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Let $G$ be a finite group acting on a commutative ring $R$ via ring maps. In doubt, one can assume $R$ to be noetherian or regular if one wants. Let $P$ be a $1$-dimensional free $R$-module with a $G$ …

My current office mate and I found out that this question has a rather simple answer, namely: Yes. I use the same notation as in the question. Let $(x)$ be a basis of $P$. Consider the $|G|$-fold te …

It is well known and more or less proven in Hartshorne's 'Algebraic Geometry' (p. 209) that for every noetherian scheme $X$ and every collection of abelian sheaves $\mathcal{F}_i$ the canonical map $ …

I asked myself whether there is an extension of this theorem to (Artin/Deligne-Mumford) stacks. … More precisely: Question: Can one characterize the class of (Artin/Deligne-Mumford) stacks $X$ such that $H^i(X; \mathcal{F}) = 0$ for all quasi-coherent sheaves $\mathcal{F}$ on $X$ and all $i>0$? …

I always found Algebraic Stacks by Tomas Gomez to be a very readable quick introduction. …

The stacks should "come up in nature". Obvious examples are classifying stacks of etale group schemes and, more interestingly, the moduli stack of elliptic curves. Are there more examples? … Edit: I'm especially interested in stacks that naturally arise as "moduli stacks of something" (although this is, of course, not a well-defined mathematical category). …

This seems to be an important example of an algebraic stack, but I am unable to find a reference where basic facts about weighted projective stacks are proven. …