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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 …

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. …

You can combine Adeel Khan's answer with Proposition 6.9 of my paper with David Gepner to prove that there are these kinds of localization sequences in a great deal of generality. (Note that our propo …

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 …

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 …

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 …

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 …

The answer to your more specific question is yes, there are tilting objects that involve complexes not quasi-isomorphic to any vector bundle. Consider the blowup $X$ of $\mathbb{P}^2$ at a single poin …

One trivial example is the derived category of a central simple algebra $D(A)$ where the class of $A$ in the Brauer group is non-zero. A non-zero simple left ideal $I$ of $A$ is a tilting complex, but …

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 …

I would suggest the lectures of Friedlander and Weibel: "An overview of algebraic K-theory" in Algebraic K-theory and its applications (Trieste 1997), 1999; MR. The later lectures include the modern p …

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 …

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 …

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 may not be particularly helpful, but when $X=\mathrm{spec} k$, for a field $k$, then $CH^{2n}(X,n)=K_n^M(k)$, the Milnor $K$-theory of $k$. I do not know if there are any other useful characteriz …