When $A \to B$ is faithfully flat, the Amitsur complex

$0 \to A \to B \to B \otimes_A B \to B \otimes_A B \otimes_A B \to \cdots$

is exact. This is the key technical fact underlying faithfully flat descent.

For the proof: if $A \to B$ has a section $s: B \to A$ (e.g., $A \to A \times C$ with projection), then the complex has a contracting homotopy $h^n: B^{\otimes(n+1)} \to B^{\otimes n}$ given by $h^n(b_0 \otimes \cdots \otimes b_n) = s(b_0) \cdot b_1 \otimes \cdots \otimes b_n$. In the general faithfully flat case, we reduce to this case by the faithful flatness of $A \to B$ (the complex is exact after tensoring with $B$, and faithful flatness detects exactness).

Let $A \to B$ be faithfully flat. Then many properties of $A$ can be detected from $B$:

• $A$ is Noetherian $\iff$ $B$ is Noetherian (and $A \to B$ is of finite type).
• $A$ is reduced $\iff$ $B$ is reduced.
• $A$ is normal $\iff$ $B$ is normal.
• $A$ is regular $\iff$ $B$ is regular (when both are Noetherian).

More precisely, these properties "descend" along faithfully flat morphisms: if $B$ has the property, then $A$ has the property. The converse ("ascent") also holds for most of these.

This is stronger than the Affine Communication Lemma (which handles only Zariski covers): faithfully flat descent works for arbitrary faithfully flat covers, including etale and fppf covers.

A vector bundle on $X$ is a locally free $\mathcal{O}_X$-module of finite rank. By faithfully flat descent:

If $f: Y \to X$ is an fpqc cover and $\mathcal{E}$ is a locally free sheaf on $Y$ with descent data, then $\mathcal{E}$ descends to a unique locally free sheaf on $X$.

In particular, a rank-$n$ vector bundle on $X$ that is trivial on $Y$ is classified by a 1-cocycle $g \in GL_n(\mathcal{O}(Y \times_X Y))$ satisfying $p_{23}^*(g) \cdot p_{12}^*(g) = p_{13}^*(g)$. This gives

$\{\text{rank-}n \text{ vector bundles on } X, \text{ trivial on } Y\} / \cong \; = \; \check{H}^1(\{Y \to X\}, GL_n).$

Let $L/K$ be a Galois extension with group $G$. By faithfully flat descent (since $K \to L$ is faithfully flat), the category of $K$-vector spaces is equivalent to the category of $L$-vector spaces with $G$-semilinear action.

Concretely, given an $L$-vector space $V$ with semilinear $G$-action (i.e., $\sigma(\lambda v) = \sigma(\lambda) \sigma(v)$ for $\sigma \in G, \lambda \in L$), the $K$-vector space $W = V^G$ (the invariants) satisfies $W \otimes_K L \cong V$.

Example: Let $L = \mathbb{C}$, $K = \mathbb{R}$, $G = \{1, \sigma\}$ with $\sigma$ = complex conjugation. A $\mathbb{C}$-vector space $V$ with a semilinear involution $\sigma$ descends to $W = V^\sigma$. If $V = \mathbb{C}^2$ and $\sigma(z_1, z_2) = (\bar{z}_2, \bar{z}_1)$, then $W = \{(z, \bar{z}) : z \in \mathbb{C}\} \cong \mathbb{R}^2$.

Let $L/K$ be a finite field extension and $X$ a scheme over $L$. The Weil restriction (or restriction of scalars) $\operatorname{Res}_{L/K}(X)$ is a scheme over $K$ with the property:

$\operatorname{Res}_{L/K}(X)(T) = X(T \times_K L)$

for any $K$-scheme $T$. When $L/K$ is separable, $\operatorname{Res}_{L/K}(X)$ exists and can be constructed using etale descent.

For $X = \mathbb{G}_{m,L}$ (the multiplicative group over $L$), $\operatorname{Res}_{L/K}(\mathbb{G}_m)(T) = (T \otimes_K L)^\times$. When $L/K$ is Galois with group $G$, we have $\operatorname{Res}_{L/K}(\mathbb{G}_m) \times_K L \cong \prod_{g \in G} \mathbb{G}_{m,L}$, and the descent datum encodes the Galois action permuting the factors.

Let $X_0$ be a scheme over a field $k$. A twisted form (or $k$-form) of $X_0$ is a scheme $X$ over $k$ such that $X \times_k k^{\text{sep}} \cong X_0 \times_k k^{\text{sep}}$ (they become isomorphic over a separable closure).

By etale descent, twisted forms of $X_0$ are classified by

$H^1_{\text{et}}(\operatorname{Spec}(k), \operatorname{Aut}(X_0))$

where $\operatorname{Aut}(X_0)$ is the sheaf of automorphisms of $X_0$.

Example: Twisted forms of $\mathbb{P}^{n-1}$ are Brauer-Severi varieties, classified by $H^1(k, PGL_n) \cong \operatorname{Br}(k)[n]$ (the $n$-torsion of the Brauer group). A Brauer-Severi variety $V$ is trivial iff it has a $k$-rational point.

Example: Twisted forms of $M_n(k)$ (the matrix algebra) are central simple algebras of degree $n$. Their isomorphism classes form $\operatorname{Br}(k)[n]$.

Let $f: X \to Y$ be a morphism and $g: Y' \to Y$ a faithfully flat quasi-compact morphism. Let $f': X' \to Y'$ be the base change ($X' = X \times_Y Y'$). Then many properties of $f$ can be checked on $f'$:

$f \text{ has property } \mathcal{P} \iff f' \text{ has property } \mathcal{P}$

for $\mathcal{P} \in \{\text{flat, smooth, étale, unramified, proper, finite, affine, quasi-finite, surjective, open immersion, closed immersion, isomorphism, ...}\}$.

This is immensely powerful: to check a morphism is smooth, it suffices to check it after any faithfully flat base change.

Let $f: Y \to X$ be an fpqc cover. Then closed subschemes of $X$ correspond bijectively to closed subschemes of $Y$ with descent data. The descent datum for a closed subscheme $Z \hookrightarrow Y$ is an equality $p_1^{-1}(Z) = p_2^{-1}(Z)$ as closed subschemes of $Y \times_X Y$.

In terms of ideal sheaves: descent for closed subschemes corresponds to descent for quasi-coherent ideal sheaves $\mathcal{I} \subseteq \mathcal{O}_Y$, which is a special case of descent for quasi-coherent modules.