4
$\begingroup$

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.

$\endgroup$

1 Answer 1

2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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