This should be a very simple question but I can't seem to figure it out. If we have a morphism of schemes $f: X \rightarrow Y$, I am trying to understand what fiber is above some $\operatorname{spec} \kappa(f(p))$ for $p$ a point in $X$. Let's just work with affine schemes for now. If we have an $A$-algebra $\phi: A \rightarrow B$ which induces a morphism of schemes $f: \operatorname{spec}B \rightarrow \operatorname{spec}A$. Then firstly, if $p \in \operatorname{spec}B$ is a point, then what does the pullback along the morphism $\operatorname{spec} \mathcal{O}_{f(p)} \rightarrow \operatorname{spec}A$ look like? If $\mathfrak{p}$ is the prime ideal of $B$ corresponding to the point $p$, then obviously the resulting scheme is the tensor product $\operatorname{spec}(B \otimes_{A} A_{\phi^{-1}(\mathfrak{p})})$. But what does this scheme actually correspond to? Is there some simplification of this? I used to wrongly think that this was the scheme $\operatorname{spec}\mathcal{O}_{p}$ but I have realised there is no reason to think that is true. So once I understand that, I would like to pullback further along the morphism $\operatorname{spec}\kappa(f(p)) \rightarrow \operatorname{spec}\mathcal{O}_{f(p)}$. I know this should be quite easy but I'm having trouble seeing it.
Suppose I am given a morphism of schemes $f: X \longrightarrow Y$ and suppose that for any point $y \in Y$, $\kappa(y)$ denotes the residue field. I would like to show that the fibered product $$X \times_{Y} \text{Spec}(\kappa(y))$$ is homeomorphic to the preimage $f^{-1}(y)$ in the induced topology.  The following is my attempt so far:  The question is obviously affine on the target, so without loss of generality we reduce to the case of a morphism $f: X \longrightarrow \text{Spec }A$. Then we have  $$ f^{-1}(\text{Spec }A) = \bigcup_{i \in I} \text{Spec }B_{i} $$ via morphisms $$ \phi_{i}: A \longrightarrow B_{i}. $$ We can then write, \begin{align*} f^{-1}(\text{Spec } A) \times_{\text{Spec } A}\text{Spec } \kappa(y) &amp;= \bigcup_{i \in I} \left( \text{Spec } B_{i} \times_{\text{Spec }A} \text{Spec }\kappa(y) \right) \\  &amp;= \bigcup_{i \in I} \text{Spec }  \left( B_{i} \otimes_{A} \kappa(y)  \right). \end{align*} Now let $\mathfrak{p}$ be the prime ideal of $A$ corresponding to the point $y \in \text{Spec } A$. Then $\kappa(y) = (A/ \mathfrak{p})_{\mathfrak{p}}$. Then for any of the $B$ (dropping the index for brevity), we have \begin{align*} B\otimes_{A} (A/ \mathfrak{p})_{\mathfrak{p}} &amp;\simeq B \otimes_{A} A / \mathfrak{p} \otimes_{A} A_{\mathfrak{p}} \\ &amp;\simeq B_{\mathfrak{p}} \otimes_{A} A / \mathfrak{p} \\ &amp; \simeq B_{\mathfrak{p}} / \mathfrak{p}B_{\mathfrak{p}} \end{align*}  Now the prime ideals in $B_{\mathfrak{p}} / \mathfrak{p}B_{\mathfrak{p}}$ correspond precisely to the prime ideals of $B_{\mathfrak{p}}$ containing $\mathfrak{p}B_{\mathfrak{p}}$. Further, the prime ideals of $B_{\mathfrak{p}}$ correspond precisely to the prime ideals of $B$ contained in $\phi(\mathfrak{p})B$. That is, we have \begin{align} \text{Spec }B_{\mathfrak{p}} &amp;= \{  \mathfrak{q} \subseteq B :  \phi^{-1}(\mathfrak{q})  \subseteq \mathfrak{p}  \} \\ &amp;=  \{ \mathfrak{q} \subseteq B : f(\mathfrak{\mathfrak{q}}) \subseteq \mathfrak{p}      \}. \end{align} Note that in the very last line we have abused notation slightly: There is not really any notion of being "inside" $\mathfrak{p}$ when it is being treated as a point in the scheme. So really we mean that $f(\mathfrak{q})$ is a point in the scheme whose corresponding prime ideal is a subset of $\mathfrak{p}$. Now as we said above, the prime ideals of $B_{\mathfrak{p}}/\mathfrak{p}B_{\mathfrak{p}}$ are the following $$ \text{Spec }(B_{\mathfrak{p}}/\mathfrak{p}B_{\mathfrak{p}}) = \{ \mathfrak{q} \subseteq B : f(\mathfrak{\mathfrak{q}}) \subseteq \mathfrak{p}      \text{ and } ???  \} $$ This is the point at which I get stuck. Ideally I would like to make some argument from the statement "Now the prime ideals in $B_{\mathfrak{p}}/\mathfrak{p}B_{\mathfrak{p}}$ correspond precisely to the prime ideals of $B_{\mathfrak{p}}$ containing $\mathfrak{p}B_{\mathfrak{p}}$" to conclude that the question marks are the condition $\mathfrak{p} \subseteq f(\mathfrak{q})$. Could someone point me in the right direction as to how to get this condition?   Is the proof up until this point correct? Different sources online seem to do this many different ways, so it's difficult for me to check.
