## A coherent sheaf with connection is locally free

This post will be rather technical, but I think it’s worthwhile to write down the following “folklore” proof, which doesn’t seem to be laid out carefully in the literature. Let be a smooth variety over (I think the proof also goes through on a complex manifold), a coherent sheaf on , and a connection on , not assumed flat. We will prove that is then locally free, or in other words the sheaf of sections of a vector bundle on . This shows that the presence of a connection strongly “rigidifies” the underlying sheaf. The proof is suggested by Malgrange in his article “Regular connections after Deligne,” although he gives no details there. I have seen a more sophisticated proof, although it assumes the connection is flat: then one can view as a -module and apply Kashiwara’s theorem (see Proposition 5.13 in Gaitsgory’s notes on geometric representation theory).

Since is coherent, it suffices to show that is locally free at any (closed) point . To this end, choose a minimal set of generators for as an -module, and consider a relation

where . Note that we must have by the minimality of , and if not all these coefficients are zero there is a unique such that for all but for some . Choose so that is minimal among all such relations.

Denote by the tangent sheaf on and recall that induces a “covariant derivative”

,

and the Leibniz identity for implies that for all , , and we have

.

We assume for the moment a crucial lemma, namely that there is a tangent vector such that (this is automatic) but . Applying to our relation yields

,

and now if we write and regroup we obtain

.

Now by the assumption on we have , and our choice of guarantees that each coefficient of this new relation lies in , but when not in . This contradicts the minimality of .

Finally, we prove the lemma. Choose local coordinates , meaning these germs represent a basis of , so by our assumption that is smooth we have as graded algebras. Moreover, if we write for the dual basis, then the map induced by a tangent vector is identified with the usual action of partial differential operators on polynomials. Thus we have reduced ourselves to the case that is the affine space, is the origin, and is a homogeneous polynomial of degree . But now since the claim is obvious: if appears with nonzero exponent in any monomial in , then has degree .

The proof of this “folklore” fact is laid out clearly by dimension induction early in Deligne’s SLN book on regular singular points.. In fact, Deligne’s book does much better, simultaneously proving the relative version for smooth morphisms in the category of complex analytic spaces allowing the base to be arbitrary (no smoothness hypothesis on it!).

Thanks for the reference! I once had a copy of this book and lost it while I was on vacation in Florida. Maybe someday I’ll get another one.