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This may be relevant, although I'm not sure if it answers your question directly. The Russell cubic x+x^2y+t^2+z^3=0 is diffeomorphic to affine space A^3. But, it is known not to be algebraically isom …

One of them was answered in response to question 1558 on when quotients of schemes by free group actions exist. When the group is finite, they exist as algebraic spaces. But, there are examples where …

The theorem of Bridgeland-King-Reid says that if $M$ is a smooth quasi-projective complex variety of dimension at most $3$ on which a finite group $G$ acts such that the canonical sheaf $\omega_M$ is …

This is not a general answer to your question, but evidence of the intriguing connection between Brauer groups of surfaces and elliptic curves. Let $X$ be a K3 surface over the complex numbers $\mathb …

See this paper of mine and Gabriele Vezzosi. We prove that HKR holds in particular for smooth proper schemes $X$ of dimension at most $p$, the characteristic prime. In particular, it holds for smooth …

Motivic cohomology computes Chow groups. And, motivic cohomology is representable in the A^1-category. More specifically, CH^p(X)=H^2p(X,Z(p)). The cohomology groups on the right are representable by …

It looks like you probably need a stronger condition for the lements of the fiber. Specifically, it is not enough that residue fields should be isomorphic. There also must be a morphism on the Nisnevi …

This got a bit long for a comment, so here's an answer. The answer is it is almost never injective. In general, for compact $X$, $Br(X)$ is infinite (at least if $H^2(X,O_X)\neq 0$), while $Br(U)$ is …

It seems like the answer to your question is no, at least without further clarification. If you could define the tensor product structure on $D^b(X)$ just from the triangulated structure and the Serre …

Suppose that $Br(X)$ is representable in the following sense: there exists a $k$-scheme $B_X$ such that for each $k$-scheme $S$ there is a natural bijection $B_X(S)=Br(X_S)$, or perhaps we should rigi …

I'm not sure off hand what the situation is for $A_\infty$-algebras, but for $\mathbb{E}_\infty$-algebras it's worth noting that in many cases generic formality does not imply formality at each fiber. …

Edited based on Sasha's answer: I will assume that we are interested in the thick subcategory generated by $A$. Under this assumption, the desired statement is closely related to a theorem of Neeman …

In general, these presheaves are not sheaves, even on the etale sites of fields. As an easy example, $K_2(\mathbb{C})$ is non-torsion divisible, but $K_2(\mathbb{R})$ has a $2$-torsion element given i …

In a paper from 1985, Raeburn and J. Taylor describe how to view all elements of H^2(X,Gm) (etale cohomology) as coming from non-unital Azumaya algebras. The construction relies on bimodule theory for …

By 'work' I would like the correspondence between fiber functors (to finitely generated projective modules) and algebraic groups to be the same as in the field case. Specifically, if $A$ is an affine …