Let $X = U_1 \cup U_2$ be a Zariski open cover. To glue sheaves $\mathcal{F}_1$ on $U_1$ and $\mathcal{F}_2$ on $U_2$ into a sheaf $\mathcal{F}$ on $X$, we need:

• An isomorphism $\varphi_{12}: \mathcal{F}_1|_{U_1 \cap U_2} \xrightarrow{\sim} \mathcal{F}_2|_{U_1 \cap U_2}$ (the gluing datum).

For three opens $U_1, U_2, U_3$, we additionally need the cocycle condition on triple overlaps:

$\varphi_{13} = \varphi_{23} \circ \varphi_{12} \quad \text{on } U_1 \cap U_2 \cap U_3.$

This data $(\mathcal{F}_i, \varphi_{ij})$ satisfying the cocycle condition is precisely a descent datum for the Zariski cover $\{U_i \to X\}$. The classical gluing lemma says such data always produces a unique sheaf $\mathcal{F}$ on $X$, i.e., Zariski descent is effective for sheaves.

The line bundle $\mathcal{O}(n)$ on $\mathbb{P}^1 = U_0 \cup U_1$ (where $U_i = \operatorname{Spec}(k[t_i])$ with $t_0 = 1/t_1$) is constructed by gluing the trivial line bundles $\mathcal{O}_{U_0}$ and $\mathcal{O}_{U_1}$ via the transition function

$\varphi_{01}: \mathcal{O}_{U_0}|_{U_0 \cap U_1} \to \mathcal{O}_{U_1}|_{U_0 \cap U_1}, \quad f \mapsto t_1^n \cdot f$

on $U_0 \cap U_1 = \operatorname{Spec}(k[t_0, t_0^{-1}]) = \operatorname{Spec}(k[t_1, t_1^{-1}])$. The cocycle condition is automatic for two opens. Different integers $n$ give non-isomorphic line bundles, and $\operatorname{Pic}(\mathbb{P}^1) \cong \mathbb{Z}$.

Let $N$ be an $A$-module and $M = N \otimes_A B$ the base change. Then $M$ has a canonical descent datum $\varphi: (N \otimes_A B) \otimes_A B \xrightarrow{\sim} B \otimes_A (N \otimes_A B)$ given by

$n \otimes b_1 \otimes b_2 \mapsto b_1 \otimes n \otimes b_2$

(i.e., swapping the $B$-factors past $N$). This is the trivial descent datum. The cocycle condition is immediate from the associativity of tensor products.

A descent datum $(M, \varphi)$ is effective if it is isomorphic to a trivial one, i.e., if $M \cong N \otimes_A B$ for some $A$-module $N$ and $\varphi$ corresponds to the canonical isomorphism.

Let $A \to B$ be faithfully flat. A descent datum on $M = B$ (a free $B$-module of rank 1) amounts to a $B \otimes_A B$-module automorphism $\varphi: B \otimes_A B \to B \otimes_A B$ satisfying the cocycle condition. Such an automorphism is multiplication by a unit $u \in (B \otimes_A B)^\times$.

The cocycle condition says that in $(B \otimes_A B \otimes_A B)^\times$:

$p_{23}^*(u) \cdot p_{12}^*(u) = p_{13}^*(u)$

This is a 1-cocycle in the Amitsur complex $B^\times \to (B \otimes_A B)^\times \to (B \otimes_A B \otimes_A B)^\times \to \cdots$, and the cohomology $H^1(B/A, \mathbb{G}_m)$ classifies isomorphism classes of such descent data, which correspond to rank-1 projective $A$-modules (i.e., elements of $\operatorname{Pic}(A)$).

For a free $B$-module $M = B^n$ of rank $n$, a descent datum amounts to a matrix $g \in GL_n(B \otimes_A B)$ satisfying the cocycle condition:

$p_{23}^*(g) \cdot p_{12}^*(g) = p_{13}^*(g) \in GL_n(B \otimes_A B \otimes_A B)$

Two descent data $g, g'$ are isomorphic if $g' = p_2^*(h) \cdot g \cdot p_1^*(h)^{-1}$ for some $h \in GL_n(B)$.

The set of effective descent data modulo isomorphism is $H^1(B/A, GL_n)$, classifying rank-$n$ projective $A$-modules (or rank-$n$ vector bundles on $\operatorname{Spec}(A)$).

Faithfully flat descent for modules (Grothendieck): Let $A \to B$ be a faithfully flat ring homomorphism. Then the functor

$\{\text{$A$-modules}\} \to \{\text{descent data for $B$-modules}\}, \quad N \mapsto (N \otimes_A B, \text{canonical } \varphi)$

is an equivalence of categories. The quasi-inverse sends $(M, \varphi)$ to

$N = \{m \in M \mid \varphi(m \otimes 1) = 1 \otimes m \text{ in } B \otimes_A M\} = \ker(M \rightrightarrows B \otimes_A M).$

This is the module of invariants (or descent module): the elements of $M$ that are "invariant" under the descent datum.

Moreover, $M$ is finitely generated (resp. finitely presented, projective) as a $B$-module if and only if $N$ is finitely generated (resp. finitely presented, projective) as an $A$-module.

Faithfully flat descent is also effective for $A$-algebras: if $A \to B$ is faithfully flat and $(C, \varphi)$ is a $B$-algebra with descent datum, then $C_0 = \ker(C \rightrightarrows B \otimes_A C)$ is an $A$-algebra with $C_0 \otimes_A B \cong C$.

This extends to: descent is effective for quasi-coherent $\mathcal{O}_X$-algebras along faithfully flat quasi-compact morphisms $Y \to X$. This is the key to "descending" affine morphisms.

