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Exercise VI.8.3 from Bourbaki's Algèbre Commutative gives am example of a hensel discrete valuation ring and a finite (inseparable) extension of its fraction field for which the normalisation is not …

This seems like the kind of thing an expert should be able to answer off the top of their head: Recall that a valuation ring is an integral domain $A$ such that for every $a \in Frac(A)$ we have eith …

Is there a name for a morphism of schemes which is an inverse limit of schemes such that the transition maps are all open immersions? To be more precise, I have a scheme $S$, and a functor $U: I \to …

I have the following Grothendieck pretopologies on the category of schemes. The first one, the covers are families of morphisms $\{ U_i \to X \}$ such that for every point $x \in X$ there exists some …

In the setting described in Bernstein, Beilinson, and Deligne, associated to a scheme $X$, a closed subscheme $i: Z \to X$ and its open complement $j: U \to X$ we have six functors between the corresp …

This feels like something I should know but I can't find an answer in Liu or in Atiyah-MacDonald, or a counter-example. To state the question again: let $A$ be an integral Noetherien ring of Krull di …

Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence $$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to …

How about this: $Spec (\mathcal{O}_{X,x})$ is the sub-locally ringed space of $X$ whose underlying topological space consists of all the points $y$ of $X$ such that $x$ is in the closure of $y$. The …

I have local hensel ring $A$ and a finite flat $A$-algebra $B$ (which is therefore a direct product of local henselien $A$-algebras) and I would like a section of the canonical map $B \to B / N$ where …

If $S$ is the Zariski-Riemann space of a noetherian subring $k$ of a field $K$, Zariski-Samuel prove that $S$ is quasi-compact. If $S'$ is the subspace of valuations that are discrete (i.e. that valua …

It was a stupid question. And obviously not very clearly stated. By $F(x)$ I mean't $F(Spec\ k(x))$ where $k(x) = \mathcal{O}_{X,x} / \mathfrak{m}_x$ is the residue field of the point $x$ in the schem …

Here is a probably stupid question : If $F$ is a sheaf on the big Nisnevich site, then is the morphism $F(X) \to \amalg F(x)$ injective where the sum is over ALL the points of $X$ (not just the closed …

Does anyone know an example of a curve $X$ over a perfect field $k$ such that if $\tilde{X}$ is its noramlisation, there exists a point $x \in X$ and a point $y \in \tilde{X}$ over $x$ such that $k(y) …

Here is the question: if $X$ is a separated, finite type scheme over a perfect field (but not necassarily smooth) is the map $KH_n(X) \to \prod_{x \in X^{(0)}} KH_n(k(x))$ injective? If $X$ is smoot …

It looks like this question has already been asked, but after reading some mistitled previous questions you can see it actually doesn't appear in the "Related Questions" list. Is there a reference fo …