In remark 5.16.1 the claim is made that if $i\colon X\to \mathbb{P}^r_Y$ is a closed immersion, then $i^*(\mathcal{O}_Y)$ is a very ample invertible sheaf on $X$, but I cannot determine why it is locally of rank 1. I tried to appeal directly to the definition of $i^*(\mathcal{O}(1))$, but wasn't able to make any sort of meaningful progress.
states that if $f:X \rightarrow Y$ is a morphism of ringed space, $G$ is a locally free $O_Y$-module, then $f^*G$ is a locally free $O_X$ module.     Suppose that $G$ is a locally free $O_Y$ module, hence it is a free $O_Y|_U=O_U$ module. We have an induced map  $$f^{-1}(U) \rightarrow U$$ given by restriction of morphism of ringed spaces.   Hence, as $\bigoplus O_U \simeq G|_U$, $f^*:Mod(O_U) \rightarrow Mod(O_{f^{-1}(U)})$ is functorial, $f^*(G|_U) = (f^*G)|_{f^{-1}(U)}$, we have isomoprhism,  $$ \bigoplus O_{f^{-1}(U)} \simeq (f^*G)|_{f^{-1}(U)} $$
Currently I have some code that is being generated dynamically.  In other words, a C# .cs file is created dynamically by the program, and the intention is to include this C# file in another project.  The challenge is that I would like to generate a .DLL file instead of generating a C# .cs file so that it could be referenced by any kind of .NET application (not only C#) and therefore be more useful. PHP: How do I loop through every XML file in a directory? I'm reading the Algebra book by Knapp and he mentions in passing that an orthonormal set in an infinite dimension vector space is "never large enough" to be a vector-space basis (i.e. that every vector can be written as a finite sum of vectors from the basis; such bases do exist for infinite dimension vector spaces by virtue of the axiom of choice, but usually one works with orthonormal sets for which infinite sums yield all the elements of the vector space).  So my question is - how can we prove this claim? (That an orthonormal set in an infinite dimension vector space is not a vector space basis).    Edit (by Jonas Meyer): Knapp says in a footnote on  of Basic algebra:     In the infinite-dimensional theory the term "orthonormal basis" is used for an orthonormal set that spans $V$ when limits of finite sums are allowed, in addition to finite sums themselves; when $V$ is infinite-dimensional, an orthonormal basis is never large enough to be a vector-space basis.   Thus, without explicitly using the word, Knapp is referring only to complete infinite-dimensional inner product spaces. Proving that $\gcd(2^m - 1, 2^n - 1) = 2^{\gcd(m,n )} - 1$ Somewhere on Stack Exchange I saw the equation     $$\gcd(2^m-1,2^n-1)=2^{\gcd(m,n)}-1.$$   I had never seen this before, so I started trying to prove it. Without success...  Can anyone explain me (so actually prove) why this equation is true?  And can we say the same when replacing the '$2$' by any integer number '$a$'? StringPattern to remove markers around text How can I Remove .DS_Store files from a Git repository? How can I remove those annoying Mac OS X .DS_Store files from a Git repository? How to justify the definition of summation $s_n=\sum_{i=1}^n a_i=a_1+\cdots+a_n$? I read  and found that they define summation via recursion, so I decided to formalize the proof that that this definition is actually valid. I've two questions:   Does my proof contain any error? Are there other simple ways to define summation?   Thank you so much!     Suppose that $(a_1,\cdots,a_n)$ is a finite sequence in $\mathbb N$. Show that there is a sequence $(s_1,\cdots,s_n)$ such that $s_1=a_1$ and $s_{i+1}=s_i+a_{i+1}$ for all $1\leq i&lt;n$.   My attempt:  We define mapping $f$ as follows: $f: \mathbb N\times\mathbb N\to\mathbb N\times\mathbb N: (i,a)\mapsto\begin{cases} (i+1,a+a_{i+1})&amp;\text{if }i&lt;n\\ (i+1,a)&amp;\text{if }i\geq n \end{cases}$  By recursion theorem, there is a unique sequence $(p_i\mid i\in\mathbb N)$ such that $p_0=(1,a_1)$ and $p_{i+1}=f(p_i)$. Let $\pi:\mathbb N\times\mathbb N\to\mathbb N$ be the projection to the second co-ordinate i.e. $\pi(i,a)=a$. Let $s_i=\pi(p_i)$ for all $1\leq i\leq n$, then $(s_i\mid 1\leq i\leq n)$ is the required sequence. It's clear from the definition of $s_i$ that $s_{i+1}=s_i+a_{i+1}$ for all $1\leq i&lt;n$. How can I change the language for dictation on the fly? With English not being my first language, I usually need support for at least two languages when using features like auto-correct and dictation.  