Hartshorne-Serre's correspondence in higher codimension 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?
 A: In codimension 3, you can hope for your varieties to be cut out (scheme theoretically) by the Pfaffians of an alternating map of vector bundles on $X$.
If you assume that $S$ is subcanonical, which you have to assume for Serre's correspondence (by the way, traditionnally, this is not called Hartshorne-Serre, but only Serre's correspondence) that $X = \mathbb{P}^N$ and some small extra-assumptions on $S$, then it is a theorem of Walter that $S$ is cut out by such Pfaffians (see : Walter, Pfaffians subschemes, Journal of Algebraic Geometry, 1996).
If $X$ is not the projective space, you can still hope for some structure theorems, but they are more difficult to explain (see : http://arxiv.org/pdf/math/9906170.pdf).
In codimension $4$ there is this : http://arxiv.org/pdf/1304.5248.pdf (which seems to me even much more complicated) and after (codimension $5$ or bigger) I guess nothing is known. 
A: 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).
A: As far as I have ever heard (and I have asked around), there is no general analogue of Serre's construction in higher codimension.
