It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ to be the category of quasi-coherent sheaves on $U$. Given a morphism $f : U \to V$, we have a functor $f^* : Qcoh(V) \to Qcoh(U)$. However we don't have $(gf)^*=f^*g^*$ on the nose, but there is a canonical natural equivalence between them. See Vistoli's notes Section 3.1 and 3.2.1.

Now let $X$ be a scheme and $U_i$ be an open cover of $X$. We have the following cosimplicial diagram of categories $$ \prod Qcoh(U_i)\rightrightarrows\prod Qcoh(U_i\times_X U_j)\text{triple arrows}\prod Qcoh(U_i\times_X U_j\times_X U_k)\ldots $$ Keep in mind that this diagram is only a pseudo diagram in $Cat$, the $2$-category of categories, i.e. the cosimplicial identities only hold up to canonical natural equivalence.

It is also well-known that the descent data of quasi-coherent sheaves is given by a collection $(\xi_i,\phi_{ij})$, where $\xi_i$ is a quasi-coherent sheaf on each $U_i$ and $\phi_{ij}$ is an isomorphism $pr_2^*\xi_j\to pr_1^*\xi_i$ in $Qcoh(U_{i}\times_X U_j)$ which satisfies the cocycle condition $$ pr_{13}^*\phi_{ik}=pr_{12}^*\phi_{ij}\circ pr_{23}^*\phi_{jk}. $$ on $U_i\times_X U_j\times_X U_k$. See Vistoli's notes Section 4.1.2.

The above descent data is given "by hand". On the other hand, I've heard that descent data is a kind of (homotopy) limit. Nevertheless Vistoli's notes doesn't consider homotopy limit.

$\textbf{My question}$ is: is there any reference which studies the (pseudo) homotopy limit (I'm not sure whether we should call it 2-categorical limit) of the above cosimplicial pseudo-diagram and show that it coincides with the descent data given in the literature?

  • 2
    $\begingroup$ If you truncate the cosimplicial diagram above degree 2 and unfold the definition of pseudocone you will see that it is more or less the same as the definition of descent data. So there's not really much to prove at all. If that seems obscure, you may want to first think about defining the set of matching families of elements of a presheaf as a limit of a cosimplicial diagram truncated above degree 1. $\endgroup$ – Zhen Lin Oct 21 '15 at 7:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.