Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every field extension $L/\Bbbk$, and reduced if its underlying ring is reduced.  Separable always implied reduced, and I found a result saying the converse is true over perfect fields. However, I would like an instructive counterexample of a reduced $\Bbbk$-algebra which is not separable. How can such a thing happen?
Let $A$, $B$ be two reduced $k$-algebras. Then if an element of the form $$\sum a_{i}\otimes b_{j}$$ is nilpotent, we can compose it with any $k$-homomorphism $f$ from $A$ to $k$ to get a homomorphism from $A\otimes B$ to $B$. This map's image is $$\sum f(a_{i})\otimes b_{j}$$must be nilpotent as well. So if we assume $b_{j}$ is linearly independent in the first place, we may reason that $f(a_{i})=0$. This showed that $a_{i}$ are in the Jacobson radical of $A$. However, we only know $\mathrm{Nil}(A)=0$, and it is not clear to me how to pass from the Jacobson radical to the nilradical without using some commutative algebra machinery.   Here is what I think that might work. If $A$ is of finite type, then by Nullstellensatz we can assert the above is true. But if $A$ is not of finite type, then I can write $A=k[x_{1},...,x_{n}][X]$, where $n$ is the transcendence degree of $k(A)$ and $k[X]$ is finite over $k$. Then I should be able to pass from $k[X]$ to $A$ by inductive argument on the transcendental degree of $k(A)$, since prime ideals are maximal in a UFD (I am working with $k[x]$ by introducing $[x]$).   The proof I have for the case of finite type is not satisfying. Given $I\subset k[X]$, using Nullstellensatz one can show that $$I_{X}V_{X}[I]=\sqrt{I}$$ where $V_{X}[I]$ is the set of all points in $X$ on which $I$ vanishes, and $I_{X}(V_{X}[I])$ be the ideal in $k[X]$ which vanishes on $V_{X}[I]$. Granting this, let $I=(a_{i})$, then we want to show $I=0$, which is $V_{X}[I]=X$. This follows since if $U\subset W$, then $V_{X}[U]\supset V_{X}[W]$. Since $V_{X}[I]\supset V_{X}[m_{x}]$ for all maximal ideal corresponding to points, it must be $X$. This proof is unnecessarily winding by using the ideal-variety correspondence. Ideally I should be able to construct a pure algebraic proof.   The thing not clear to me is whether the above strategy really works. Namely I am not sure if I can really induct on the transcendental degree of $k(A)$. To make matters simple assume we showed the statement holds for $A$ with transdental degree $&lt;n$. For $A$ with transdental degree $n$ we can thus write it as $k[A'][x]$. Then any maximal ideal $m'$ of $k[A'][x]$ must correspond to a maximal ideal $m=m'\cap k[A']$ in $k[A']$. But it seems not possible to show that any prime ideal is maximal. Indeed considering the simple example of $k[x,y]$, where $(x)$ is a prime ideal but not maximal since $k[x,y]/(x)=k[y]$ is not a field. The problem is in the inducting process the base ring is not neccessarily a field, and so this strategy cannot work.   I am thinking maybe I can show $m'=m[f(x)]$ with $f(x)$ some irreducible polynomial in $k[A'][x]$. Because we know $k[A'][x]$ is noetherian by Hilbert's basis theorem, so in particular $m'$ must have a finite generating set. And we can reduce the number of irreducible polynomial to 1 since $m'$ is a maximal ideal. But I still do not know how to use this to finish the proof.
