2
$\begingroup$

Over projective space, it is well-known that given a graded $S^\bullet$-module $M_\bullet$, where $S^\bullet = k[x_0, \dots, x_N]$, there is a unique minimal free resolution $$ \cdots \to \bigoplus_q S^\bullet(-q) \otimes B_{p, q} \to \cdots \to \bigoplus_q S^\bullet(-q) \otimes B_{1, q} \to \bigoplus_q S^\bullet(-q) \otimes B_{0, q} \to M \to 0 $$

Over an arbitrary variety $X$ over $k$ (assumed to be projective and smooth if necessary), if I define a locally free resolution of a coherent $\mathcal O_X$-module $\mathcal F$ $$ \mathcal E^\bullet \to \mathcal F \to 0 $$ to be minimal if for each local ring $\mathcal O_{X, x}$, the complex $\mathcal E^\bullet \otimes_{\mathcal O_{X, x}} \kappa(x)$ has zero differentiations (Matsumura, (18.E)).

I wonder whether a minimal locally free resolution of $\mathcal F$ exists, and whether it is unique if exists.

$\endgroup$
4
$\begingroup$

This is not a good definition, because the complex $\mathcal{E}^\bullet \otimes_{\mathcal{O}_{X,x}} \kappa(x)$ computes $\mathrm{Tor}_i^{\mathcal{O}_{X,x}}(\mathcal{F}, \kappa(x))$, so if this complex has zero differential and at least two terms, it follows that $$ \mathrm{Tor}_1^{\mathcal{O}_{X,x}}(\mathcal{F}, \kappa(x)) \ne 0 $$ for any point $x \in X$. But any coherent sheaf is locally free at general point, which contradicts the above inequality.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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