# Global obstructions for being a quotient of a rank $d$ vector bundle

In this recent question (which now has an answer), Richard Thomas asked whether any projective $$k$$-scheme $$X$$ of (local) embedding dimension $$d(X)$$ can be embedded in a smooth $$k$$-scheme of dimension $$d(X)$$. If $$i \colon X \hookrightarrow Y$$ is such an embedding, then in particular we get a surjection $$i^*\Omega_Y \twoheadrightarrow \Omega_X$$. My (so far unsuccessful) strategy was to obstruct such a surjection from existing.

For a coherent sheaf $$\mathscr F$$, write $$d_x(\mathscr F) = \dim_{\kappa(x)} \mathscr F_x \otimes_{\mathcal O_{X,x}} \kappa(x)$$ and $$d(\mathscr F) = \max \left\{d_x(\mathscr F)\ |\ x \in X\right\}.$$

Question. If $$X$$ is a quasi-projective $$k$$-scheme, and $$\mathscr F$$ a coherent sheaf, does there exist a surjection $$\mathscr E \twoheadrightarrow \mathscr F$$ from a locally free sheaf of rank $$d(\mathscr F)$$?

Already if $$X = \mathbf A^n$$ this seems false to me; for example there should exist finite modules $$M$$ with $$d(M) = 2$$ that cannot be generated by $$2$$ elements (here I am using the Quillen–Suslin theorem that a finite projective module on $$\mathbf A^n$$ is free). But I don't know so many ways to prove that something is not generated by $$2$$ elements, except for a local obstruction $$d_x(\mathscr F) > 2$$.

I think it should be possible to give a negative answer to Thomas's question along these lines, by exhibiting a finite flat cover $$\pi \colon X \to \mathbf A^n$$ such that $$\pi_*\Omega_X$$ does not admit a surjection from a vector bundle of rank $$\deg(\pi) \cdot d(\Omega_X)$$. A great answer would incorporate something like this, but I would already be very happy with some global obstruction to surjecting from a vector bundle of a given rank.

• You may want to look at the literature around Foster-Swan theorem. – Hailong Dao Jan 28 at 20:23

Let me explain a simple example.

Let $$C \subset \mathbb{P}^3$$ be a twisted cubic curve. It is a locally complete intersection of codimension 2, hence its ideal $$I_C$$ is locally generated by two sections. Let me show that there are no surjections $$E \twoheadrightarrow I_C$$ from a locally free sheaf $$E$$ of rank 2.

Indeed, assume such a surjection exists. Its kernel is a reflexive sheaf of rank 1, hence is a line bundle, so we have an exact sequence $$0 \to L \to E \to I_C \to 0.$$ Restricting to $$C$$ we obtain an exact sequence $$0 \to \det N^* \to L\vert_C \to E\vert_C \to N^* \to 0,$$ where $$N^*$$ is the conormal bundle. But $$N^*$$ is locally free of rank 2, hence the surjection $$E\vert_C \to N^*$$ is an isomorphism, hence the middle arrow is zero, hence $$\det N^* \cong L\vert_C.$$ But the adjunction fromula shows that $$\det N^* \cong \mathcal{O}_C(-10)$$, and this line bundle does not restrict from $$\mathbb{P}^3$$ (because 10 is not divisible by 3). This contradiction proves that no surjection from $$E$$ as above exists.

Of course, the same argument works for many other lci of codimension 2.

• Could you say a word about where $\det N^*$ comes from? (I can do a $\mathscr Tor$ computation locally, but the Koszul resolution doesn't globalise by the very statement you're trying to prove.) – R. van Dobben de Bruyn Jan 29 at 15:12
• In fact, it does (sometimes this is called the fundamental local isomorphism): $Tor_i(\mathcal{O}_Z,\mathcal{O}_Z) \cong \wedge^iN^*$ for any lci scheme $Z$. – Sasha Jan 29 at 15:53
• I don't mean that the resolution generalizes, but its consequence (the isomorphism for $Tor_i$) does. – Sasha Jan 29 at 16:10
• See "Les K-groupes d'un schéma éclaté..." by Thomason (Inventiones 1993), Lemme 3.2 (the indexing is correct). – Riza Hawkeye Jan 29 at 21:03
• Maybe it's worth mentioning that in many "practical" situations, by the Serre construction, the existence of an isomorphism $\text{det} N^*\cong L|_C$ is exactly what is needed to give a global resolution of $I_C$. – Yosemite Stan Jan 31 at 6:58