Statement: The category of $K$-vector spaces is equivalent to the category of $L$-vector spaces with semilinear $G$-action.

Given a $K$-vector space $W$, the $L$-vector space $V = W \otimes_K L$ has a natural semilinear $G$-action: $g(w \otimes \lambda) = w \otimes g(\lambda)$.

Conversely, given $(V, \{g\}_{g \in G})$, we recover $W = V^G = \{v \in V \mid g(v) = v \; \forall g \in G\}$.

Concrete example: $L = \mathbb{Q}(i)$, $K = \mathbb{Q}$, $G = \{1, \sigma\}$ where $\sigma(i) = -i$. The $\mathbb{Q}(i)$-vector space $V = \mathbb{Q}(i)^2$ with semilinear $\sigma$-action $\sigma(z_1, z_2) = (\bar{z}_1, \bar{z}_2)$ descends to $W = \mathbb{Q}^2$.

But a different $\sigma$-action, say $\sigma(z_1, z_2) = (\bar{z}_2, \bar{z}_1)$, also descends to a $\mathbb{Q}$-vector space of dimension 2. This gives an isomorphic $W$, reflecting that $H^1(G, GL_2(L)) = 1$ (non-abelian Hilbert 90).

Let $L/K$ be Galois with group $G$. A smooth projective curve $C$ over $K$ can be studied via its base change $C_L = C \times_K L$. Conversely, a curve $D$ over $L$ with a $G$-equivariant structure (compatible semilinear $G$-action on $D$) descends to a curve over $K$ if the action is "geometric" (respects the scheme structure).

Example: The curve $x^2 + y^2 = 1$ over $\mathbb{R}$ has a natural $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$-equivariant structure on its complexification. Complex conjugation acts on $\mathbb{C}$-points by $(x, y) \mapsto (\bar{x}, \bar{y})$, and the fixed points are exactly the $\mathbb{R}$-points.

Example: The conic $x^2 + y^2 + z^2 = 0$ in $\mathbb{P}^2_\mathbb{R}$ has no $\mathbb{R}$-points (it is a non-trivial Brauer-Severi curve). Over $\mathbb{C}$, it becomes isomorphic to $\mathbb{P}^1_\mathbb{C}$, but the descent datum is non-trivial: it corresponds to the unique non-trivial element of $\operatorname{Br}(\mathbb{R})[2] = \mathbb{Z}/2\mathbb{Z}$.

Quadratic forms over $K$ can be studied via Galois descent. A quadratic form $q$ over $K$ extends to $q_L$ over $L$, which is easier to classify (e.g., over algebraically closed fields, every non-degenerate form is equivalent to $\sum x_i^2$).

The set of $K$-forms of the standard quadratic form $\sum_{i=1}^n x_i^2$ is $H^1(G_K, O_n)$ where $O_n$ is the orthogonal group. The exact sequence $1 \to SO_n \to O_n \to \mu_2 \to 1$ gives a long exact sequence relating quadratic forms to the discriminant and the spinor norm.

For $n = 2$ over $\mathbb{Q}$: quadratic forms $ax^2 + by^2$ with $ab \neq 0$ are classified (up to scaling) by $a \in \mathbb{Q}^\times/(\mathbb{Q}^\times)^2$. The form $x^2 + y^2$ is isomorphic to $x^2 - y^2$ over $\mathbb{Q}(\sqrt{-1})$ but not over $\mathbb{Q}$.

Let $X$ be a scheme with an action of a finite group $G$. The quotient $X/G$ always exists as an algebraic space (though not always as a scheme). If $G$ acts freely and $X$ is separated, then $X/G$ is a separated algebraic space.

Concrete example: Let $X = \mathbb{A}^2_k \setminus \{0\}$ and $G = \mathbb{Z}/2\mathbb{Z}$ acting by $(x,y) \mapsto (-x, -y)$. The quotient $X/G$ is a smooth algebraic space, but it is NOT a scheme (it is the "affine plane with doubled origin" quotient). The etale cover $X \to X/G$ exhibits $X/G$ as an algebraic space via etale descent.

More generally, if a finite group $G$ acts on a scheme $X$ and the stabilizers are trivial (free action), then $X \to X/G$ is a finite etale $G$-torsor, and $X/G$ is the algebraic space obtained by etale descent.

The classical example (due to Hironaka) of an algebraic space that is not a scheme: start with a smooth threefold $X$ over $\mathbb{C}$ with an involution $\sigma$ that acts freely outside a codimension-2 subvariety. The quotient $X/\sigma$ exists as an algebraic space but cannot be a scheme (it fails to have enough open affine subsets).

Another example: the etale quotient of $\mathbb{A}^1$ by the equivalence relation that identifies $0$ and $1$ (but this one fails to be separated, so it requires more care).

