Silly question about descent

Question

Most sources say descent is defining an object over $S$ using objects over $U_i$ for some cover $\left\{ U_i \right\}$ of $S$. If I replace the covering family with a single arrow $\coprod _i U_i\rightarrow S$, then it's not at all true that giving an object over each $U_i$ is the same as giving an object over $\coprod _iU_i$. How to resolve this?

Answer 1

You should be much more precise about what type of covers and categories you are working with, as @KevinCarlson suggests. For example, in the fpqc topology and quasi-coherent sheaves, these are the same questions. See Chapter 6 of Neron Models for a more detailed example. Other great references are the Stacks Project and for the fancier cohomological version Notes by Brian Conrad.

In general, a descent is phrased on a cover in some topology as you mentioned, or rather one defines a cover to have descent. If we recall the definition, it should be more or less clear how to resolve in the case of sheaves which I assume is the case you are interested in.

To be precise, fix a category $C$ with fiber products and a topology on this category $\tau$. For an object $X$ and any $\tau$-cover $\mathfrak{U}_I := \{ U_i \to X\}$ an induced functor from $\tau$ Sheaves on $X$ to descent datum, specifically a choice of sheaf $\mathcal{F}_i$ on $U_i$ (with its induced $\tau$-topology) and identifications $\varphi_{i,j} \colon pr_0^* U_i \cong pr_1^* U_j$ along each $U_i \times_X U_j$ which also satisfies a messy cocycle condition along triple fiber products. Ultimately its not that enlightening to read the statement of it, so I refer you the references, but it essentially is just saying that various pullbacks to triple fiber products amount to the same data. The point is that each sheaf on $X$ when pulled back to each $U_i$ automatically satisfies these conditions.

For the novices, it is important to first write this all down using the Zariski topology. Here one should make sure to see that this is nothing more than the statement that one is gluing sheaves along an open cover of $X$. In fact, its a great exercise to try to glue up $X$ from its affine open cover using these types of statements.

One makes a category out of descent datum $Data_{\mathfrak{U}_I}$ with maps between $(\mathcal{F}_i, \varphi_{i,j})$ and $(\mathcal{F}_i', \psi_{i,j})$ that make various obvious squares commute. As @KevinCarlson noted, the property that this cover has descent is that statement that the natural functor from Sheaves over $X$ to descent datum is fully faithful. We say the cover has effective descent provided the functor is an equivalence of categories -- that is sheaves glue in the $\tau$-topology. The derived version is basically a derived analogue of this statement.

Now, your question is about comparing covers $\mathfrak{U}_I = \{ U_i \to X\}$ and the cover $\bigsqcup U_i := U \to X$. Note that any sheaf on $U$ defines a collection of sheaves on each $U_i$. Moreover, any sheaf on $U$ is canonically determined by what its restrictions are to the $U_i$. In particular there is no overlap. Thus so long as you are phrasing things about a descent theorem of sheaves (or equivalently any object which is already determined locally in the topology in question), then there is no difference between the two versions of the question you asked.

I would write more details, but really the chapter in Neron Models is very detailed and incredibly well-written. Once one understands fpqc descent and the proofs there, it is very easy to switch to other topologies and objects and see what happens.

If you try to phrase a very general version of descent, and attempt to glue objects together which are not really local in nature in the first place (meaning determined up to isomorphism locally), then maybe there difficulties can arise, but its impossible to determine how to resolve unless you phrase the descent very carefully. In practice, one always has to have a good knowledge of at least the types of things one is gluing and to my knowledge every application of descent I've ever seen has been to glue some kind of locally determined objects together in some funky cover. If anyone knows of cases/applications which contradict this, please share!

Answer 2

If I replace the covering family with a single arrow $\coprod _i U_i\rightarrow S$, then it's not at all true that giving an object over each $U_i$ is the same as giving an object over $\coprod _iU_i$

Sure it is. For example, in any extensive category (e.g. sets, topological spaces, schemes), giving a morphism $Y \to \coprod_i U_i$ is the same as giving a tuple of morphisms $Y_i \to U_i$. The $Y_i$ here are pullbacks along the coproduct inclusions $U_i \to \coprod_i U_i$. Similarly, if the $U_i$ are schemes, then giving a quasicoherent sheaf on $\coprod_i U_i$ is the same as giving a quasicoherent sheaf on each $U_i$.