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Yes: if $X$ is quasi-compact and quasi-separated, then the category of quasi-coherent sheaves on $X$ is canonically equivalent to the category of ind-objects on the (essentially small) category of coh …

I assume the question holds in contexts where we can glue open immersions and proper morphisms to produce $f_!$ for $f$ separated of finite type. In particular, we shall have $f_!=f_\ast$ for $f$ prop …

Maybe you might have a look at Toën's paper: "On motives for Deligne-Mumford stacks", IMRN No. 17 (2000), 909-928. He defines Chow groups of a Deligne-Mumford stack X with coefficients in the characte …

Let $k$ be a field of characteristic $p>0$. Consider the field $k((t))$ of Laurent series, endowed with its $t$-adic valuation. Let $L$ be a subextension of $k((t))/k$ which is of finite type over $k$ …

There is an homotopy theory associated to any geometry. This can be done in various ways. but a rather systematic point of view is the one of Morel and Voevodsky. The idea is that a geometry should de …

The triangulated category $\mathsf{D}_{\mathsf{B}}(\mathsf{A})$ can be promoted to a stable $\infty$-category. One of the many interests of working with stable $\infty$-categories is that we have a re …

In one sentence: the theory of cohomology theories on algebraic varieties and the idea that there is a universal such thing. Of course, this is not a very satisfying answer, unless we specify what a …

The answer to question 1) is yes. However, Chow groups do not form what we should call a cohomology theory, but are part of a Borel-Moore homology theory. This ambiguity comes from the fact that, by P …

With rational coefficients, the answer is yes. The first case to understand is when $E$ is a finite algebraic extension of $F$. In the case when moreover $E$ is purely inseparable, then the extension …

I don't know any alternate geometric construction of the commutativity contraint, but there is a natural way to explain this, as a simple fact of homological algebra (and assuming very optimistic conj …

Florian Marty studied this question in his thesis. The relevant chapter is available as arXiv:0712.3676 (otherwise, the thesis is available here). He describes the space $|\mathrm{Spec}(A)|$ as the se …

To keep things simple, let us assume we work over a perfect field. The easiest part of motivic cohomology which we can get is the Picard group (i.e. the Chow group in degree 1). This works essentially …

Even if your question has a negative answer in general, it is related to Artin's approximation theorem (see Publ. IHES, vol 36), which can interpreted as follows. Let S be nice scheme (classically, th …

If you have an abelian category $A$ with enough injective, for any functor defined on bounded below cochain complexes $\Phi\colon C^+(A)\to D$ sending cohain homotopy equivalences between degree-wise …

To prove that the standard conjectures are true for any Weil cohomology over a given field $k$, it suffices to prove them for an arbitrary chosen Weil cohomology over each subfield of finite type of $ …