# 3.3. 下降

The idea of descents should be that, they are like sheaf axiom for fibered categories.

例 3.3.0.1. Let $X$ be a scheme and $C$ be the category $Op(X)$. Then consider $F=(Vect)→C$ where $F(U)$ be the category of vector bundles on $U$. Then, if $U=⋃_{i}U_{i}$, a vector bundle on $U$ is not equivalent to $E_{i}$ on $U_{i}$ with double intersections isomorphic (i.e. $σ_{ij}:E_{i}∣_{U_{ij}}∼ E_{j}∣_{U_{ij}}$).

In this case, the naive sheaf axioms fit into the picture, i.e. we get$F(U)→i∏ F(U_{i})⇒i,j∏ F(U_{ij})$and this diagram is not exact.

We are missing the “cocycle” condition (to make the above diagram exact/equalizer). This means that, on $U_{ijk}=U_{i}∩U_{j}∩U_{k}$, we get diagram

In other word, the right “exact” diagram we need will be something like

Therefore, we want to formalize this triple arrow thing in fibered categories.

Let $p:F→C$ be a fibered category. Given $f:X→Y$ in $C$, let $F(Xf Y)$ be the category defined as follows (this is called category of descent data). The object should be $(E,σ)$ with $E∈F(X)$ andwhere the triple arrows are $p_{12},p_{13},p_{23}$ and the double arrows are $p_{1},p_{2}$, and $σ:p_{1}E→p_{2}E$ is an isomorphism in $F(X×_{Y}X)$ such that we get the following commutative diagramwhere $=$ means canonical isomorphism. This is called the cocycle condition.

注 3.3.0.2. Here is just a brife recall of what all the above notations (i.e. $p_{i}E$, $p_{ij}$, etc) means.

First, recall that $p_{i}E$ are defined as the pullback of the following diagramSimilarly, $p_{jk}p_{i}E_{∗}$ are defined as the pullback of the following diagram

Second, we note $p_{ij}$ are projections come from the universal property of (fibered) products. In other word, note we would define $X_{1}×_{S}X_{2}×_{S}X_{3}$ as the unique object satisfies the following diagram

Finally, a word on the isomorphisms $p_{jk}p_{i}E$. Continue with the above diagram (where now we let $X_{1}=X_{2}=X_{3}=X$), we get the followingwhere the two pullbacks along $p_{13}$ and $p_{12}$ must be the same object living over $X_{1}×X_{2}×X_{3}$ , hence the canonical isomorphisms between $p_{13}p_{1}E≅p_{12}p_{1}E$. The others are similar.

The idea is that, if we have $F(Y)ϵ F(Xf Y)$, then stack would be $F$ fibered in groupoids where $ϵ$ is an equivalence^{1}.

Last time, we let $p:F→C$ be a fibered category (not necessarily fibered in groupoids). Given $Xf Y$ in $C$, we defined a category $F(Xf Y)$ be the category of descent data.

The objects of this category is $(E,σ)$ where $E∈F(X)$ and $σ:p_{1}E→p_{2}E$ is an isomorhism between $p_{1}E∼ p_{2}E$. Here we have a diagramand hence the two pullbacks of $E$ should be isomorphic, and we just choose a particular $σ$. However, this $σ$ cannot be arbitrary, as we need one additional condition called the cocycle condition.

The cocycle condition says that, all the ways to pullback $E$ should all be commuting (we have three arrows $p_{ij}$ from $X×X×X$ to $X×X$, and two arrows from $X×X$ to $X$, then we want $p_{23}σ∘p_{12}σ=p_{13}σ$).

The point is that, we then get $F(Y)ϵ F(Xf Y)$ by$F↦(f_{∗}F,σ_{can})$This is because, $p_{1}f_{∗}F$ and $p_{2}f_{∗}F$ are pullbacks of $F$ along $g:X×_{Y}X→Y$. For example, we get $p_{1}f_{∗}F$ by the following diagram (where $g$ is equal both $fp_{1}$ and $fp_{2}$ at the same time)

Therefore we get a canonical map $σ_{can}:p_{1}f_{∗}F∼ p_{2}f_{∗}F$.

In this case, we say $f$ is an effective descent morphism if $ϵ$ is equivalence of categories. In this case we also say $F$ satisfies descent for $f$.

Before we say explain what this means, we recall the morphisms of $F(Xf Y)$ is the following. From $(E_{′},σ_{′})$ to $(E,σ)$, a morphism is $α:E_{′}→E$ in $F(X)$ such that we get the following commuting diagram

More generally, we can define $F({X_{i}f_{i} Y}_{i∈I})$ as $({E_{i}}_{i∈I},{σ_{ij}}_{i,j})$ such that $E_{i}∈F(X_{i})$ and $σ_{ij}:p_{i}E_{i}∼ p_{j}E_{j}$ is an isomorhism, whereWe also need the cocycle condition $σ_{jk}∘σ_{ij}=σ_{ik}$.

Normally, we do not need to think about this more general case because of the following lemma.

引理 3.3.0.3 (Olsson, Lemma 4.2.7). Assume coproducts exist in $C$ and coproducts commute with fiber products when they exists. Assume for all sets of objects ${X_{i}}_{i∈I}$ in $C$ the natural map $F(∐X_{i})→∏F(X_{i})$ is an equivalence. Then if ${X_{i}→Y}$ are morphisms in $C$ and $Q=∐_{i}X_{i}$, then $F(Y)→F({X_{i}→Y})$ is equivalence (of categories) if and only if $F(Y)→F(Q→Y)$ is equivalence (of categories).

Thus, the point is that, we can always think $F({X_{i}→Y})$ as $F(∐X_{i}→Y)$, which means we back to the first case.