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A useful form of the Seesaw theorem determines a line bundle on a product from its fibre restrictions. Let $X$ be a complete geometrically integral variety with a base point $x_0$, and let $L$ be a line bundle on $X\times Y$ trivial on each geometric fibre $X\times \{y\}$. For a reduced variety $Y$, constancy of the fibrewise section dimension and cohomology and base change imply that $L$ is pulled back from $Y$; restricting to $\{x_0\}\times Y$ identifies the bundle being pulled back. If that transverse restriction is also trivial, then $L$ is trivial. This is useful for verifying normalized universal bundle identities by checking fibres; matching only one fibre in each direction is insufficient.

For the defining representation $V=V({\omega }_1)$ of ${\mathfrak{sl}}_3$,

If $\alpha ,\beta$ are the roots of $T^2-aT+q$, where $a=q+1-\#E({\mathbb{F}}_q)$, then

For an elliptic curve over a nonarchimedean local field, $E_0$ consists of points with nonsingular reduction, $E_1$ is the kernel of reduction, and $E_r$ for $r\ge 1$ consists of points whose formal parameter has valuation at least $r$. The successive quotients for $r\ge 1$ are copies of the additive residue-field group.

For a proper morphism of Noetherian schemes and a coherent sheaf flat over the base, the fiber dimension $s\mapsto {\dim }_{\kappa (s)}{H}^p(X_s,{\mathcal{F}}_s)$ is upper semicontinuous, and the fiberwise Euler characteristic is locally constant.

For fixed $\xi >0$,

The scheme-theoretic image of $f:X\to Y$ is the smallest closed subscheme of $Y$ through which $f$ factors. For an affine morphism $\mathrm{Spec}B\to \mathrm{Spec}A$ induced by $\phi :A\to B$, it is $\mathrm{Spec}(A/\ker \phi )$, whose underlying set is the closure of the set-theoretic image.

The construction retains the scheme structure on the closed subscheme, not merely the set of points in an image.

Every closed subset $Z\subseteq X$ has a canonical reduced closed-subscheme structure defined affine-locally by $\mathrm{Spec}(A/\sqrt{I})$ when $Z=V(I)\subseteq \mathrm{Spec}A$. It is the smallest closed subscheme with underlying set $Z$.

The regulator is the determinant of the canonical-height pairing on a basis of the free part of the Mordell-Weil group. A unimodular basis change leaves it invariant.

When the homological Bockstein satisfies ${\beta }^2=0$, it makes $H_{\ast }(C\otimes \mathbb{Z}/n)$ into a chain complex. Its homology is the Bockstein homology $\beta H_{\ast }(C;n)$.

For simple roots $(1,0)$ and $(-1,1)$, the fundamental weights are ${\omega }_1=(1/2,1/2)$ and ${\omega }_2=(0,1)$. The representation $[1,0]$ is the four-dimensional spin representation with weights $(\pm 1/2,\pm 1/2)$; $[0,1]$ is the five-dimensional vector representation with weights $(\pm 1,0),(0,\pm 1),(0,0)$.

The root systems $B_2$ and $C_2$ are isomorphic after interchanging long and short simple-root labels. The classification of complex simple Lie algebras therefore gives ${\mathfrak{so}}_5(\mathbb{C})\cong {\mathfrak{sp}}_4(\mathbb{C})$.

The punctured affine plane is obtained by deleting the closed point $(x,y)$ from $\mathrm{Spec}k[x,y]$. Its regular functions still form $k[x,y]$, whilewhich has basis represented by $x^{-a}{y}^{-b}$ for $a,b\ge 1$.

For direct primitive vectors $(1,0)$ and $(-1/2,\sqrt{3}/2)$, reciprocal primitive vectors areThe reciprocal Wigner--Seitz cell is a regular hexagon of area $8{\pi }^2/\sqrt{3}$ and circumradius $4\pi /3$.

For the principal specialization, let $h_{\mathrm{pr}}=2{\rho }^{\vee }$, so every simple root takes value two on $h_{\mathrm{pr}}$. The q-character isIt is obtained from the Weyl character formula by substituting $e^{\mu }=q^{\mu (h_{\mathrm{pr}})}$.

The dimension obtained by evaluating the Weyl character formula at the identity is

A singular vector is a nonzero weight vector annihilated by the positive nilpotent subalgebra ${\mathfrak{n}}^{+}$.

The weight space $V_{\mu }$ consists of all weight vectors of weight $\mu$, together with zero.

This is the simultaneous eigenspace belonging to a Weight.

A weight of a representation is a functional $\mu$ whose weight space is nonzero.

Minimality here concerns the discriminant valuation of an integral Weierstrass equation of an elliptic curve over a discretely valued field.

Over a discretely valued field, a minimal Weierstrass equation has integral coefficients and has least possible discriminant valuation among integral Weierstrass equations related by admissible changes of variables.