[Math] Hartshorne-Serre’s correspondence in higher codimension

Question

There's a well-known correspondence (traditionally called Hartshorne-Serre) between codimension 2 smooth subvarieties $S\subset X$ of a smooth algebraic variety $X$ and certain rank two vector bundles on $X$. Is there anything similar for higher codimensional subvarieties of $X$, i.e. codimension 3 and more?

Answer 1

To get some idea of the difficulty in generalizing Serre's correspondence to higher codimension, bear in mind that what makes a resolution $0 \rightarrow \mathcal{F} \rightarrow \mathcal{E} \rightarrow \mathcal{I}_{Z|X} \rightarrow 0$ plausible when $\mathcal{F},\mathcal{E}$ are vector bundles on $X$ and $Z \subset X$ is smooth of codimension 2 is that $Z$ is locally Cohen-Macaulay. If $Z$ is assumed to be locally Cohen-Macaulay of possibly higher codimension, then $\mathcal{F}$ is at best reflexive, and even if we pass to a resolution of $\mathcal{I}_{Z|X}$ by vector bundles, recovering $Z$ from a series of vector bundle maps is a highly nontrivial matter.

EDIT: As indicated below by Libli's informative answer, what is more relevant is that $Z$ is at least locally Gorenstein (which in codimension 2 is equivalent to being a local complete intersection).