I believe this should be a well known result, but I wasn’t able to prove or find a good reference for it.


Let $E$ and $F$ be $n$-regular, respectively $m$-regular vector bundles in the sense of Castelnuovo-Mumford, the is it true that $E\otimes F$ is at most $n+m$-regular?


The result is true for modules, let’s say $M$ and $N$ provided that $Tor^1(M,N)$ is zero, due to a result by Caviglia. This is the case here locally, so I would like to claim it for the vector bundle case, but I am not sure if this holds, because of the difference in the definition of the regularity for modules and vector bundles.


Any help or reference will be appreciated.



Castelnuovo-Mumford regularity for tensor products of vector bundles
Yes, furthermore, your statement holds even if $E$ is locally free and $F$ is coherent.


You can apply the following fact to prove it:



> 
> Suppose a coherent sheaf $\mathscr F$ on $\mathbf P$ is resolved by a
> long exact sequence $$\cdots \rightarrow \mathscr F\_2\rightarrow
>  \mathscr F\_1\rightarrow \mathscr F\_0\rightarrow \mathscr F\rightarrow
> 0$$ of coherent sheaves on $\mathbf P$. If $\mathscr F\_i$ is
> $(m+i)-$regular for every $i\geq0$, then $\mathscr F$ is $m-$regular.
> 
> 
> 


You can prove this fact by chasing through the long exact sequence.


