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Hi Stefan! Torsion-free (coherent) sheaves are pure sheaves of codimension 0; I suggest that you try "The Geometry of Moduli Spaces of Sheaves", by Daniel Huybrechts and Manfred Lehn, where pure sheav …

It does generalize to Chow classes and to (classical or étale) rational cohomology. For Chow classes this is part of the basics of intersection theory of stacks, for example, in my paper Intersection …

This is true, if you take étale cohomology with coefficients in a finite abelian group of order not divisible by the characteristic. You can embed $Y$ in the corresponding line bundle $L \to X$. Then …

The treatment in EGA is indeed intimidating, but in fact over a field the formula is not hard to prove. You only need $X$ and $Y$ to be separated schemes over $k$, and $\mathcal{F}$ and $\mathcal{G}$ …

I assume that you are over an algebraically closed field. If $G$ is connect, the action is trivial, because any affine algebraic group is rational, so every point can be connected via a chain of open …

The split Grothendieck group for vector bundles on a complete variety appears in Nori's PhD thesis on the fundamental group scheme; this was published in the Proceedings of the Indian Academy of Scien …

I don't think this is true. Take $X = \mathbb P^1 \times \mathbb P^2$. Let $C$ be $\mathbb P^1 \times 0$, and let $C_1$ be the first infinitesimal neighborhood of $C$. The curve $C_1$ is the relative …

If by $V/G$ you mean the space of orbits, this is not true. Consider $\mathbb C^*$ acting on the affine space $\mathbb A^1$ by multiplication; the space of orbits has two points, but the only variety …

The notion of anti-analytic involution is perfectly well defined for general analytic spaces: it is an involution of locally ringed spaces, that is antilinear with respect to complex scalars. Any pro …