# A derived category of formal sheaves

Grothendieck (EGA I 0.7 & 1.10) defined a category of (topologically Noetherian) "formal rings" and a corresponding global category of formal schemes. Roughly, a formal ring is a topological commutative ring $R$ whose topology is $I$-adic for $I\subset R$ an open ideal. Its formal Spec is $Spf(R) : = Spec(R_{red})$ for $R_{red}$ the "maximal topologically reduced quotient", equivalently the reduced quotient of $R/I$. In the special case $I=0$ (discrete topology), we get the category of ordinary schemes as a full subcategory of formal schemes.

Given a formal ring $R$, we have a category $Mod(R)$ of suitably nice topological modules; these have a theory of completed tensor product and pullback. Again, this globalizes to a category of formal quasicoherent sheaves on a formal scheme.

Now in order to have pushforwards for formal sheaves, we need to allow new topological modules over ordinary rings. The fundamental example is the topological module $k[[t]]$ over $k[t]$ (the pushforward of the constant sheaf on the formal neighborhood of $0$). Since the completed stalk of $k[[t]]$ is zero at all points $0\neq x\in \mathbb{A}^1,$ it makes sense to think of this as a "formal sheaf on $\mathbb{A}^1$ supported at $0$".

I would like to understand a category of topological coherent sheaves on an ordinary scheme which includes such "formal" objects, and especially a derived category of such objects (I'm particularly interested in a category of "constructible" formal sheaves, which I would define as repeated extensions of pushforwards of coherent sheaves on formal completions of closed subschemes). For some reason, I can't seem to find anyone who has written about this, except for in the very general context of ind-schemes à la Gaitsgory-Rozenblyum. Do people here know of a good source?

• Could you provide some details or references on what the category $Mod(R)$ for a formal ring is, and what the formal quasi-coherent sheaves on a formal scheme are? – Leonid Positselski Aug 2 '16 at 19:28
• The reference is the same two chapters of EGA (see I.10.11 in particular). I think the definitions there are equivalent to saying that a sheaf is a sheaf of topological modules which is $I$-adically complete and has locally a fundamental collection of neighborhoods of zero consisting of submodules. A sheaf is coherent (again in my re-interpretation) if it is locally a quotient of a finitely-generated free module by an open submodule. – Dmitry Vaintrob Aug 2 '16 at 20:03
• OK, I see. Thanks. I was surprised because your question mentioned "formal quasicoherent sheaves". EGA I 10.11, however, only discusses coherent sheaves, not quasicoherent ones. With coherent sheaves on a formal scheme, there is no surprise. – Leonid Positselski Aug 2 '16 at 20:19
• But if the question concerns references to expositions about categories of coherent and quasi-coherent sheaf-like structures over formal schemes generally, then you may want to be aware of the existence of my paper arxiv.org/abs/1503.05523 and its more elementary counterpart arxiv.org/abs/1605.03934 . These are relevant in connection with the quasi-coherent (infinitely generated) modules/sheaves/complexes, and only treat affine formal schemes. – Leonid Positselski Aug 2 '16 at 20:52

For a start, you have an approach to ind-coherent sheaves and its derived category in

Duality and Flat Base Change on Formal Schemes Contemp. Math. 244 (1999), pp. 3-90.

On line version, with some corrections incorporated.

The theory gets interesting because this derived category contains two subcategories, one of quasi-coherent torsion sheaves and another one of complete complexes, where Grothendieck duality takes a specially nice form. These categories turn out to be equivalent, but the equivalence does not respect the inclusion functors.

It might be interesting to point out that the full subcategory of the derived category of complexes with coherent homology is contained in the subcategory of complete complexes.

All of this is discussed in the cited paper.

• Thanks! This looks very much like what I was looking for. I also didn't know about Deligne's appendix to Residues and Duality that the work references. – Dmitry Vaintrob Aug 4 '16 at 21:28
• You are wellcome. There are further developments on duality, see work by Neeman, for instance – Leo Alonso Aug 4 '16 at 21:43