Is there a destructor for Java? I'm using Xubuntu 14.04 64 bit. I know that question been asked a billion times, but all of them consist of editing following file:  /etc/polkit-1/localauthority/50-local.d/com.ubuntu.enable-hibernate.pkla   That file does not exist on my system. I tried com.xubuntu.enable-hibernate.pkla but doesn't exist either, locate command doesn't find it. How do I repair a corrupted Firefox places.sqlite database? I had some problems with my RAM (bluescreen several times, Windows XP) and now are my Firefox databases damaged. Firefox is working, but my history is gone and it's reporting several inconsistencies and errors when executing pragma integrity_check on places.sqlite:     database disk image is malformed   Now the question, how do I repair SQLite-Databases? Is $2^\sqrt{2}$ irrational? Is it transcendental? Trying to figure what language this PDA (pushdown automata) accepts I have the following PDA and I can't figure our what words it accepts, would like to get some help with figuring this out.    Of course it only accepts if that stack is empty in the accepting state. There seems to be something wrong with the sorting of Stack Exchange sites by name:    elementary OS and Русский язык appear in the wrong place at the end of the list. Seeking GeoServer tutorials? What are sources where I can learn how to use GeoServer?  I know of these two sites:    - which in fact is not working (at least at the moment)  - not only (tutorials) for GeoServer   But I am interested in more complex information, and not to read only documentation. In my question  I asked about the definition and usage of the word "model", for which I was told the following definition:     A formula of propositional logic is true under an interpretation iff   the interpretation assigns the truth value T to that formula. If a   formula is true under an interpretation, then that interpretation is   called a model of that formula.   An interpetation satisfies a formula φ iff it is a model of the formula. Same for a set $\Gamma$ of formulas, if an interpretation satisfies all the formulas in $\Gamma$ then it is a model of $\Gamma$. Then if we say $\Gamma \vDash \varphi$ then $\varphi$ is a semantic consequence of $\Gamma$ / is satisfied in all "models" of $\Gamma$ (true whenever everything in $\Gamma$ is true).  This also matches the definitions for model as seen on the Wiki for  and .  All good so far. But when I get really confused when I see posts like  where they use "models" as a word meaning "satisfies all the axioms in a theory."  From Asaf's answer:     Models are structures, and structures are models. But when we say   "model" we mean that there is a particular theory which holds in the   structure, and when we say "structure" we are mainly interested in an   arbitrary interpretation of the language.   From Metin's answer:     A structure is a set with some interpretable symbols(constants,   relations and functions) within a fixed language. You do not ask for   more from a structure.      However...      A model (of a theory) is a structure which satisfies the axioms of the   theory. It makes more "structural sense"...   And then from one of Asaf's comments:     When I say "a model of 𝖹𝖥𝖢" they also know, immediately, that I am   talking about a model which satisfies all the axioms of 𝖹𝖥𝖢.   So I am unclear on which definition is correct or if this is the same word being used in two different ways or if this is "logical model" versus "semantic model" or something else entirely.  On the one hand we seem to have "model" meaning "any interpretation that satisfies a formula or set of formulas." On the other hand we have "model" meaning "Interpretation that satisfies all the axioms of a theory".  From the Wiki article on  we see also:     For a given theory in model theory, a structure is called a model, if   it satisfies the defining axioms of that theory.   So which is it exactly? Does "model" have a different meaning in propositional logic compared to model theory or first-order logic or set theory etc? Are split infinitives grammatically incorrect, or are they valid constructs? When I check for updates, I get a "Failed To Download Repository Information" error.   This is what comes up under details:   W: Failed to fetch gzip:/var/lib/apt/lists/partial/us.archive.ubuntu.com_ubuntu_dists_natty_main_source_Sources  Hash Sum mismatch,  E: Some index files failed to download. They have been ignored, or old ones used instead. Since canonical is  Ubuntu One, what are the possible alternatives?  I currently use Ubuntu One as a backup service for my Documents folder and various other folders I deem important. Of course I have backups on an external hard drive as well, but nevertheless, I find Ubuntu One to be very useful. What I'm especially looking for is a cloud service that does not need a special folder (such as a dropbox or google drive folder); something that is able to integrate into my own file system and syncs just any folder I want to.   