The following properties descend along etale surjective morphisms $f: Y \to X$:

Properties of schemes: reduced, normal, regular, Cohen-Macaulay, locally Noetherian, S$_n$ (Serre's condition), R$_n$ (regularity in codimension $n$).

Properties of morphisms: if $g: Z \to X$ is a morphism and $g': Z \times_X Y \to Y$ is its base change, then $g$ is flat (resp. smooth, etale, unramified, proper, finite, affine, quasi-finite, a closed immersion, an open immersion) if and only if $g'$ is.

Properties of sheaves: a coherent sheaf $\mathcal{F}$ on $X$ is locally free of rank $r$ if and only if $f^*\mathcal{F}$ is locally free of rank $r$ on $Y$.

These results follow from etale descent being a special case of fpqc descent, combined with the fact that etale morphisms are open maps.

To verify that a morphism $g: Z \to X$ is smooth, it suffices to check after an etale base change. For instance, consider $Z = \operatorname{Spec}(k[x,y]/(y^2 - x^3 - x)) \to X = \operatorname{Spec}(k)$ (an elliptic curve over $k$ with $\operatorname{char}(k) \neq 2, 3$).

After the etale cover $\operatorname{Spec}(\bar{k}) \to \operatorname{Spec}(k)$, the morphism becomes $E_{\bar{k}} \to \operatorname{Spec}(\bar{k})$, which is smooth (the curve $y^2 = x^3 + x$ is non-singular since $4 \cdot 1^3 + 27 \cdot 0 = 4 \neq 0$). By etale descent of smoothness, the original morphism is smooth.

For a surjective etale morphism $f: Y \to X$ and an abelian sheaf $\mathcal{F}$ on $X_{\text{et}}$, there is a descent spectral sequence (or Cech-to-derived spectral sequence):

$E_2^{p,q} = \check{H}^p(\{Y \to X\}, \mathcal{H}^q(\mathcal{F})) \implies H^{p+q}_{\text{et}}(X, \mathcal{F})$

where $\mathcal{H}^q(\mathcal{F})$ is the presheaf $U \mapsto H^q_{\text{et}}(U, \mathcal{F})$ and the Cech cohomology is computed using the simplicial scheme $Y^{\times_X \bullet}$.

When $f$ is a Galois cover with group $G$ (e.g., $\operatorname{Spec}(L) \to \operatorname{Spec}(K)$ for a Galois extension $L/K$), the spectral sequence becomes the Hochschild-Serre spectral sequence:

$E_2^{p,q} = H^p(G, H^q_{\text{et}}(Y, \mathcal{F})) \implies H^{p+q}_{\text{et}}(X, \mathcal{F})$

This is one of the most important computational tools in etale cohomology.

Let $k$ be a field, $X = \operatorname{Spec}(k)$, $Y = \operatorname{Spec}(k^{\text{sep}})$, $G = G_k = \operatorname{Gal}(k^{\text{sep}}/k)$. For $\mathcal{F} = \mathbb{G}_m$:

$E_2^{p,q} = H^p(G_k, H^q_{\text{et}}(\operatorname{Spec}(k^{\text{sep}}), \mathbb{G}_m))$

Since $H^0(\operatorname{Spec}(k^{\text{sep}}), \mathbb{G}_m) = (k^{\text{sep}})^\times$ and $H^q = 0$ for $q > 0$ (Spec of a separably closed field), the spectral sequence degenerates to

$H^p_{\text{et}}(\operatorname{Spec}(k), \mathbb{G}_m) = H^p(G_k, (k^{\text{sep}})^\times)$

This gives $H^1 = 0$ (Hilbert 90) and $H^2 = \operatorname{Br}(k)$ (the Brauer group equals the Galois cohomology $H^2(G_k, (k^{\text{sep}})^\times)$).

For a scheme $X$ and an etale cover $\{U_i \to X\}$, the Picard group $\operatorname{Pic}(X)$ can be computed via the Cech complex:

$\prod_i \mathcal{O}(U_i)^\times \to \prod_{i,j} \mathcal{O}(U_i \times_X U_j)^\times \to \prod_{i,j,k} \mathcal{O}(U_i \times_X U_j \times_X U_k)^\times$

with $\operatorname{Pic}(X) = \check{H}^1(\{U_i\}, \mathbb{G}_m)$. This is the standard description: line bundles are classified by transition functions (1-cocycles) modulo coboundaries.

For $X = \operatorname{Spec}(\mathbb{Z})$ with the etale cover $\operatorname{Spec}(\mathbb{Z}[\sqrt{-5}]) \to \operatorname{Spec}(\mathbb{Z})$ (after inverting appropriate primes), one can compute $\operatorname{Pic}(\mathbb{Z}[\sqrt{-5}]) = \mathbb{Z}/2\mathbb{Z}$ (the class group of $\mathbb{Q}(\sqrt{-5})$).