I have a question concerning part (c) of Exercise 5.1 in Chapter II of Hartshorne: He asks to prove that if $\mathcal{E}$ is a locally free $\mathcal{O}_X$-module of finite rank and $\mathcal{F},\mathcal{G}$ are arbitrary $\mathcal{O}_X$-modules, then $$ \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{E}\otimes_{\mathcal{O}_X}\mathcal{F},\mathcal{G})\cong \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\mathcal{Hom}_{\mathcal{O}_X}(\mathcal{E},\mathcal{F})). $$ My question is: is it really necessary to suppose that $\mathcal{E}$ is locally free of finite rank for this to hold? Because I seem to be able to construct canonical maps in both directions which are mutually inverse, without using that $\mathcal{E}$ is locally free. Namely $$ F:\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{E}\otimes_{\mathcal{O}_X}\mathcal{F},\mathcal{G}) \to \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\mathcal{Hom}_{\mathcal{O}_X}(\mathcal{E},\mathcal{F}))\\ \varphi\mapsto \left(\substack{F(\varphi):\mathcal{F}\to\mathcal{Hom}_{\mathcal{O}_X}(\mathcal{E},\mathcal{F})\\ b\in\mathcal{F}(U)\mapsto F(\varphi)(U)(b)\in\mathcal{Hom}_{\mathcal{O}_X|_U}(\mathcal{E}|_U,\mathcal{F}|_U)}\right) $$ where we define for open sets $V\subseteq U\subseteq X$ and sections $a\in\mathcal{E}(V)$ and $b\in\mathcal{F}(U)$ $$ [F(\varphi)(U)(b)](V)(a):=\varphi^{\text{pre}}(V)(a\otimes b|_V). $$ Here $\varphi^{\text{pre}}$ is the precomposition of $\varphi$ with the canonical map from the presheaf tensorproduct to the sheaf tensor product. On the other hand, we also have the map $$ G:\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F},\mathcal{Hom}_{\mathcal{O}_X}(\mathcal{E},\mathcal{F}))\to\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{E}\otimes_{\mathcal{O}_X}\mathcal{F},\mathcal{G})\\ \psi\mapsto G(\psi) $$ where on simple tensors $a\otimes b$ with $a\in\mathcal{E}(U)$ and $b\in\mathcal{F}(U)$ (and I believe the values on simple tensors suffice to specify all of $G(\psi)$) we define $$ G(\psi)(U)(a\otimes b):=[\psi(U)(b)](U)(a). $$ It seems to me that these two constructions are mutually inverse and don't use any assumption on $\mathcal{E},\mathcal{F},\mathcal{G}$. Am I mistaking somewhere?
Does the hom-tensor adjunction hold for $O_X$ modules also? With sheaf hom and sheaf tensor product, the statement would consist of a natural transformation $Hom_O (M \otimes_O N, K)\cong_{nat} Hom_O(M, Hom_O(N, K))$, $O = O_X$ is the structure sheaf and $M,N,K$ are sheaves of $O_X$ modules.  If true I think I can check this by hand, by describing sheaves in terms of compatible germs, defining this adjunction on the germs and checking that they glue together. It is easier in the quasi-coherent sheaves on a scheme case because then one can just work in an affine cover and the distinguished base.  I am little bit concerned that one might need a finite presentation hypothesis somewhere, in order for $Hom$ to localize well. ($Hom_{R[S^{-1}]}(M[S^{-1}], N[S^{-1}]) \cong Hom_R(M,N)[S^{-1}]$ needs $M$ finitely presented.)