How to get exact camera view? I'm currently a beginner when it comes to 3D modelling but doing okay so far. I'm struggling a bit when it comes to the final renders of my scene, specifically on the camera view I want. Sometimes its to zoomed in and sometimes to zoomed out that it's gone out of the environment of the blender. I realized the settings for the camera on blender are for actual cameras / photography and I'm no expert on that too. A tip on how to control them precisely would be nice. Thank you. When trying to execute a symbolic link which points to an exe file (on Windows 7), the "Open with" windows appears. When renaming it to a ".exe" file, it says "The specified path does not exist. Check the path, and then try again.".  Symlinks to other destinations seem to work fine.  So is there any way to execute a symlink?  EDIT: In the command prompt, I execute the following and the MyProgram.exe runs both times.  However, if I double-click on the symbolic link in Windows Explorer it fails with the above error message.  C:\Temp&gt;mklink SomeProgram.exe MyProgram.exe symbolic link created for Backup.exe &lt;&lt;===&gt;&gt; Backup-1.0.1.exe  C:\Temp&gt;SomeProgram.exe  C:\Temp&gt;del SomeProgram.exe  C:\Temp&gt;mklink SomeProgram.exe C:\Temp\MyProgram.exe symbolic link created for Backup.exe &lt;&lt;===&gt;&gt; C:\Temp\Backup-1.0.1.exe  C:\Temp&gt;SomeProgram.exe Problems with \input{table.tex}\hline after 2020 fall LaTeX release Since updating to the 2020 Fall release (MikTeX on Windows but also tested with TeXLive), I can no longer compile files where tables are imported from an external file using \input and followed by \hline (or \bottomrule). Things work perfectly with older releases. Note that it works if the last \\ is outside the file to be imported, but this is not a feasible option if the tables are created automatically from other softwares. Here is a MWE:  \begin{document}  \begin{table}   \caption{Table in text}   \begin{tabular}{lcc}     a &amp; 1 &amp; 2\\     b &amp; 1 &amp; 2\\     \hline   \end{tabular} \end{table}  \begin{table}   \caption{Table in other file 1}   \begin{tabular}{lcc}     \input{table1}     \hline   \end{tabular} \end{table}  \begin{table}   \caption{Table in other file 2}   \begin{tabular}{lcc}     \input{table2}\\     \hline   \end{tabular} \end{table}  \end{document}  with external files: table1.tex:     a &amp; 1 &amp; 2\\     b &amp; 1 &amp; 2\\  table2.tex:     a &amp; 1 &amp; 2\\     b &amp; 1 &amp; 2  Here is the log when it fails: This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 20.10) entering extended mode (MWE.tex LaTeX2e &lt;2020-10-01&gt; patch level 1 L3 programming layer &lt;2020-10-05&gt; xparse &lt;2020-03-03&gt; (&quot;C:\Program Files\MiKTeX\tex/latex/base\article.cls&quot; Document Class: article 2020/04/10 v1.4m Standard LaTeX document class (&quot;C:\Program Files\MiKTeX\tex/latex/base\size10.clo&quot;)) (&quot;C:\Program Files\MiKTeX\tex/latex/l3backend\l3backend-pdftex.def&quot;) (MWE.aux) (table1.tex) ! Misplaced \noalign. \hline -&gt;\noalign                    {\ifnum 0=`}\fi \hrule \@height \arrayrulewidth \futurelet... l.18     \hline                 ! You can't use `\hrule' here except with leaders. \hline -&gt;\noalign {\ifnum 0=`}\fi \hrule                                           \@height \arrayrulewidth \futurelet... l.18     \hline                 ! Missing number, treated as zero. &lt;to be read again&gt;                     \futurelet  l.18     \hline                 ! Illegal unit of measure (pt inserted). &lt;to be read again&gt;                     \futurelet  l.18     \hline                 (table2.tex) [1{C:/Users/Gouel/AppData/Local/MiKTeX/pdftex/config/pdftex.map}] (MWE.aux) ) (see the transcript file for additional information)&lt;C:/Program Files/MiKTeX/fo nts/type1/public/amsfonts/cm/cmr10.pfb&gt; Output written on MWE.pdf (1 page, 14750 bytes). Transcript written on MWE.log.  With an older release under Linux, it works perfectly: This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Debian) (preloaded format=pdflatex)  restricted \write18 enabled. entering extended mode (./MWE.tex LaTeX2e &lt;2020-02-02&gt; patch level 2 L3 programming layer &lt;2020-02-14&gt; (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2019/12/20 v1.4l Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo)) (/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdfmode.def) (./MWE.aux) (./table1.tex) (./table2.tex) [1{/var/lib/texmf/fonts/map/pdftex/up dmap/pdftex.map}] (./MWE.aux) )&lt;/usr/share/texlive/texmf-dist/fonts/type1/publi c/amsfonts/cm/cmr10.pfb&gt; Output written on MWE.pdf (1 page, 13788 bytes). Transcript written on MWE.log. Can only search social network messages in the Dash after 13.10 upgrade There are four coins in the main hallway, and a number of them in the Boos in the yard. Is there anything to gain from them other than health? Do I get anything for collecting them, perhaps in a particular order? An equivalent condition with $\{0\}$ being the only nilpotent ideal Is there a terminal command that will log you out of your current desktop session and take you back to the login screen? Time traveler asks archaeologist for help in transporting a girl from dinosaur Earth to present day Many decades ago, I read a short story about a time traveler who asks an archaeologist for help.  