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Let $A$ be a smooth affine variety of dimension $n$. Are there any facts known on $K_{\ast}(A)$ and its $\gamma$-filtration which do not hold for $K_*(V)$ for an arbitrary smooth $V$ (of the same dim …

Is there a long exact sequence for the K-theory of (coherent sheaves on) blow-ups of compact complex manifolds? Does it split? What can one say on (possibly, singular) complex analytic spaces here? …

For a (bounded) double complex (of abelian groups or vector spaces) one can consider two spectral sequences that converge to the cohomology of the totalization: one can first compute either the cohomo …

Let $F:A\to A'$ be a (full) exact embedding of abelian categories. When $D(F):D(A)\to D(A')$ (or its bounded version) is a full embedding also? I would be interested in any necessary or sufficient c …

Let $A\to B$ be a full embedding of exact categories that induces an embedding $D^b(A)\to D^b(B)$. My question is: what can one say about the relation of the homotopy cofibre $K(A)\to K(B)$ (the relat …

It is well-known (and was proved by Gabber?): if $R$ is a regular henselian local ring containing a field of characteristic prime to $l$, $k$ is its residue field, then $K_\ast(R,\mathbb{Z}/l)\cong K …

Let $A$ be a differential graded algebra, $S\subset H^*(A)$. I would like to 'kill $S$ in a canonical way'. Is it possible to do it as follows: consider the $A_\infty$-algebra structure on $H^\ast(A)$ …

I am interested in the following assumption on left $R$-modules: for a module $I$ and all injective homomorphisms $A\to B$ of finitely generated (or possibly finitely presented) modules I want the hom …

Let $P$ be a smooth projective variety. For an object $X$ of $D_{perf}(P)$ (i.e. a bounded perfect complex of $\mathfrak{O}_P$-module sheaves) we consider its class $[X]$ in $K_0(P)\otimes \mathbb{Q}( …

How do people call an additive functor from a triangulated category $C$ to an abelian one that converts distinguished triangles into long exact sequences. Does one usually call a covariant functor o …

Let $X$ be a Voevodsky's motif (over a perfect field) that belongs to the positive part of the homotopy $t$-structure (i.e. its cohomology as an object of $D^-(ShSmCor)$ is zero in negative degrees). …

In my previous question The vanishing of non-connective K-theory in negative degrees I asked when one can be sure that the negative non-connective $K$-groups of a differential graded category vanish. …

Some not very clever questions on closed model categories. For a (left or right) Quillen functor $F:C\to D$ what arguments does one usually use for proving that $Ho F$ is fully faithful when restric …

It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In particu …

Given a ringed topological space $S$ one can easily define its K and K'-theory as the K-theories of the categories of locally free sheaves and of the category of coherent sheaves on it, respectively. …