Is it safe to change the lens on my new DSLR? I recently bought myself a shiny new DSLR camera. Before I can use it, I had to do something kind of terrifying: mount the lens.  Now of course, half the advantage of SLR cameras is that the lenses are interchangeable. But sitting there holding an extremely expensive brand new camera and staring into its bare innards, I'm just terrified that if I make one wrong move, a hair or a piece of dust could fall in there, and then the camera will be irreparably damaged and I'll never be able to use it again.  This might sound like a stupid question, but&hellip; is it actually &ldquo;safe&rdquo; to change the lens on the camera? Is this likely to destroy it?  (Obviously it goes without saying that you should limit the time spent with no lens attached to the minimum number of seconds possible. But even if you have lightning fingers, the camera will still be lenseless for some number of seconds. And it's not like I'm standing in a class 100 cleanroom here! Besides, you don't want to be too rough with the delicate lens mounting either&hellip;) If a lens focuses all incoming light to a point, how do we get 2D images? How do lenses produce 2-dimensional images, if a lens bends all incoming rays of light to intersect at the focal point? Shouldn't this produce a single dot of light on a screen placed at the focal length?  This is basically the standard diagram that always shows up in textbooks:    I know this doesn't happen in real life--I used to use telescopes pretty frequently for work. The most in-focus image would be the one with the smallest diameters for the stars, which, of course, we think of point sources at infinity. But--the light from all stars (despite being at infinity) is not all focused at a single point. Instead, each star's light is focused at its own $(x,y)$ point on a 2D image.  I can't reconcile the theory as I understand it with any of my real-life experience with optics. What am I missing? How to use `ybar interval` without skipping the last value Petersen graph is not a Cayley graph How can I show that the Petersen graph is not a Cayley graph?  I don't know very much about Cayley graphs, I know that they are vertex-transitive, but so is the Petersen graph. It probably has to do with the group structure of $\Gamma$ in $Cay(\Gamma,S)$ (which is a group of order $10$, i.e., it is either the cyclic group $C_{10}$ or the dihedral group $D_{10}$). Then I am guessing we go for some sort of contradiction(?). My group theory is a bit rusty, I would very much appreciate someone showing me how to do this one! What's the right course of action for cross-site duplicates?  There are two flavors:   A question is posted which belongs on another site and ends up already having been asked there. I can't find an example right now, but it's bound to happen soon if it hasn't already. The same question is posted in two places at the same time. Take  for example, which is a duplicate of , posted within a half hour of each other by the same user.   Possible actions:   Move it to the proper site and close it once it gets there as a duplicate. Provide a way to close it right away as a cross-site duplicate, without moving it first. Flag it and let a moderator merge them. Close it as "no longer relevant" or something else.   Additionally, flavor 2 is a bit of a special case; perhaps the question really spans both sites in terms of content and intended audience. Should multiple-postings in this case be allowed? If so, should they somehow be linked to each other? If not, how do we avoid a migration war? How do users get started on new Stack Exchange sites? How do users on the new Stack Exchange sites get started? Each user starts with 1 rep, and can't upvote until they have 15 rep.  It struck me that this creates a bit of a deadlock. If a new user asks a question and a new user answers, neither can upvote the other.  Are we relying solely on existing SE users to join these exchange sites (with their bonus 100 rep) so that they can get started? Ubuntu 18.04 can't resume after hibernate Hibernate worked correctly when I had been using Ubuntu 17.04. After upgrade to 17.10 I couldn't resume because of black screen after resume ().   After upgrade to 18.04 if I try to start my computer after hibernation it boots as if there was no hibernation done. What are Null Pointer Exceptions (java.lang.NullPointerException) and what causes them?  What methods/tools can be used to determine the cause so that you stop the exception from causing the program to terminate prematurely? Query the post which have a thumbnail I've got a highlight section on the header of my website which shows the title and the thumbnail of the last X posts having a thumbnail.  Is there any custom query which allows me to retrieve only the last X post having a thumbnail?  At the moment I'm retrieving the last X * 10 posts (because not all the posts have a thumbnail) hoping that in this result I'll have X posts having a thumbnail. After doing my query I add a condition: if (has_post_thumbnail()).  This solution is really uncertain and ugly, a custom query with a thumbnail parameter would be great but I couldn't find any hint about it. Is "has or will read" grammatical? I have an ASUS N550JK laptop (BIOS version 204) running Ubuntu 14.04 (3.13.0-43-generic kernel).  My issue is that the battery indicator ALWAYS shows that it's using battery power - even when plugged in and fully charged. So even though I have selected to show the battery status in the menu bar only when charging or in use (in power settings), it is ALWAYS displayed. When it reaches a full charge, the indicator icon will either show just the percentage (usually 99% or 100%), or sometimes a very big number for time (like 22:14) along with the percentage. If I go to the power settings menu, it always says "Using battery power", with something like 40 hours remaining. This is all while plugged in and fully charged.  