Update: never mind, This question has been asked before  Let $f:E \to S$ be an elliptic curve (precise definition is given below from Hida’s book Geometric Modular Forms and Elliptic Curves). Is $f_*\Omega_{E/S} \cong \mathcal{O}_S$?  I would think so by Grothendieck-Serre duality and (2.15) in the image below, but few pages later Hida says (the push forward in question) is invertible on S, i.e locally free on S - while I seem to think it is globally free.
If $f:E\rightarrow S$ is an elliptic curve over a scheme $S$ (so $f$ is proper and smooth of relative dimension one with geometrically connected fibers of genus one, equipped with a section $0:S\rightarrow E$), then is the sheaf $\underline{\omega}_{E/S}:=f_*\Omega_{E/S}^1$ actually free of rank one? According to the statement of Grothendieck-Serre duality in Hida's book Geometric Modular Forms and Elliptic Curves, there should be a canonical isomorphism $\underline{\omega}_{E/S}\cong\mathcal{Hom}_{\mathscr{O}_S}(R^1f_*\mathscr{O}_E,\mathscr{O}_S)$, and the first result on elliptic curves in this book is that $R^1f_*\mathscr{O}_E\cong\mathscr{O}_S$. It's also not clear to me whether $S$ is being assumed (locally) Noetherian, but if that's necessary for Grothendieck-Serre duality to hold, then I'm fine with assuming it. I can't find another reference with a statement of Grothendieck-Serre duality in this generality which does not use the language of derived categories (which I unfortunately don't understand).  The reason I'm kind of skeptical about this is that in Hida's book, as well as in Katz-Mazur, it is said that $\underline{\omega}_{E/S}$ is invertible, so that an $\mathscr{O}_S$-basis $\omega$ for $\underline{\omega}_{E/S}$ can be found locally on $S$. If the invertible sheaf in question were really trivial then one would be able to choose a global $\mathscr{O}_S$-basis for $\underline{\omega}_{E/S}$, and there would be no reason to talk about doing so locally. Hida goes on to say that, choosing an $\mathscr{O}_S$-basis $\omega$ locally on $S$ allows one to regard $(\Omega_{E/S}^1,\omega)$ as a relative effective Cartier divisor in $E/S$, which also doesn't make complete sense to me because if we can only find $\omega$ locally, how are we getting a global section $\omega\in H^0(E,\Omega_{E/S}^1)=H^0(S,\underline{\omega}_{E/S})$ (unless $\underline{\omega}_{E/S}$ really is trivial)?
How is dark energy consistent with conservation of mass and energy? I initially thought that dark energy must in some way violate conservation of mass and energy since the component of the energy density of space that comes from dark energy is constant, and space is expanding. Therefore, as space expands the energy in the universe that comes from dark energy would increase. I presumed the source of this energy was not coming from the conversion of other types of energy to dark energy, so it must violate conservation.  I decided to Google this and came upon this article:    It says that dark energy does NOT violate conservation and quotes Carroll, Press, and Turner (1992):  "…the patch does negative work on its surroundings, because it has negative pressure. Assuming the patch expands adiabatically [i.e. without loss or gain of heat], one may equate this negative work to the increase of mass/energy of the patch. One thereby recovers the correct equation of state for dark energy: P = – ρ c2. So the mathematics is consistent."  Is there a way to explain this in layman's terms? (The blog attempted to do this, but it was very unclear to me.) More specifically, can you explain where my initial train of thought described above fails when I erroneously concluded that dark energy violates conservation?  Thank you. Homeomorphism of the real line-Topology I want to do a survey of textbooks in set theory. Amazon returns 3582 books for the keywords "set theory". A small somewhat random selection with number of references in Google scholar is the following.   Two questions:    What textbooks in set-theory are considered standard?  What is a good place to aggregate standard references (of textbooks)?    (I'm only interested in text-books, which as a format already provide a kind of standardization)  Textbooks on set theory (author: title (year) #Google Scholar citations )   Thomas J. Jech: Set Theory 3rd Edition (1978) #2441  Kenneth Kunen: Set Theory (1980) #1881  P. R. Halmos: Naive Set Theory (1974) #1079 AA Fraenkel, Y Bar-Hillel, A Levy: Foundations of set theory (1973) #696 K Kuratowski, A Mostowski, M Mączyński: Set theory (1976) #609 Suppes: Axiomatic set theory  (1972) #589 Quine: Set theory and its logic (1969) #442 A Levy: Basic set theory (1979) #442 Herbert B. Enderton: Elements of Set Theory (1977) #349 Karel Hrbacek, Thomas J. Jech: Introduction to set theory (1999) #284 RR Stoll: Set theory and logic (1979) #281  N. Bourbaki: theory of sets (1970) #211 K Devlin: The joy of sets: fundamentals of contemporary set theory (1994) #182 Ciesielski: Set theory for the working mathematician (1997) #156 YN Moschovakis: Notes on set theory (1994) #138 FW Lawvere: Sets for mathematics (2003) #118 G Takeuti, WM Zaring: Introduction to axiomatic set theory (1982) #106 Kaplansky: Set Theory and Metric Spaces (2001) #103 Potter: Set Theory and Its Philosophy: A Critical Introduction (2004) #91 PT Johnstone: Notes on logic and set theory (1987) #87 Judith Roitman: Introduction to Modern Set Theory (1990) #47 Derek Goldrei: Set Theory: For Guided Independent Study (1996) #31 Basic set theory: A Shen, NK Vereshchagin (2002) #14 Ralf Schindler: Set Theory (2014) #13 Foreman, Kanamori: Handbook of Set Theory (2009) #38   Note:  Please feel free to add textbooks to the list, as long as they are not just on a smaller subset of set theory. (Some of the dates might not be the first published edition.) If I have the following enum:  public enum ReturnValue{     Success = 0,     FailReason1 = 1,     FailReason2 = 2     //Etc... }   Can I avoid casting when I return, like this:  public static int main(string[] args){     return (int)ReturnValue.Success; }   If not, why isn't an enum value treated as an int by default? How to convert java.util.Date to java.sql.Date? I am trying to use a  as input and then creating a query with it - so I need a .    