Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors (Q1729902)

The authors generalize the Galois correspondence and existence of Galois closure of a field to certain algebraic stacks over a field. More precisely, they introduce the notions of pseudo-properness and inflexibility of stacks. Then they consider a pseudo-proper and inflexible algebraic stack \(\mathcal X\) of finite type over a field \(k\) and an essentially finite cover \(f: {\mathcal Y}\to {\mathcal X}\). They also need some additional assumptions if \({\mathrm char} \, k>0\): either \(f\) is étale or \(\dim H^1 ({\mathcal X} , E)<\infty\) for all vector bundles \(E\). Then they show that there exists a unique (up to equivalence) finite map to the Nori fundamental gerbe \(\Pi ^{\mathrm N}_ {{\mathcal X}/k}\) of \({\mathcal X}/k\), whose base change along \({\mathcal X} \to \Pi ^{\mathrm N}_{{\mathcal X}/k}\) gives \(f\). They also prove some additional criteria on when \({{\mathcal Y}/k}\) is inflexible in case \(f\) is étale or a torsor. As a corollary they get a Galois correspondence between pointed essentially finite covers \(({\mathcal Y}, y)\to ({\mathcal X}, x)\) with inflexible \(\mathcal Y\) and subgroups of finite index in the Nori fundamental group of \(\mathcal X\). The authors prove also existence of a Galois closure for (pointed) towers of torsors under finite group schemes over a pseudo-proper and inflexible algebraic stack of finite type over a field. They also show that previous attempts to construct such closures fail and their assumptions in positive characteristic are necessary. In particular, the construction provided in [\textit{M. A. Garuti}, Proc. Am. Math. Soc. 137, No. 11, 3575--3583 (2009; Zbl 1181.14053)] is incorrect. Part of the paper is devoted to extension of the above results from the Nori set-up to the so called S-fundamental gerbes that are defined using numerically flat bundles.