I'm trying to prove the snake lemma for sheaves of abelian groups, namely:     If we have the following commutative diagram with exact rows:            Then we have an exact sequence:   $$0\to\text{ker}(f)\to\text{ker}(g)\to\text{ker}(h)\to\text{coker}(f)\to\text{coker}(g)\to\text{coker}(h)\to 0$$   The part $0\to\ker(f)\to\ker(g)\to\ker(h)$ is easily constructed from $i,j$ and it's easy to verify it is exact. Now let's call $\mathcal{F}'',\mathcal{G}'',\mathcal{H}''$ the presheaves defined by $U\mapsto \mathcal{F}'(U)/f(\mathcal{F}(U))$, $U\mapsto \mathcal{G}'(U)/g(\mathcal{G}(U))$ and $U\mapsto \mathcal{H}'(U)/h(\mathcal{H}(U))$, whose sheafifications are $\mathcal{F}''\to\text{coker}(f)$, $\mathcal{G}''\to\text{coker}(g)$ and $\mathcal{H}''\to\text{coker}(h)$ respectively. The presheaf morphisms $\mathcal{F}''\to\mathcal{G}''\to\mathcal{H}''$ are naturally induced by $i',j'$ and, using the universal property of sheafification, we get the following commutative diagram:        If I'm not mistaken, we can conclude $\text{coker}(g)\to\text{coker}(h)$ is an epimorphism directly by the fact that $\mathcal{G}'\to\mathcal{H}'$ is an epimorphism.   Now I'm stuck trying to proving that $\text{coker}(f)\to\text{coker}(g)\to\text{coker}(h)$ is exact and I have no idea how to build the connection map $\ker(h)\to\text{coker}(f)$.  Any ideas?
Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A&#39; \mathrel{\overset{f&#39;}{\to}} B&#39; \mathrel{\overset{g&#39;}{\to}} C&#39; \to 0 $$ and morphisms $a : A \to A&#39;, b : B \to B&#39;, c : C \to C&#39;$ making the obvious diagram commute. The snake lemma states that there is then an exact sequence $$0 \to \ker a \to \ker b \to \ker c \to \operatorname{coker} a \to \operatorname{coker} b \to \operatorname{coker} c \to 0$$ where the morphisms between the kernels are induced by $f$ and $g$ while the maps between the cokernels are induced by $f&#39;$ and $g&#39;$.  It is not hard to show that the morphisms induced by $f, g, f&#39;, g&#39;$ exist, are unique, and that the sequence is exact at $\ker a, \ker b, \operatorname{coker} b, \operatorname{coker} c$. With the use of a somewhat large diagram shown , we can even construct the connecting morphism $d : \ker c \to \operatorname{coker} a$. However, I'm stuck showing exactness at $\ker c$ and $\operatorname{coker} a$. I thought Freyd might have had an element-free proof in his book, but it turns out he proves it by diagram chasing and invoking the Mitchell embedding theorem [pp. 98–99]. Is there a direct proof?
How can I restrict various roles with limit in creating nodes of a particular content type Is it true that $(R\times S)[G]\cong R[G]\times S[G]$? Possible to click on an object in the viewport and the object list hierarchy automatically goes to that object also? I can see that this works for any integer $n$, but I can't figure out why this works, or why the number $42$ has this property. How to make sense out of the $\epsilon-\delta$ definition of a limit? The informal intuition for the limit of a function is this:     What is the value of the function $f$ as $x$ gets infinitely close to $c$?   How on earth does this monster  $$ \lim_{x \to c} f(x) = L  \iff  (\forall \varepsilon &gt; 0)(\exists \ \delta &gt; 0) (\forall x \in D)(0 &lt; |x - c | &lt; \delta \ \Rightarrow \ |f(x) - L| &lt; \varepsilon)$$ capture our intuition?  I want the answer to explain and motivate every bit of this formal statement. Remember: I am able to parse the statement, I just don't see how it captures our informal intuition. Provide an easily discoverable way to get the full URL to an answer , clicking &quot;link&quot; &quot;share&quot; for anwers yields a short (traceable) URL in the &quot;share a link to this answer&quot; dialog, such as †:  These short URLs show no clue about the linked question whatsoever. Also, when different people use a short URL for the very same post, then the user id suffix makes those be different URLs for a browser. Such links will then not be shown as visited‡. I know how to get URLs that have a proper slug. But as I see  pop up on these sites (where referrals ), I assume many others just don't realize full URLs would be better for use on these sites themselves? Or many don't want to do the little effort to get the full URL? So: can we please have an easy way to get the full URL as well, in a way that is easily found for those who don't know about the different links? (Currently for questions, clicking &quot;link&quot; has the same effect. But then one can also right-click the title, or take the URL from the location bar, to get the full human-readable URL. Once, Jeff .)  † The short links for answers are not (yet) [automatically converted][6] into a clickable question title either, unlike links such as `https://meta.stackexchange.com/questions/74274/privacy-leak-in-permalink#75963`, which renders as https://meta.stackexchange.com/questions/74274/privacy-leak-in-permalink#75963 But I guess that will change soon.  ‡ On Meta the CSS for posts doesn't indicate such difference anyway, which is a different issue. How to determine the amount of experience needed for leveling up Related questions under this topic might include:   How should I base how much experience points it takes to get to the next level? What factors are involved in this? How do I keep the player not entertained long enough till he levels up? How can I vary the way the player gets experience points? Search among posts you've voted for Related posts:    These posts are different from mine in that they merely ask for the &quot;votes&quot; tab back or ask if it is at all possible to get such information, while I propose how to implement this as a search feature. Background I recently found myself facing a problem very similar to one that I had solved before. I remembered that I found the solution on SO and I remembered upvoting the post containing the solution. I wasn't able to find that particular post by just searching with relevant keywords. It would have been nice if I could search only among the questions I have voted on. I searched on meta and found that there had been a tab for looking at what you've voted on, but that it was removed due to lack of interest. This is understandable, since just listing the hundreds/thousands of posts you've voted on isn't particularly useful. Proposal What I want is to be able to search for posts I've voted on similarly to the way I'm able to search on my posts within a specific tag. For instance: user:310121 [iphone] apple  This will return all my posts tagged iPhone, where I mention Apple. This is brilliant. This is just how I want search to work. Now what I want to do is to extend this to include votes. The exact syntax isn't relevant, but I had to make one up to show you what I mean: To show all posts you've voted on: user:310121 {votes}  To show all posts you've upvoted: user:310121 {upvotes}  To show all posts you've downvoted: user:310121 {downvotes}  You could probably omit the the user part, since you're only allowed to see your own votes. (Unless it's desirable to be able to look at other peoples votes. Maybe you should be allowed to do this if you have enough rep? Well, that's another discussion.) I guess that even if you're allowed to see other peoples votes it would make sense to display your own votes if the user part is omitted. So in my case I could have done the following search and found the post I was looking for: {upvotes} keyword  Lastly I just want to suggest that clicking on your vote count will work just like clicking on a tag. If I click on a tag in my profile the following search will be done: user:310121 [tag]  So if I click on my upvote count the following search should be made: user:310121 {upvotes}  The same goes for downvotes: user:310121 {downvotes}  This would be a very useful feature and it wouldn't be intrusive in any way. The users that don't care for this feature might never know that it's there. And just for the record the favorite system doesn't replace this feature. Singular or plural verb with two subjects Possible Duplicate:          I'm writing an interrogative sentence questioning someone else's writing:  "Is grammar and spelling correct?"  Can I use the singular verb "is" or must it be the plural verb "are"?  (The sentence is one of a list of interrogative questions about the quality of writing, e.g. "Are words in the right order?" "Does the copy flow?") Proving that the set of $\nu$ measurable sets $M\subseteq P(X)$ is an algebra How to install bicycle pedals without a notch on the spindle? I bought a pair of bicycle pedals that to my surprise had no notches on the pedal spindle to use when tightening them. I managed to screw them in by hand, but they are not quite tight since I can't use a spanner on them. So what would be the correct way to tighten them? Are there any special tools or similar that I'm supposed to use?  The offending pedals can be seen below. The spindle is, as you can see, completely smooth. I would like to align the view straight on to a face, perpendicular to the normal. How do I align the 3D viewport to the face of a mesh? How can I decide between the medium sized intuos tablet or the small one? I am a high school student with an interest for drawing tablets for some hobby. I went to check on a store that sell Intuos, a sales representative (or something) of Wacom told me that most students buy the medium sized tablets. Even the staff said that since it is always bought the supplies of the medium size are a lot more than the small size.  I am on a tight budget (which is why I am buying the Intuos), and the small size seems small to me (when my hands are open, my thumb and my pinky would reach the sides). But then, why would Wacom sell the Manga variant in small? Is the staff just saying that so I would buy the more expensive one?  I have enough savings from my small allowance to buy the small size right now, should I wait for another month so that I can buy the medium size? Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$.  Then why does this ring not contain any two-sided ideal?    Thanks for any clarification, and this is an exercise from the notes of Commutative Algebra by Pete L Clark, of which I thought as simple but I cannot figure it out now. How to install OneDrive on ubuntu 13.10 x64 Display historical characters "uͤ", "oͤ" and "aͤ"? Duplicate notifications I receive every push notification from the iOS app twice on iPhone. The second comes immediately after the first. This happens every single time.   I also have the app installed on an iPad and signed in to the same account - could this be a possible cause?   App Version: 1.4.0 Device: iPhone7,2 OS Version: Version 9.1 (Build 13B143) $f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$. Assume that $f_n → f$ uniformly on $S$ and each $f_n$ is continuous on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.  I'm stuck in thinking about it and I don't manage to prove it neither to find a counterexample. Thanks for your help. In "Tasksel" how do I select a package? I found a video on YouTube that shows someone using the command tasksel from the server's boot menu after a fresh install of Ubuntu Server. Simple enough it looked like, but when I go to the options I want to choose for installation it has no information on how to select them, and when I click Enter on one of them it takes me back to the command prompt.  So how do I make a selection of the items listed under tasksel?  I have tried Shift++ to add, but no luck. Anyone know?  I want to install Ubuntu Desktop, as Ubuntu GNOME GUI is not an option in the listing for Ubuntu 17.10 Server. When do I use "I" instead of "me?"