Descent is effective for affine morphisms along fpqc covers. More precisely: let $f: Y \to X$ be an fpqc morphism, and let $Z \to Y$ be an affine $Y$-scheme equipped with descent data $\varphi$. Then there exists an affine $X$-scheme $W$ with $W \times_X Y \cong Z$ (compatibly with $\varphi$), and $W$ is unique up to unique isomorphism.

This follows from the effectiveness for quasi-coherent algebras: an affine $Y$-scheme $Z = \operatorname{Spec}(\mathcal{A})$ corresponds to a quasi-coherent $\mathcal{O}_Y$-algebra $\mathcal{A}$, and the descent datum on $Z$ corresponds to a descent datum on $\mathcal{A}$. Descending $\mathcal{A}$ to an $\mathcal{O}_X$-algebra $\mathcal{A}_0$ gives $W = \operatorname{Spec}(\mathcal{A}_0)$.

For any fpqc morphism $f: Y \to X$, descent for morphisms is effective: the sequence above is always an equalizer. This is equivalent to saying that representable presheaves are sheaves for the fpqc topology (the fpqc topology is subcanonical).

Concretely: given schemes $W_1, W_2$ over $X$ and a morphism $g: W_1 \times_X Y \to W_2 \times_X Y$ over $Y$ such that the two pullbacks to $Y \times_X Y$ agree, there is a unique morphism $\bar{g}: W_1 \to W_2$ over $X$ with $\bar{g} \times_X Y = g$.

Let $L/K$ be a finite Galois extension with group $G$. Descent for morphisms along $\operatorname{Spec}(L) \to \operatorname{Spec}(K)$ says: given $K$-schemes $W_1, W_2$, a $K$-morphism $W_1 \to W_2$ is the same as an $L$-morphism $W_{1,L} \to W_{2,L}$ that commutes with the Galois action (i.e., $g^* \circ f = f \circ g^*$ for all $g \in G$).

This is the scheme-theoretic version of Galois descent: to give a $K$-rational map between $K$-varieties, it suffices to give a $\bar{K}$-rational map that is Galois-equivariant.

Descent is not always effective for arbitrary schemes (or algebraic spaces) along fpqc covers. The classical counterexample is due to Hironaka:

There exists a proper non-projective threefold $X$ over $\mathbb{C}$ with an involution $\sigma$ such that the quotient $X/\sigma$ exists as an algebraic space but not as a scheme. The descent datum for $X \to X/\sigma$ (as a scheme over $X/\sigma$) is a scheme over $X/\sigma$ by descent, but the "descended" object $X/\sigma$ is only an algebraic space.

More precisely: descent along etale covers is effective for separated schemes (by a theorem of Artin), but effectiveness can fail for non-separated schemes. This failure is one motivation for enlarging the category of schemes to algebraic spaces and stacks.

Descent for vector bundles (locally free sheaves of finite rank) is effective along fpqc covers. This is because vector bundles are affine over the base (locally $\operatorname{Spec}(\operatorname{Sym}(\mathcal{E}^\vee))$), and descent is effective for affine schemes.

However, descent for projective bundles $\mathbb{P}(\mathcal{E})$ is more subtle. A projective bundle $\mathbb{P}^{n-1}$ with descent data gives a Brauer-Severi variety, and the obstruction to effectiveness is an element of the Brauer group $\operatorname{Br}(X) = H^2(X, \mathbb{G}_m)$.

Concretely, a Brauer-Severi variety $V$ over a field $k$ is a variety that becomes isomorphic to $\mathbb{P}^{n-1}$ over $\bar{k}$. It is trivial (i.e., already isomorphic to $\mathbb{P}^{n-1}$) if and only if it has a $k$-rational point. Its class in $\operatorname{Br}(k)$ is the obstruction.

Let $L/K$ be a finite Galois extension with group $G = \operatorname{Gal}(L/K)$. For the fibered category of vector bundles:

$\operatorname{DD}(\operatorname{Spec}(L)/\operatorname{Spec}(K))$ has objects: pairs $(V, \{\varphi_g\}_{g \in G})$ where $V$ is a finite-dimensional $L$-vector space and $\varphi_g: V \to g^*V = V$ (with $L$-action twisted by $g$) are isomorphisms satisfying $\varphi_{gh} = g^*\varphi_h \circ \varphi_g$.

This is precisely a semilinear representation of $G$ on $V$: each $\varphi_g$ is $g$-semilinear ($\varphi_g(\lambda v) = g(\lambda) \varphi_g(v)$) and $\varphi_{gh} = \varphi_g \circ \varphi_h$.

By Hilbert's Theorem 90 (in the generalized form), descent is effective: every semilinear representation comes from a $K$-vector space $W$ with $V = W \otimes_K L$.

Let $G$ be a group scheme over $X$ and $f: Y \to X$ an fpqc cover. A $G$-torsor $P$ over $X$ becomes trivial over $Y$ if and only if $P(Y) \neq \emptyset$ (i.e., $P$ has a section over $Y$).

Assuming $P \times_X Y \cong G \times_X Y$ (trivial torsor over $Y$), the descent datum for $P$ is encoded by an element $g \in G(Y \times_X Y)$ satisfying the cocycle condition $p_{23}^*(g) \cdot p_{12}^*(g) = p_{13}^*(g)$. This is precisely a Cech 1-cocycle, and isomorphism classes of torsors trivial over $Y$ are classified by $\check{H}^1(\{Y \to X\}, G)$.

Taking the colimit over all covers gives $H^1(X, G)$ (first cohomology, or first nonabelian cohomology if $G$ is not commutative).