One motivation for studying infinity categories is that the derived category does not satisfy Zariski descent, although the infinity categorical version does.


I would like to see an example of Zariski descent failing for the (one categorical) derived category of coherent sheaves on a scheme.


That is I would like an example of an (preferably affine) algebraic variety $X$, an open cover $\{U\_i \hookrightarrow X\}$ and a descent datum in the derived category which does not glue uniquely to an element of the derived category of $X$.



The derived category does not satisfy descent - example
Consider the canonical functor $H$ from the homotopy category of homotopy coherent descent data in the ∞-category of coherent sheaves
to the category of descent data in the derived category of coherent sheaves.


Nonuniqueness of gluing amounts to $H$ not being a full functor
and is observed for covers of cardinality 3 and higher.


Nonexistence of gluing amount to $H$ not being essentially surjective
and is observed for covers of cardinality 4 and higher.


For covers of cardinality 0, 1, and 2, the functor $H$ is an equivalence.


Let's analyze first the case of covers of cardinality 3.
Given three opens $U\_1$, $U\_2$, $U\_3$,
an object in the domain of $H$
will have chain complexes of coherent sheaves $F\_i$ over $U\_i$,
isomorphisms $t\_{i,j}:F\_i→F\_j$ of their restrictions to $U\_i∩U\_j$ ($1≤i<j≤3$, setting $t\_{j,i}=t\_{i,j}^{-1}$),
and a choice of a homotopy $h$ (or rather its homotopy class) from the composition $t\_{3,1}t\_{2,3}t\_{1,2}$ to the identity map on the restriction of $F\_1$ to $U\_1∩U\_2∩U\_3$.
In particular, objects with the same data of $F\_i$ and $t\_{i,j}$, but nonhomotopic $h$ are nonisomorphic.


An object in the codomain of $H$ has a similar description, except that $h$ is merely required to exist, but its data is not included.


It is now clear how to show that $H$ is not full: pick two objects in the domain of $H$ with the same data of $F\_i$ and $t\_{i,j}$,
but different (nonhomotopic) $h$.
Their images under $H$ will see the data of $h$ discarded, so the two objects become isomorphic in the codomain of $H$,
despite not being isomorphic in the domain of $H$.


Therefore, to construct a concrete example, one needs to specify $U\_i$, $F\_i$, and $t\_{i,j}$
such that the identity map $p$ on $F\_1$ restricted to $U\_1∩U\_2∩U\_3$
admits a homotopy $h$ from $p$ to itself
that is not homotopic to the identity homotopy from $p$ to itself.
That is to say, we want $\def\Aut{\mathop{\sf Aut}\nolimits}\def\End{\mathop{\sf End}\nolimits} π\_1(\Aut(F\_1))$ to be nontrivial.
For example, we can take $\def\cO{{\cal O}} F\_1=\cO[0]⊕\cO[1]$, then $$\End(F\_1)=\cO[-1]⊕\cO[0]⊕\cO[0]⊕\cO[1],$$ and a nontrivial 1-cocycle in $\Aut(F\_1)$ yields such a homotopy $h$.


The case of covers of cardinality 4 is similar.
As before, for an object in the domain of $H$ we have $U\_i$ and $F\_i$ ($1≤i≤4$), $t\_{i,j}$ ($1≤i<j≤4$), $h\_{i,j,k}$ ($1≤i<j<k≤4$),
and now on $⋂\_i U\_i$ there is a homotopy $q$ from a certain composition of homotopies $h$ to the identity homotopy on the identity map on the restriction of $F\_1$ to $⋂\_i U\_i$.


The functor $H$ discards the data of $h$ and $q$.
In the codomain of $H$, some maps $h$ must exist (but their choice is not given to us),
but $q$ need not exist at all.
Indeed, by modifying the above example appropriately, we can construct descent data for the derived category
in which the composition of homotopies $h$ is not homotopic to the identity homotopy,
which means that the existence of gluing is violated for the derived category.