Is it possible to toggle the language which is used for dictation via a shortcut?  As of now, I always have to manually navigate through the menus in the System Preferences which if far from convenient. I've tried to see if there are any pre-defined actions in Automator, but found none. In the neural network literature, often we encounter the word "tensor".  Is it different from a vector? And from a matrix? Have you got any specific example that clarifies its definition?  I'm a bit confused about its definition. Wikipedia doesn't help and sometimes I have the impression that its definition depends on the specific machine learning environment used (TensorFlow, Caffee, Theano). When I have Excel running and I double click an Excel file in Windows explorer, I'd like it to launch in a new window by default so that I can move documents to separate monitors. Is there any way to get this behavior? Why are my render times 30+ minutes, but other people's are ~3 minutes? Texture is missing in Cycles? Because I am pretty new to blender I watched the tutorial  from BlenderCookie.com.  I loaded  onto a sphere but then in all views the texture is just some kind of blurry with no details. How is Russell’s Paradox equal to the universal set? Is there a general method (which can be implemented by hand) to finding the $n$-th roots of $2 \times 2$ matrices? Is there a similar method for a general $m \times m$ matrix? (for $n &gt; 1$ and $n\in\mathbb{Z}$) How to cut the non-overlapping part of 2 shapes in Illustrator CS6 I have 2 shapes (see below) a circle and a shape of the US. I want to remove the part of the US that falls outside the circle. How is this possible?    Thanks I have created a model which requires some faces to be dual sided.  I intend to import this model into Unity. Reading about this, it seems that the recommendation is to duplicate the polygons and then flip the normals.  What I have done so far is simply to duplicate the all of the relevant faces using Shift+D and then flip the normals. I don't like this method though as it hampers any future alterations to the model itself.  What is the best way of doing this? Question involving the Cauchy-Goursat Theorem Show $\int_C \frac{dz}{(z-z_0)^n} = 2\pi i$ when $n = 1$ and 0 when $n \gt 1$   I'm not entirely sure how to approach this one.  Any help would be much appreciated. If you are hurrying to reply, System → Administration → StartUp Disk Creator -- no, that's not what I'm talking about.   I want to try Ubuntu 11.04's Unity without touching my existing Ubuntu install.  To do this, I need to install the nVidia drivers first (sigh).  To do this, I need changes to persist a reboot.  To do this, I need to really install Ubuntu on a USB key.  How do you do that?    What I tried   I tried to make a USB key from , then boot from it, then choose "Install Ubuntu." The installer refused to install to the installation media itself. I tried, from my installed copy of Ubuntu:  sudo kvm /dev/sdb --cdrom .cache/testdrive/iso/ubuntu_natty-desktop-i386.iso   ...but the installer didn't detect the disk properly. When should I use an em-dash, an en-dash, and a hyphen? I generally know how to use a hyphen, but when should I use an en-dash (–) instead of an em-dash, or when should I use a hyphen (-) instead of an em-dash (—)? Which of the following is most correct?      The distance of the shortest path must not be higher than 10km.   The distance of the shortest path must not be greater than 10km.   The distance of the shortest path must not be bigger than 10km.   Or is there an even better word?