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 Apr 12 asked Image of map of étale fundamental groups Feb 22 awarded Nice Answer Feb 16 awarded Favorite Question Feb 11 asked Monomorphism in the category of schemes Feb 10 comment Residue fields of schemes of finite type (over $\mathbb{Z}$) @AriyanJavanpeykar, sure, but my point is that if $k(x) \rightarrow K$ is a strictly bigger field, than $x$ is a $K$-rational point whose residue field is not $K$. So your stated equality of sets between the set of closed points with residue field $\mathbb{F}_q$ and the set $X(\mathbb{F}_q)$ of $\mathbb{F}_q$-rational point need not hold in general. For example if $q = p^2$ and $X$ contains a point $x$ with residue field $\mathbb{F}_p$, than this point is not in the first set but it occurs in fact twice in the second set. Am I right? Feb 9 comment Residue fields of schemes of finite type (over $\mathbb{Z}$) Hi @AriyanJavanpeykar, great answer, thanks! Just one question: you say "the set of (closed) points with residue field $\mathbb{F}_q$ is the set $X(\mathbb{F}_q)$ of $\mathbb{F}_q$-rational points of $X$". But isn't this only true if $q$ is itself a prime number? If not, if $q = p^n$ with $n > 1$, then a point with residue field $\mathbb{F}_p$ will be in $X(\mathbb{F}_q)$, right? Jan 12 awarded Good Question Jan 10 awarded Nice Question Nov 8 awarded Nice Answer Oct 30 awarded Popular Question Oct 27 comment An example of a map that is not unramified in a specific way It is basically your example Georges but then with $\mathbb{Z}_p$ replaced by $k$. Thanks in any case for your great answer! Oct 27 comment An example of a map that is not unramified in a specific way Hi Georges and @AlexYoucis , how about the map $k \rightarrow k[x]$, the generic point of $\mathbb{A}^1$ is mapped to the unique point of $\operatorname{Spec}(k)$, resulting in $k = k_{(0)} \rightarrow k[x]_{(0)} = k(x)$ where 1. is satisfied but 2. not since the extension is infinite? Oct 27 accepted An example of a map that is not unramified in a specific way Oct 20 comment An example of a map that is not unramified in a specific way Thanks! But this extension is still finite right? I was actually looking for an infinite extension (not necessarily inseparable). Oct 20 comment An example of a map that is not unramified in a specific way @AlexYoucis Woops I messed up the definition of finite type maps. ^^ But that makes me think, since the map $\kappa(x) \rightarrow \kappa(y)$ is of finite type doesn't that already imply it is finite? Or am I being silly again? Oct 20 comment An example of a map that is not unramified in a specific way Hi @AlexYoucis ! Thanks for your comments. But I just realised something, how about the map induced by $\mathbb{C} \rightarrow \mathbb{C}(T)$? Isn't it an example? Its fibers are obviously finite, which is quasi-finiteness of the map by definition right? So why does your logic not apply in this case? Oct 20 asked An example of a map that is not unramified in a specific way Jul 19 comment How to replace addition with multiplication to find the next integer value? Hi Vita. First of all let me congratulate you with your mathematical curiosity! Keep asking yourself questions like these, and never worry about whether or not they are stupid questions! Jul 18 awarded Yearling Jun 30 comment What are the most prominent uses of transfinite induction outside of set theory? @CiaPan could you point me to a reference for that statement? Sounds intriguing..