item[{\(\bullet\)}] The base change under any purely inseparable field extension \(k'\) of \(k\) yields an equivalence of categories \({\mathcal{C}}_k \to {\mathcal{C}}_{k'}\) (Theorem 3.11). \item[{\(\bullet\)}] The category \({\mathcal{C}}_k\) is artinian and noetherian. Its simple objects are the additive group \(\mathbb{G}_{a_k}\), the simple tori, and the simple abelian varieties (Proposition 3.2). \item[{\(\bullet\)}] A description of the projective objects (Theorem 5.14) and the injective objects (Theorem 5.16) of the category \({\mathcal{C}}_k\). \item[{\(\bullet\)}] The product functor \({\mathcal{T}}_k \times {\mathcal{U}}_k \to {\mathcal{L}}_k\) yields an equivalence of categories (Proposition 5.9). \item[{\(\bullet\)}] If char\((k)=0\), then \({\mathcal{C}}_k\) is equivalent to \({\mathcal{V}}_k \times_{{\mathcal{A}}_k} {\mathcal{S}}_k\) (Proposition 5.13), where \({\mathcal{V}}_k\) is the full subcategory of \({\mathcal{C}}_k\) with objects the vector extensions of all abelian varieties (see \S 5.1). \item[{\(\bullet\)}] If char\((k)>0\), then the product functor \({\mathcal{S}}_k \times {\mathcal{U}}_k \to {\mathcal{C}}_k\) yields an equivalence of categories (Proposition 5.10). If in addition \(k\) is locally finite, then the product functor \({\mathcal{T}}_k \times {\mathcal{A}}_k \to {\mathcal{S}}_k\) yields an equivalence of categories as well (Proposition 5.8). \end{itemize}} Let us note that in [``Commutative algebraic groups up to isogeny. II'', \url{arXiv:1612.03634}], the author develops a representation-theoretic approach to the category \({\mathcal{C}}_k\) and that in [``Homological dimension of isogeny categories of commutative algebraic groups'', \url{arXiv:1709.00175}] he determines the homological dimension of various other isogeny categories of commutative algebraic groups over \(k\).
scientific article