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In general suppose I have an object $E$ living over a field $K$ (this could be a $K$-algebra of some kind or a scheme over $K$) such that I can talk about its extension of scalars $E_L$ along a field …

If you remember the tensor product then in nice cases $X \mapsto \text{QCoh}(X)$ is fully faithful, even for some nice stacks; this is a generalization of Tannaka-Krein duality due to Lurie in Tannaka …

This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). Vic …

Here is a concrete example which may be useful to think about. It is not exactly the situation you describe but is similar. Let $k$ be a field of characteristic zero and let $R = k[x, y]/(y^2 - x^3 - …

This is an extended comment on KConrad's discussion of symmetry groups. We can think of $k$-forms on a vector space $V$ (homogeneous polynomials of degree $k$) abstractly as elements of the symmetric …

Here is an alternate approach, more algebraic and less geometric. As in Noam's answer we'll consider the more general equation $x^2 - Dy^2 \equiv 1 \bmod p$. Consider the $\mathbb{F}_p$-algebra $A = \ …

We have $$\log \sum_{k \ge 0} T_k t^k = \sum_{i=1}^n \log \frac{x_i t}{1 - e^{-x_i t}}$$ so if we write $$\log \frac{x_i t}{1 - e^{-x_i t}} = \log \sum_{k \ge 0} B_k^{+} x_i^k \frac{t^k}{k!} = \sum_{k …

They are not. $H^0(\Gamma, \text{Aut}(X_{\bar{k}}))$ computes the subgroup of automorphisms which are Galois-invariant, which is equivalently the automorphism group of $X$, and similarly for $Y$, so t …

This is true if $X$ is geometrically irreducible by the Lang-Weil bound, which gives us that the size of $|X \cap Y|$ is $(c(X \cap Y) + O_c(p^{-1/2})) p^{\dim (X \cap Y)|}$ where $c(X \cap Y)$ is the …

I am also very far from an expert here, but I think there's a case to be made that the "correct notion" involves actions on categories of sheaves, as Donu says in the comments. Consider the following …

To my mind it's cleaner to take opposite categories and talk about coproduct decompositions of affine schemes, where $\text{Spec } \mathbb{Z} \times \mathbb{Z}$ is just "two points" (the coproduct $2 …

The object you've defined is not the group of automorphisms of $\mathbb{P}^n$; among other things, it is a group-valued functor, not a group. Here is a simpler example of this sort of thing: In any c …

The category $[C^{op}, \text{Ab}]$ of $\text{Ab}$-valued presheaves on any (small, for simplicity) $\text{Ab}$-enriched category is about as nice as it gets - locally finitely presentable, Grothendiec …

One version of the question you might be asking is where such equivalences come from in general, as opposed to a proof of this one in particular. The general setup is "derived Morita theory": here is …

Write $BGL_n$ for the classifying stack of rank $n$ vector bundles; in principle it is not necessary to know what a $GL_n$ torsor is in order to say this. $E$ is represented by a map $f : S \to BGL_n$ …