I am aware that this question is rather subjective, but I do guess many people who have used Ubuntu One are now in the dark. Where can I access OS X keyboard shortcuts, say for Dropbox sync? How to enable bullet points when using the presento beamer theme? I have the problem that when I use an itemize list and the , there are no bullet points.  Here is a minimal working example (requires the  not available on CTAN to be downloaded first):  \documentclass{beamer} \usepackage{config/presento} \begin{document} \begin{frame} \begin{itemize} \item ... \item ... \item ... \end{itemize} \end{frame} \end{document}     How to fix this problem? mkfs on raw software raid device or partition I am making a huge RAID10 device (8 TB) for storage on CentOS. I have created the 4 partitions on my 4 physical hard disks and have created the raid device /dev/md2 out of those 4 partitions.  Now it's time to put a filesystem on it. I am wondering whether I should do just:  mkfs.ext4 /dev/md2   which totally works (I tried it), or I should first create a single partition on that device (with fdisk, gdisk or parted?), that takes up all the space, and then do the mkfs stuff on that partition.  Googling the Web, I found another person asking the very same question here:  but, although a responder said:      Partitions go on the array members.   Filesystem goes on the array.   they didn't give him a definitive, non-ambiguous answer or an explanation when one approach should be preferred over the other. Forgotten PostgreSQL Windows password Compute: $\lim_{x\to 0}\frac{\sqrt[m]{\cos(\alpha x)}-\sqrt[n]{\cos{(\beta x)}}}{x^2},\;m,n\in\mathbb N,\alpha,\beta\in\mathbb R$ Compute: $$\lim_{x\to 0}\frac{\sqrt[m]{\cos(\alpha  x)}-\sqrt[n]{\cos{(\beta x)}}}{x^2},\;m,n\in\mathbb N,\alpha,\beta\in\mathbb R$$   My attempt:  $$L=\lim_{x\to 0}\frac{\sqrt[m]{\cos(\alpha x)}-\sqrt[n]{\cos{(\beta x)}}}{x^2}=\lim_{x\to 0}\frac{\sqrt[m]{\cos(\alpha x)}-1+1-\sqrt[n]{\cos(\beta x)}}{x^2}\\\displaystyle=\lim_{x\to 0}\frac{\left(1+(\cos(\alpha x)-1)\right)^{\frac{1}{m}}-1-\left(\left(1+(\cos(\beta x)-1)\right)^{\frac{1}{n}}-1\right)}{x^2}$$  Transformed expression: $$-\frac{(1+(\cos(\alpha x)-1))^{\frac{1}{m}}-1}{\cos(\alpha x)-1}\cdot\frac{1-\cos(\alpha x)}{\alpha^2x^2}\alpha^2+\frac{(1+(\cos(\beta x)-1))^{\frac{1}{n}}-1}{\cos(\beta x)-1}\cdot\frac{1-\cos(\beta x)}{\beta^2x^2}\beta^2$$  I applied the standard limits from the table: $$\lim_{x\to 0}\frac{(1+x)^a-1}{x}=a\;\&amp;\;\lim_{x\to 0}\frac{1-\cos x}{x^2}=\frac{1}{2}$$ $$L=-\frac{1}{m}\cdot\frac{1}{2}\alpha^2+\frac{1}{n}\cdot\frac{1}{2}\beta^2=\frac{m\beta^2-n\alpha^2}{2mn}$$ Is this correct? How do you hide back faces so that you only see faces when its normal is facing the camera? When exporting it for use in a game, the face is rendered in only the direction of its normal, so this would be useful to have something similar. $S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective? $S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is  a continuous function.  I've already shown that $$T_{\mathrm{max}} := \mathrm{sup}\{ T(x) : x \in S^2\}  \: \: \mathrm{ and }\: \: T_{\mathrm{min}}:= \mathrm{inf}\{ T(x) : x \in S^2\}$$ are reached by $T$ with the argument that $S^2$ is closed and bounded and therefore after the Heine-Borel-Theorem compact, and therefore since T is continuous it assumes its max and min in $S^2$.   Now i have answer the following:     1) Is there a point $x_0 \in S^2$ with $T(x_0)=T(-x_0)$ ?      2) Is there a value $T \in(T_{\mathrm{min}}, T_{\mathrm{max}})$ which is only assumed at exactly one point of $S^2$ ?   I wanted to show that $T$ is injective to get that 1) is false and 2) is true but i didn't get anywhere. Any ideas? Thanks in advance! Place label in middle of line with TikZ What is the best way to place a label in the middle of a line with TikZ? I mean smack dab in the middle of the line, not above or below it. The following is close  \begin{tikzpicture} \draw[|-&gt;, color=red] (-4cm, 0.5cm) -- (2.5cm, 0.5cm) node[midway, color=black] {$\tau_{r}$}; \end{tikzpicture}   But the line crosses right through the node and its text, which is bad.  I don't know if there is an option to choose the background color of a node, which I think would solve the problem (e.g. background=white). fill doesn't seem to work. How to change boot splash screen in 18.04 I have done this in the past in 16.04 and 17.10 but in 18.04 I can not get plymouth manager to work.  Downloaded it from the software center but it doesn't work.  Is there a guide available that will let me do it in terminal?