$\underline {Background}$:Let,$X$ be a projective scheme over a field $K$.

Let, $\phi:\mathcal F\to\mathcal F\otimes\mathcal L$ be a morphism of $\mathcal O_X$ modules for some vector bundle $\mathcal F$ of rank $n$ and for some line bundle $\mathcal L$.

Let us denote $T$:=total space of line bundle $\mathcal L$=relative spectrum of(Sym($\mathcal L^*$))

By universal property we have a morphism $p:T\to X$

$\underline {Question}$:(1) What is the meaning of the statement "$\phi:\mathcal F\to\mathcal F\otimes\mathcal L$ corresponds to an action of Sym($\mathcal L^*$) on the sheaf $\mathcal F$ "

(I mean what is the precise meaning of action of one sheaf over another?)

(2) How do we construct a coherent sheaf of modules over $T$ out of this action?

Any help from anyone is welcome.


(1) The action of a quasicoherent sheaf of $\newcommand\calO{\mathcal{O}}\calO_X$-algebras, $\newcommand\calR{\mathcal{R}}\calR$ on a quasicoherent sheaf $\newcommand\calF{\mathcal{F}}\calF$ is a morphism of $\calO_X$-modules from $\calR$ to $\newcommand\shHom{\operatorname{\mathcal{H}om}}\shHom(\calF,\calF)$. At least, this is the definition that appears obvious to me, based on what such things usually mean.

Now given $\phi$, how do we get an action of $\newcommand\calL{\mathcal{L}}\newcommand\Sym{\operatorname{Sym}}\Sym(\calL^*)$ on $\calF$?

Well, we can define the morphism $\tau: \Sym(\calL^*)\to \shHom(\calF,\calF)$ on open affine sets $U$, and on degree one elements of $\Sym(\calL^*)$, $\alpha\in\Sym(\calL^*)(U)_1$, define $\tau_U(\alpha) = \mu_U\circ (1\otimes\alpha)\circ \phi_U$, where $\mu:\calF\otimes \calO_X\to \calF$ is the usual natural isomorphism.

You do have to check that the images of degree one elements commute in $\shHom(\calF,\calF)$ to make sure that the morphism is well defined. This isn't hard to check though.

(2) To construct the corresponding coherent sheaf of modules over $T$, there are a couple ways to go about it. The first is to see if you can convince yourself that if $\mathcal{A}$ is a quasicoherent sheaf of $\calO_X$-algebras and $\mathcal{M}$ is a quasicoherent $\calO_X$-module that is also a $\mathcal{A}$ module, then it lifts to a module on $\newcommand\relSpec{\operatorname{\mathbf{Spec}}}\relSpec\mathcal{A}$ in exactly the same way that modules over rings correspond to quasicoherent sheaves on affine schemes. (This is a general idea. Relative constructions should be thought of as behaving more or less identically to their nonrelative cousins. Indeed the nonrelative versions can be thought of as being the same as the relative versions over $\newcommand\Spec{\operatorname{Spec}}\Spec\newcommand\ZZ{\Bbb{Z}}\ZZ$.)

Anyway, in our particular case, the concrete construction goes like this.

Let $p:T\to X$ be the structure map. Fix $\{U_i\}$ an open cover of $X$. $p\newcommand\inv{^{-1}}\inv(U_i)=\Spec (\Sym(\calL^*)(U_i))$, and $\calF(U_i)$ is conveniently a module over $\Sym(\calL^*)(U_i)$, so it defines a quasicoherent sheaf on the spectrum, and the restriction maps coming from the sheaf structure of $\calF$ induce gluing maps for the corresponding modules on $T$, allowing the sheaves corresponding to $\calF(U_i)$ to glue together into a quasicoherent sheaf on $T$.

Then you just need to check that, let's call it $\tilde{\calF}$, is in fact coherent, not just quasicoherent. However, this follows since $K$ is noetherian, so $X$ is noetherian, so $T$ is noetherian, hence the fact that $\calF$ is finitely generated over $\Sym(\calL^*)$ (which contains $\calO_X$) implies that it is coherent.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.