Introduction to algebraic stacks (Q2875828)

The lecture note provides a self-contained introductory course on the theory of algebraic stacks. More than a half of the notes is devoted to describe the motivation behind the notion of stack and moduli stack in a very intuitive manner. The intuition for stacks is provided by considering families of triangles. The notion of moduli space for triangles is defined and then it is explained why a fine moduli space for triangles does not exist. The reader learns intuitively that the obstruction for the existence of such a fine moduli space is caused by the symmetric objects, namely, the objects with non-trivial symmetries (in this context, isosceles triangles): using the symmetries, one can cut a family of triangles at the symmetric objects and glue in such a way that the new family has the same moduli map as the original one but is not isomorphic to it. On the other hand, it is shown that there exists a fine moduli space for scalene triangles, i.e., triangles with no (non-trivial) symmetry. The notion of stack for families of triangles is described and motivated naturally and through various pictorial examples. Consider the set \(N=\{(a,b,c)\in \mathbb{R}^{3}\mid a+b+c=2, a,b,c<1\}\) and the family of triangles \(\mathcal{N}/N\) over it, where each fiber \(\mathcal{N}_{(a,b,c)}\) is the triangle with side lengths \(a,b\) and \(c\). It is justified that \(\mathcal{N}/N\) is a versal family for triangles: Every family over any base is a pull-back of \(\mathcal{N}/N\). During the course of showing this, it is motivated in a natural and intuitive manner what a versal family is and what task should it fulfill. If \(\mathcal{F}/T\) is a family of triangles over a base \(T\), one can consider the cover \(T^{\prime}\to T\) to be the \(S_3\)-cover related to 6 different labelings of triangles in the family. This defines a continuous map \(f:T^{\prime}\to N\) which maps a labelled triangle to the triple of lengths of its sides. In order to have a versal family, one needs to replace the usual notion of moduli space by that of \textit{generalized moduli maps} \((T^{\prime}/T,f)\). Any family \(\mathcal{F}/T\) is completely determined by its generalized moduli map. A stack which admits a versal family, and the stacks that are considered in the notes are so, is essentially equal to the stack of ``generalized moduli maps''. The second half of the lecture note is devoted to the formalism of stacks. This part is a quite standard introduction and covers several types of stacks from topological to algebraic ones.NEWLINENEWLINEFor the entire collection see [Zbl 1286.14002].