2
$\begingroup$

Let $X$ be a smooth $k-$ variety and $F$ a coherent sheaf on it. One knows that the sheaf of differentials $\Omega^1_X$ of $X$ is locally free.

Hence one has by standard isomorphisms

$Hom(F,F\otimes \Omega^1_X) \simeq H^0(X, \underline{Hom}(F,F\otimes \Omega^1_X)) \simeq H^0(X,\underline{Hom}(F,F)\otimes \Omega^1_X)\simeq End(F) \otimes \Omega^1_X(X)$

where in the Hom's I always mean $\mathcal O_X$-linear.

Now I am a bit confused: How do I practically get this Iso?

I mean: if I have an element $\phi \otimes \omega$ in the last vector space, how do I get my sheaf map $F \rightarrow F\otimes \Omega^1_X$ ? And also the converse isn't clear to me.

  • 0
    @Martin: my question was motivated by the fact that the difference of two connections yield an element in the left hom. People then always write that this corresponds to an element in $\Omega^1(End(F))$; I thought that with this last one one simply means the tensor product I wrote on the right. But if not, what is it then?2011-11-01

1 Answers 1

0

The natural morphism $End(F) \otimes H^0(X,\Omega^1_X) \to Hom(F,F\otimes\Omega^1_X)$ is indeed pretty "natural": just map $\phi \otimes \omega$ to $(s \mapsto \phi(s) \otimes \omega)$.

The inverse isomorphism is not as easy to write down because it uses dualizability of $\Omega^1_X$ (the fact that it is locally free of finite rank). Take a local basis $(dx_1,\ldots,dx_n)$ of $\Omega^1_X$, then any $\psi : F \to F\otimes \Omega^1_X$ can be written $\psi(s) = \sum \psi_i(s) \otimes dx_i$ where $\psi_i \in End(F)$. You should then just check that the $\sum_i \psi_i \otimes dx_i$ glue nicely as a global section of $End(F) \otimes \Omega^1_X$.

  • 1
    Since locally the second map is the inverse of the first one, which is defined globally, the former must glue correctly.2011-11-02