I was surprised to find that it couldn't do the conversion implicitly or explicitly - but I don't even know how I would do this, as the Java API is still fairly new to me. I am a freelancer programmer and so far I have only worked on projects on Windows platform. I don't have much idea about OS X hardware. I am not looking for a very high end configuration but integrated development environment and one database server should just run fine. I have a Dell laptop i7 processor, 8 GB RAM, 1 TB HDD; I am fairly satisfied with the performance. I am mainly going to use it for app development and not for any rich graphics app development.  What are the minimum and recommended hardware requirements for Xcode development on OS X? Let $x$ and $y$ be positive integers such that $xy \mid  x^2+y^2+1$. Show that $$ \frac{x^2+y^2+1}{xy}= 3 \;.$$ I have been solving this for a week and I do not know how to prove the statement. I saw this in a book and I am greatly challenged. Can anyone give me a hint on how to attack the problem? thanks How to find relevant and new open research topics? Is Java viable for serious game development? I have had no problems exporting .fbx files for Unity except recently some of my .fbx started to rotate the mesh. The asset is correctly oriented, but the mesh is rotated 90 degrees. Is there an explanation and fix for why this happens sometimes, and works properly otherwise? How do I get my Python program to sleep for 50 milliseconds? I am trying to find examples of maps between topological space which are local homeomorphism but not covering maps. Especially, how twisted has to be such a counterexample : can it be a local diffeomorphism between connected manifolds which is not a covering map ?   I found here() a nice proposition which state that a local homeo from a compact space to a connected Hausdorff space is a covering map.   I am interested in all type of counterexamples, from non-Hausdorff spaces to surfaces, to get a better picture of the differences between covering maps and local homeomorphisms.  Thank you ! Refused entry to UK, not sure what to do I'm a writer and theater/film director who has been traveling nearly all my life without incident. In February, I decided to pull up stakes in NYC and move to London. I am a freelancer and I'm working with three different organizations in NYC so I wasn't working illegally in the UK. I was, however, seeing if there might be some opportunities in London, ie. meeting with Artistic Directors of various theatre companies to see if they might be interested in producing my work.   In April, I was invited to a conference in San Diego and then a film festival in Switzerland. I didn't think it would be a problem if I went and then bummed around in France for a few days since I hadn't broken any laws and had left the UK well under my six month leave to stay. Well, when I tried to return to London, I was denied entry at Luton. The border control officer said I didn't have enough funds for a two month stay and put a black cross over a stamp in my passport. They put me in a detention room until the next flight back to Lyon, which was the following morning. I don't know anyone in Lyon so I ended up going to a friend's place in Paris.   After a few days of not knowing what to do, a friend suggested that perhaps I should try going back to London on the Eurostar since border control is in Paris. I wondered if I ought to go to the British Embassy but was deterred by something on their website that says that a visitors visa could take up to three weeks (I can't sit around here in Paris for three weeks) and anyway, the decision ultimately is with the border control, so I decided to take a chance on the Eurostar. I bought myself a return ticket in two weeks to Paris just so I had something to show.   Well, the UK border control denied me entry after giving me a third degree. They accused me of lying and seeking work in the UK. They also insisted that the amount of money I had was not enough to live on for two weeks. So I'm back at my friend's place in Paris, wondering what to do.   Is it possible for me to ever get back to the UK? I had to cancel one meeting and I'm wondering if I have to cancel going to the Sheffield DocFest. (Two of my projects were accepted to programs there.)   Should I apply for a visa from the British Embassy? Will this really take three weeks? When I started to say something about obtaining a visa, the border control officer cut me off and said, "Why are you going to the Embassy? This has nothing to do with the Embassy. We make the decisions." How do I create an Excel (.XLS and .XLSX) file in C# without installing Microsoft Office? Definition of $\mathbb{R}^\infty$ Question: Why is the topological space $\mathbb{R}^\infty$ defined to be the subset of $\prod_{i=1}^\infty \mathbb{R}_i$ consisting of sequences $(a_i)_{i=1} ^{\infty}$ such at most finitely many $a_i\neq 0$? Why does one insist on the condition that $a_i\neq0$ for at most finitely many $i$? Using "that" and "this" interchangeably I have one bash source run.sh as follows,  #!/bin/bash if [ $# -ne 1 ]; then     exit fi ...   when I execute it in two ways, there are different behaviors. The first way is,  source run.sh   It will close the terminal after execution. The second way is,  ./run.sh   this will simply finish running the script, and stay on the terminal. I am asking if there is a command for exiting a bash scripts for both source run.sh and ./run.sh execution. I have tried return too, which does not work well under ./run.sh execution.  More generally, I am interested in why this is happening, and what's difference between using "source" and "." for script execution? Value of $x=\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}} $ and location of second digit $1$ of $x$ What is the value of the series   $\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}}$   and can one find closed forms for the sums   $\sum\limits_{n=1}^N {\frac{1}{10^n-1}}$?   I found that $\sum_{n=1}^N {\frac{1}{10^n-1}} $  =  $\sum_{n=1}^N (\lim_{m\to\infty  }{\frac{10^{-n}-10^{-nm}}{1-10^{-n}}})$  =  $0.111\ldots+0.0101\ldots+\ldots$ but I can't go farther.     The original question is: "Where does the second '1' appear?" Do you need line of sight to cast spells on someone? Linked account +100 rep bonus also applied to account with 200