He convinces the archaeologist that he can time travel by bringing him back a freshly killed dinosaur (small one.)  The time traveler needs help because when traveling back in time to the age of the dinosaurs he met a girl and wants to transport her back to the present.  However only one person can travel in time at a time so he needs the archaeologist at the other end to wait for the girl.  The archaeologist agrees to help.  Unfortunately after the traveler goes back he and the girl find themselves in the middle of a battle between aliens on Earth.  They are chased by a dinosaur and never make it back to the present.  The archaeologist meanwhile is digging for evidence that the traveler went back in time and the story or my recollection of it ends with that. How to best determine if a bike will fit without riding it Does it make sense to use a date variable in a regression? I'm not used to using variables in the date format in R. I'm just wondering if it is possible to add a date variable as an explanatory variable in a linear regression model. If it's possible, how can we interpret the coefficient? Is it the effect of one day on the outcome variable?   See my  with an example what I'm trying to do. If I have a legendary creature out, and I make a copy of it (such as by enchanting it with  or making a ), do I get enters-the-battlefield effects from the creature before I have to sacrifice it to the legend rule? Why is there no last activity message? With left nav, responsive design, and various new-user notices, this is what an anonymous user currently sees (browser window here is 1090px wide):    I realize that a lot of this content -- code of conduct, new-user intro, cookies notice -- is important.  And I get that the left nav is an important design element, and responsive design dictates the width of the right column.  But when you add it all up, Q&amp;A -- what the site is about -- gets a distressingly small portion of the screen.  Can we do better?   Maybe we can make that intro/mini-tour shorter? Maybe we can hide the right column for anonymous users? Maybe some things can wait until the user's viewed a few pages? Maybe other things I haven't thought of?   My concern is that anonymous users -- people coming from Google, people who were referred to a site by users hoping to recruit them, or others who find their way to us but aren't invested in us -- will turn away before seeing how rich and helpful our sites are.  Which would be a real shame, because our sites are awesome.  First impressions matter.  When I'm asking a rabbi to check out Mi Yodeya or a professional chef to check out Seasoned Advice or my vet to check out Pets (all of these have happened), I want to make a good impression.  I want the person to see more than 2.5 questions; I want to increase the chances the person will see something interesting or say "oh I can answer that!" and dig deeper.  Yes, the other stuff is important, but maybe not now?  Is there a better way to "stage" all this information for people we haven't yet hooked? How can you strip non-ASCII characters from a string? (in C#) How can you strip non-ASCII characters from a string? (in C#) What factors influence the spring constant? For certain springs we have been taught that Hookes law, $F=-kx$ works. For such springs, what factors determine k. For instance, k is proportional to length. Can k be expressed in terms of dimensions, mass and other fundemental quantities, or is k a fundemental property of the spring itself? I have two shapefiles, Point and Polyline. I'm trying to populate my polyline layer with the point data ID (systemID) where the lines start to where they end; eg. create two columns (from and to). The lines already have flow direction assigned to them. I am using ArcHydro tools but Arcpy is also fine.  In ArcHydro I have tried using Attribute Tools > Generate From/To Nodes for lines but it creates values for To and From that are not from my point data. My other alternative was using Feature to Vertices and joining the result to my point data table to get the systemID. But then I would have to update the polyline layer as well. I am sure there is a easier way of doing it.  Note: not all the lines have points at the vertices, blank values are fine for them. Im trying to craft a regex that only returns &lt;link&gt; tag hrefs  Why does this regex return all hrefs including &lt;a hrefs?       (?&lt;=&lt;link\s+.*?)href\s*=\s*[\'\"][^\'\"]+        &lt;link rel="stylesheet" rev="stylesheet"      href="idlecore-tidied.css?T_2_5_0_228" media="screen">     &lt;a href="anotherurl">Slash Boxes&lt;/a>   thank you What is $\sum_{n=1}^{\infty} \frac{n}{2^{\sqrt{n}}}$? Accessing the web page's HTTP Headers in JavaScript How do I access a page's HTTP response headers via JavaScript?  Related to , which was modified to ask about accessing two specific HTTP headers.     Related: BasicTeX - how to identify packages that need to be added after an upgrade I'm using the BasicTeX distribution on OS X and recently upgrade from the 2011 to the 2012 distribution. This forces me to add the packages that I had added over time to the BasixTeX 2011 installation by pulling them from the online repository with tlmgr. Is there a good way to list these packages that are not part of BasicTeX but are part of the complete TeXLive?