Are there any fixes less drastic than trying to update BIOS, or is a BIOS update necessary to fix this? The latest BIOS version is 208 according to the ASUS website, but the (very brief) descriptions of the updates don't mention anything about battery issues. If at all possible I'd like to avoid updating BIOS because of the risk involved.  I've already tried letting the battery drain all the way and then charging to full, with no luck.  Thanks in advance! How to rename partitions? I'm new at Ubuntu; I installed Ubuntu 12.10 and the partitions appear like 30 GB volume and like 78EF-2C5E in task bar. I tried to rename it but rename and remove buttons are deactivated so I tried by using disks but it gives me error:  (Error setting label: Command-line `mlabel -i "/dev/sdb8" ::"windows"' exited with non-zero exit status 1: Total number of sectors (18550784) not a multiple of sectors per track (63)! Add mtools_skip_check=1 to your .mtoolsrc file to skip this test  (udisks-error-quark, 0))   and this happen also to any attached usb to my laptop. How can I efficiently drive an LED? I understand that I can not connect an LED directly to a battery because it will draw too much current. Thus, there must be something else in the circuit to limit the current. What options are there? Are some methods more efficient than others? Is it possible now to use microtype with opentype fonts? I'm  having some trouble editing the content of a Managed Email Template, I can quite easily modify the content of the Email Template, but when I wish to refer to an unmanaged component, like this:  &lt;messaging:attachment renderAs="PDF" filename="invoice.pdf"&gt;     &lt;c:MyComponent/&gt; &lt;/messaging:attachment&gt;   I get the error message:     Error: Unknown component c:mycomponent   Does anyone know if its possible to embed an unmanaged component in a managed Email Template? Comprehensive list of tools that simplify the generation of LaTeX tables Manually hacking tables in LaTeX is one of the less funny things when preparing a document. Naturally, there are quite a few tools that promise to simplify this. In fact, I have somewhat lost track of all the tools available, and I'd like to gather your valuable experience into one big post. Some of the tools are mentioned in , but I feel there's much more around.  I'd suggest that somebody starts an overview with an alphabetic list (so that I can accept this answer and it stays on top), and that everyone replies with a "community wiki" entry, one reply per tool, and inserts the tool into the list.  Personally, I use Excel2LaTeX, but I'm unhappy with having to fire up Windows in order to generate the LaTeX code. I'm thinking of a rewrite as a standalone Python tool using the xlrd library, but I'm not too inclined to reinvent the wheel. I'm currently the proud owner of Frost the horse. What I want to know is, what will happen to him if I ride another horse?  Obviously nothing will happen as long as I have him in sight, e.g. I'll be able to ride Frost to make him my horse again.  Scenarios  But what will happen if, before I get back on Frost, I fast travel somewhere? Will he return to where I originally stole him, will he disappear forever, or can I own two horses simultaneously?  Ownership?  If he goes back to where I originally stole him, will I have to steal him again, and therefore never be his true owner again?  Two Horses?  The reason I'm asking is because I want to get Shadowmere (the Dark Brotherhood horse) while keeping Frost. Is it possible to do this, or will riding Shadowmere make me lose Frost forever? Indenting a whole paragraph How do I indent a whole paragraph at the same distance from my left margin, with the simplest command or environment? The Lorentz force on a charged particle is perpendicular to the particle's velocity and the magnetic field it's moving through. This is obvious from the equation:  $$ \mathbf{F} = q\mathbf{v} \times \mathbf{B} $$  Is there an intuitive explanation for this behavior? Every explanation I've seen simply points at the equation and leaves it at that.  I can accept mathematically why $\mathbf{F}$ will be perpendicular to $\mathbf{v}$ and $\mathbf{B}$ (assuming the equation is correct, which it is of course). But that doesn't help me picture what's fundamentally going on.  Trying to create an analogy with common experiences seems useless; if I were running north through a west-flowing "field" of some sort, I wouldn't expect to suddenly go flying into the sky.  I'm hoping there's a way to visualize the reason for this behavior without a deep understanding of advanced theory. Unfortunately, my searching for an explanation makes it seem like something one just has to accept as bizarre until several more years of study. Adding 3 electron spins I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers needed are total spin and total z-component.  But how do you add three 1/2-spins? It should yield 8 different eigenstates. Which quantum numbers do you need to characterise the 8 states?  It is not as easy as using C-G-coefficients and the usual quantum numbers as for the total momentum the doubly degenerate 1/2 state and quadruple degenerate 3/2 state can describe only 6 or the 8 states. You will need an additional quantum number for the degeneracy.  So how do you get the result?  (I actually tried out myself with a large 8x8 matrix. The total spin 1/2 is each doubly degenerate. For the additional quantum number I chose the cyclic permutation. Spin 1/2 states are neither symmetric nor antisymmetric. But what is the usual way to derive this?)  EDIT: For reference I'm adding my results for up to 4 spins from some time ago:   If you recall the basics of quantum mechanics with matrices it is actually a straightforward matrix diagonalization and requires no specialized knowledge. However, you still need to find an additional operator which breaks degeneracy. I chose the cyclic permutation, which seems to do the job. Please refer to the below answer, since I haven't checked all details.