# Is the pushforward of a locally free sheaf by an open immersion coherent?

Let $$X$$ be a quasi-projective variety, $$Y$$ a projective variety, and $$f:X \rightarrow Y$$ an open immersion. If $$\mathcal{F}$$ is a locally free coherent sheaf, what can be said about $$f_\ast \mathcal{F}$$? Is it coherent? Is it torsion free? Is it reflexive?

• $F=\mathcal{O}_X$ is a counterexample for the first and the third question. Sep 14, 2010 at 11:42
• You also might find Section 1, of Generalized Divisors on Gorenstein Schemes'' a useful read. In particular Prop 1.11 and Thm 1.12. Sep 14, 2010 at 15:34

Let $$Y$$ be a projective variety and let $$X\subset Y$$ be an open subset with complement the closed subset $$S:=Y\setminus X$$. Call $$f:X\hookrightarrow Y$$ the inclusion.
Let $$\mathcal F$$ be an algebraic coherent sheaf without torsion on $$X$$.

Theorem (Serre-Grothendieck) Suppose that $$Y$$ is normal and that $$S$$ has codimension $$\geq 2$$. Then the sheaf $$f_\ast \mathcal F$$ is coherent.

Serre, Prolongement de faisceaux analytiques cohérents, Ann.Inst.Fourier 16 (1966), 363-374

• This is close to Torsten's answer, but Serre supposes that the sheaf is without torsion, rather than reflexive.The article also considers the analytic case. Sep 14, 2010 at 14:04
• In the context of locally free F this statement (replacing projective variety with integral scheme) follows immediately from algebraic Hartogs' lemma
– Tomo
Apr 28, 2018 at 3:51

Dear Yemon,

a)The sheaf $f_\ast \mathcal{F}$ is not coherent in general since its stalk will not be finitely generated over the local ring of a point of $Y\setminus X$. For example take $P$ a point of $\mathbb P^1=Y$ and $X= \mathbb P^1 \setminus P=\mathbb A^1$. Then for $\mathcal F =\mathcal O_X$, you get $(f_\ast \mathcal{F})_P= Rat(Y)$

b) The direct image $f_\ast \mathcal{F}$ will be torsion free because an inductive limit of torsion free modules over a domain is torsion free ( I assume that variety means in particular integral scheme.)

c) I'm not sure reflexive is a reasonable concept for a non-coherent sheaf.

• Thank you for your answer. My question was motivated by the fact that I would like to construct a reflexive coherent sheave $\mathcal{G}$ on $Y$ such that $\mathcal{G}|_X = \mathcal{F}$. Is it possible? What if I suppose that $Y$ is normal and codim($Y\setminus X) \geq 2$? Sep 14, 2010 at 12:50
• This is different, you may always extend any coherent sheaf on $X$ to some coherent sheaf on $Y$ and then take its double dual. Under your supplementary conditions such an extension is equal to the direct image (which in particular is coherent and reflexive). Sep 14, 2010 at 12:56
• Indeed, for the part on finding a coherent sheaf on $Y$ that restricts to $F$ on $X$, take a look at Hartshorne, chapter II, exercise 5.15 where this construction is done step by step. Sep 14, 2010 at 15:16

By the way, assuming by varieties, you mean irreducible varieties, then for the second question, the answer is yes.

For the torsion free-ness, suppose that $r \in H^0(U, O_X)$ kills some non-zero element $z \in H^0(U, f_* \mathcal{F}) = H^0(U \cap X, \mathcal{F})$. By restriction, $r$ is a non-zero element of $H^0(X \cap U, \mathcal{O}_Z)$. We still have $rz = 0$ even in this setting, and so by restricting to an affine cover of $X$, it still happens. This will contradict the torsion-freeness (and thus in particular the locally-freeness) of $\mathcal{F}$.