Is there a reason for defining the differential forms before the vector fields ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T23:55:43Z http://mathoverflow.net/feeds/question/26947 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26947/is-there-a-reason-for-defining-the-differential-forms-before-the-vector-fields Is there a reason for defining the differential forms before the vector fields ? nicojo 2010-06-03T18:34:22Z 2010-06-03T23:41:37Z <p>Hi, my question is the following : </p> <p>In EGA IV chapter 16, given $X$ a scheme over $S$, Grothendieck defines first $\Omega^1_{X/S}$, the $O_{X}$-module of the 1-differentials. He then defines the tangent sheaf : $T_{X/S}:= Hom_{O_X} (\Omega^1_{X/S}, O_X)$, which is equal to $Der(O_X, O_X)$. Why, one does not do the opposite and first define $T_{X/S}:=Der(O_X, O_X)$ and then $\Omega^1_{X/S}:=Hom_{O_X} (T_{X/S}, O_X)$ ?</p> <p>I suspect there are several reasons for this : </p> <p>1) This new object, whose definition seems to be easier, is maybe less handy to work with.</p> <p>2) In some important cases, it gives the wrong object</p> <p>3) Some other philosophical reason</p> <p>I would like to have the opinion on this question of mathematicians who know more than me geometry and differential forms.</p> <p>Thanks.</p> http://mathoverflow.net/questions/26947/is-there-a-reason-for-defining-the-differential-forms-before-the-vector-fields/26953#26953 Answer by Donu Arapura for Is there a reason for defining the differential forms before the vector fields ? Donu Arapura 2010-06-03T19:27:19Z 2010-06-03T19:27:19Z <p>Unless $X/S$ is smooth, or something close to it, $\Omega_{X/S}^1$ may fail to equal $Hom(T_{X/S},O_X)$. The precise condition should be that $\Omega_{X/S}^1$ is reflexive. So it does appear that differentials are more fundamental.</p> http://mathoverflow.net/questions/26947/is-there-a-reason-for-defining-the-differential-forms-before-the-vector-fields/26966#26966 Answer by Heinrich Hartmann for Is there a reason for defining the differential forms before the vector fields ? Heinrich Hartmann 2010-06-03T20:13:41Z 2010-06-03T20:13:41Z <p>Note that the universal property of Kaehler differentials is just</p> <p>$Hom(\Omega_{X/S},M)=Der_S(\mathcal{O}_X,M)$</p> <p>so in a certain sense, vector fields do come first.</p> http://mathoverflow.net/questions/26947/is-there-a-reason-for-defining-the-differential-forms-before-the-vector-fields/26989#26989 Answer by Georges Elencwajg for Is there a reason for defining the differential forms before the vector fields ? Georges Elencwajg 2010-06-03T23:41:37Z 2010-06-03T23:41:37Z <p>Dear Nicojo, since you mention philosophical reasons let me remark that, since a scheme $(X,\mathcal O_ X)$ is a locally ringed space, the most primitive concepts should be as close as possible to the data, the (generalized) functions encapsulated in the sheaf $\mathcal O_X$ . </p> <p>At a point $x\in X$ , what could be more natural to consider as the COtangent Zariski space $\mathcal M/\mathcal M^2$ ? It just consists of the functions vanishing at $x$ modulo those vanishing at higher order. And the dimension $d$ of this space will already tell you if the (locally noetherian) scheme $X$ is regular or not at $x$, according as $d=dim\mathcal O_{X,x}$ or $d> dim\mathcal O_{X,x}$</p> <p>In the relative case $X/S$ the sheaf $\Omega_{X/S}$ will give you a lot of information. Just its nullity at a point already tells you (under a mild finiteness condition) everything about nonramification: </p> <p>$\Omega_{X/S,x}=0 \iff$ $f:X\to S$ is unramified at $x$ </p> <p>And this is only the beginning: by taking the top exterior product of $\Omega_{X/k}$ you get (in the smooth projective case over an algebraically closed field $k$) the canonical sheaf $\omega_{X/k}$ which is a key concept for Serre duality. This canonical sheaf also plays a fundamental role in the classification of curves, surfaces and higher dimensional varieties, arguably the very heart of classical algebraic geometry . For example to $X$ you associate its canonical ring $R$, a graded ring whose degree $m$ component is $R_m=\Gamma(X,\omega^m)$. The Kodaira dimension of $X$ is $\kappa(X)=trdeg_k (R)-1$ and general varieties are defined as those with $\kappa(x)=dim(X)$. They are supposed to be generic in some sense and have been intensively studied, in particular by the Japanese school (Kodaira, Iitaka, Kawamata, Ueno, Mori, ... )</p>