Notes on algebraic log stack (Q2831247)

The paper under review compares two different notions of stacks with a log structure, that have appeared and have been used in the literature before.NEWLINENEWLINEThe first notion, that of a log algebraic stack, is of an ``usual'' algebraic stack, together with a log structure on its lisse-étale site. The second type of object is given by stacks over the category of fine log schemes, that admit a ``smooth presentation'' of sorts (what the author calls ``algebraic log stacks'').NEWLINENEWLINEThere is a very natural functor from the first kind of objects to the second kind, whose essential image in the 2-category of fibered categories over fine log schemes has been studied by \textit{W. D. Gillam} [Int. J. Math. 23, No. 7, Article ID 1250069, 38 p. (2012; Zbl 1248.18008)], by means of what he calls minimal objects.NEWLINENEWLINEThe main result of this paper is that a stack over the category of fine log schemes (equipped with the strict fppf topology) that admits a ``flat atlas'', i.e. flat, strict, surjective, representable and locally finitely presented morphism from an ``algebraic log space'', is an algebraic log stack. This is the analogue of the classical fact that a stack for the fppf topology that has an fppf atlas, is actually algebraic.NEWLINENEWLINEFrom this result, the authors deduce that, through the functor mentioned above, the 2-category of log algebraic stacks is equivalent to the 2-category of algebraic log stacks. They furthermore prove some basic facts about algebraic log stacks, that are the close analogue of well-known facts about usual algebraic stacks.NEWLINENEWLINEOne caveat: the paper contains a mistake, in the fact that it is asserted that, through the correspondence described above, the category of log algebraic spaces is equivalent to the category of algebraic log spaces. There are counterexamples to this, for instance the quotient stack \([\mathrm{Spec } k[P]/ \widehat{P}]\) with its natural log structure, where \(P\) is a fine monoid and \(\widehat{P}\) denotes the algebraic group \(\mathrm{Hom}(P^{\mathrm{gp}}, \mathbb{G}_m)\). This stack is not an algebraic space, but the associated fibered category over fine log schemes (which is an algebraic log stack) is in fact equivalent to a functor (see Proposition 5.17 in [\textit{M. C. Olsson}, Ann. Sci. Éc. Norm. Supér. (4) 36, No. 5, 747--791 (2003; Zbl 1069.14022)]). An erratum detailing the necessary changes in the rest of the paper will appear soon.