Affine group schemes over symmetric monoidal categories
Let \(S\) be a scheme. A group scheme \(G\) over \(S\) is said to have finite order \(r\) over \(S\) if \(G\) can be written as \(\text{Spec}(A)\), where \(A\) is a sheaf of \(\mathcal{O}_S\)-algebras which is locally free of finite rank \(r\). In [\textit{J. Tate} and \textit{F. Oort}, Ann. Sci. Ec. Norm. Super. (4), No. 3, 1--21 (1970; Zbl 0195.50801)], the authors presented the following result, due to Deligne: A commutative group scheme \(G\) over \(S\) of order \(r\) is annihilated by \(r\). In the paper under review, the author proves a similar result for group schemes over symmetric monoidal categories. Namely, let \((C, \otimes, 1)\) be an abelian, closed, \(\mathbb{C}\)-linear symmetric monoidal category and let \(G\) be an affine commutative group scheme over \(C\) free and of finite rank \(r\). Then \(G\) is annihilated by \(r\). Here, an affine group scheme over \(C\) is a covariant functor from the category of algebras in \(C\) to the category of groups, with the usual compatibility conditions.NEWLINENEWLINEIt must be said here that the subject of relative algebraic geometry over a symmetric monoidal category is has been developed in the papers [\textit{P. Deligne}, in: The Grothendieck Festschrift, Collect. Artic. in honor of the 60th birthday of A. Grothendieck. Vol II, Prog. Math. No. 87, 111--195 (1990; Zbl 0727.14010); \textit{M. Hakim}, Topos anneles et schemas relatifs. Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 64. Berlin-Heidelberg-New York: Springer-Verlag. (1972; Zbl 0246.14004); \textit{B. Toen} and \textit{M. Vaquie}, J. K-Theory 3, No. 3, 437--500 (2009; Zbl 1177.14022)]. The current paper contributes another result to this theory.
scientific article; zbMATH DE number 6029225