# Smallest non-vanishing index of Local-Cohomology and Ext functor, as in the definition of depth, for not necessarily affine schemes

Let $$(Z,\mathcal O_Z)$$ be a closed subscheme of a Noetherian scheme $$(X,\mathcal O_X)$$. Then there is an ideal sheaf $$\mathcal J$$ on $$X$$ such that $$i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$$ , where $$i:Z\to X$$ is the inclusion and $$\mathcal J$$ is the kernel of $$i^{\#}: \mathcal O_X \to i_*(\mathcal O_Z)$$.

Let $$\mathcal F$$ be a quasi-coherent sheaf of $$\mathcal O_X$$-modules on $$X$$.

Now if $$X$$ is affine, then it is known that $$\inf \{n |$$ Ext$$^n(\mathcal O_X/\mathcal J,\mathcal F)\ne 0\}=\inf\{n | H^n_Z(X,\mathcal F)\ne 0\}$$

My question is: In general, when $$X$$ is not affine, is there any relation between the integers defined by the two sides of the above equality ? Also, is there any general cases known where the above equality or some analogue of it holds when $$X$$ is not necessarily affine ? (If needed, assume $$\mathcal F$$ is coherent)

EDIT: From Proposition 3.7 and 3.8 of R. Hartshorne, Local Cohomology, A seminar given by A. Grothendieck, Harvard University. Fall, 1961. Lecture Notes in Mathematics, 41. Springer, 1967. http://link.springer.com/book/10.1007%2FBFb0073971, it follows that if $$Z$$ is a closed subscheme of a Noetherian scheme $$(X,\mathcal O_X)$$ and $$\mathcal J$$ is the ideal sheaf defining $$Z$$ as above, then for any coherent sheaf $$\mathcal F$$ of $$\mathcal O_X$$-module , one has

$$\inf \{n | \mathcal Ext^n(O_X/\mathcal J, \mathcal F)\}=\inf \{n| \underline{H^n_Z}(\mathcal F)\ne 0\}=\inf_{x\in Z}$$ depth$$_{O_{X,x}} \mathcal F_x$$

where $$\underline {H^n_Z}$$ is the Local Cohomology sheaf as Mohan alluded to in the comments and are the derived functors of the functor which takes a sheaf of abelian groups $$\mathcal G$$ to the sheaf $$U\to \Gamma_{U\cap Z}(U,\mathcal G|_U)$$ and moreover $$\underline {H^n_Z}(\mathcal F)$$ can also be thought of as the sheaf associated to the pre-sheaf

$$U \to H^n_{U\cap Z}(U,\mathcal F|U )$$ (Proportion 1.2 in the above reference) .

The question I originally asked, which still stands, is a global version of this.

• When $X$ is not affine, there are two kinds of ext (as well as $H^n_Z$), local and global. Local means everything is treated as a sheaf and then there is little difference in the non-affine case for what you want. The global case can be quite different. – Mohan Jul 3 at 2:05
• @Mohan: by local and global ext, do you just mean $\mathcal Ext$ and Ext ? And I haven't heard of two kinds of local Cohomology, whatever source I've read from just treats cohomology and local Cohomology as groups (modules) only ... do you mean some kind of sheafification of local Cohomology too ? (maybe the derived functor of $\mathcal F$ goes to the sheaf $U\to \Gamma_{U\cap Z} (U, \mathcal F|_U)$ ? ) And could you elaborate on what do you mean when things are different etc. ? – sde Jul 3 at 10:18
• $\mathcal{E}xt$ is usually called local and it is a sheaf, while the other is called global and it is just a module over $\Gamma(X,\mathcal{O}_X)$. – Mohan Jul 3 at 13:57
• @Mohan: yes that's what I thought you meant by local and global ext ... however I'm still not sure about your two kinds of local Cohomology and the comment you made about little and big difference ... – sde Jul 3 at 15:05
• It is easy to construct examples where $\mathcal{E}xt^n\neq 0$, but $\mathrm{Ext}^n=0$. – Mohan Jul 3 at 15:19