Let $\kappa :G\Omega \to \Omega$ classify $G\top :1\hookrightarrow G\Omega$. In the cofree coalgebra $R\Omega$, equalize the identity and $G\kappa {\delta }_{\Omega }$. A map $\chi :X\to \Omega$ classifies a subcoalgebra exactly when $\chi =\kappa G\chi x$, since this says the subobject equals the inverse image of its image under $G$. Transposition gives precisely the displayed equalizer condition.

The canonical height pairing is $⟨P,Q⟩=(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))/2$. The height parallelogram identity makes it bilinear. It descends through torsion and is positive definite on the free part of the Mordell-Weil group. The Cauchy-Schwarz inequality gives $\hat{h}(P-Q)\le 2\hat{h}(P)+2\hat{h}(Q)$, useful for height descent.

For naive logarithmic $x$-height on an elliptic curve, $h_x(P+Q)+h_x(P-Q)=2h_x(P)+2h_x(Q)+O(1)$ uniformly. The unordered pair of sum and difference on the $x$-line defines a bidegree-$(2,2)$ morphism to ${\mathrm{Sym}}^2{\mathbb{P}}^1$, yielding the estimate. Applying it to $2^{n}P,2^{n}Q$ and taking the defining limit gives the exact identity for the canonical height of an elliptic curve.

On the standard charts of projective space, wedge the coordinate differentials in increasing order. The change $X_0/X_i=(X_i/X_0{)}^{-1}$ and $X_j/X_i=(X_j/X_0)/(X_i/X_0)$ gives the displayed Jacobian determinant. Multiplying the $i$th local generator by $(-1{)}^i$ removes this sign and yields the transition functions of the twisting sheaf on projective space with degree $-n-1$.

A cancellative commutative monoid is torsion-free in this sense when $na=nb$ implies $a=b$ for every positive integer $n$. Its Grothendieck group is then a torsion-free abelian group, because $n(a-b)=0$ forces $na=nb$. This is the condition inherited by a submonoid of a torsion-free abelian group.

The growing root of the ideal magnetorotational dispersion relation is maximized continuously at $a=v_A^2{k}^2/{\Omega }^2=15/16$, with growth $3\Omega /4$. Discrete vertical modes instead have $a_n=2n(n+1)/\beta$. The fastest one is found by comparing admissible integers on either side of the continuous maximum, rather than rounding a vertical wavenumber blindly. At $\beta =24$, $n=3$ gives $s=\Omega \sqrt{(\sqrt{17}-3)/2}$.

For a continuous chordal SLE trace started at zero, hitting the real interval $[b,\infty )$ is equivalent to finite boundary-point swallowing time for a Loewner chain at $b>0$. A boundary crosscut may swallow all points between its endpoints without visiting them individually. The Boundary-point Bessel flow for SLE determines the probability of this event.

The integral Bockstein is the connecting homomorphism of $0\to \mathbb{Z}\stackrel{n}{\to }\mathbb{Z}\to \mathbb{Z}/n\to 0$. Lift a modulo-$n$ cocycle to an integral cochain $a$; write its coboundary as $\delta a=nb$. Then $\hat{\beta }[a]=[b]$. Its image is the subgroup of integral cohomology killed by $n$.

The Bockstein homomorphism for $0\to \mathbb{Z}/n\to \mathbb{Z}/n^2\to \mathbb{Z}/n\to 0$, with injection $[a]\mapsto [na]$, is coefficient reduction $\rho$ after the integral Bockstein homomorphism. The commuting maps of coefficient sequences prove this by naturality of the connecting homomorphism.

For the coefficient sequence $0\to {\mathbb{Z}}_2\to {\mathbb{Z}}_4\to {\mathbb{Z}}_2\to 0$, the Bockstein homomorphism on infinite-dimensional real projective space is zero from an even degree and an isomorphism from an odd degree. In cellular cohomology, lift the mod-two generator to $1\in {\mathbb{Z}}_4$. The coboundary is zero in even degree and two in odd degree; identifying two with the image of $1\in {\mathbb{Z}}_2$ gives the formula.

If $X\to Y$ is a surjective morphism of schemes, the projection $X{\times }_Y{Y}^{\prime }\to Y^{\prime }$ is surjective for every $Y^{\prime }\to Y$. For a point $y\prime$, choose a point $x$ over its base image and apply the point-lifting property of a scheme fibre product. No flatness assumption is necessary.

For simple roots $(1,0)$ and $(-1,1)$, coroot integrality gives the weight lattice $P={\mathbb{Z}}^2\cup (\mathbb{Z}+1/2{)}^2$. This is the lattice for the Spin group $\mathrm{Spin}(5)$ and its Lie algebra. Representations descending to $SO(5)$ use the integer coset, because a $2\pi$ rotation in either coordinate plane must act trivially.

For each pair $f:V\to U$, $g:W\to U$, require $fh=gk$ for some arrows $h:T\to V$, $k:T\to W$. This is necessary and sufficient for stability of nonempty sieves under pullback; the other topology axioms then follow. Nonempty finite sets with arbitrary functions fail it at distinct singleton-to-two-point maps.

Use a small skeleton of nonempty finite sets and surjections, with every nonempty sieve covering. Fiber-product projections are surjections, so the atomic coverage exists. Surjections are effective quotients of their kernel pairs, making the site subcanonical. Its sheaves admit the primitive decomposition of an atomic finite-surjection sheaf.

For real bounded observables, the Cesaro limit of the correlation comes from arithmetic-progression multiple averages under weak mixing. Apply the same result on the weakly mixing product system to $f_j\otimes f_j$ to obtain its second moment. The averaged squared error tends to zero, so the mean-square criterion for convergence in density gives the displayed density limit.

For a uniform initial column, annular return flow around a slumping column competes with axial extensional stress over $\hat{z}=\sqrt{h_0^3/(2\lambda a_0)}$. With $G=\Delta \rho g/(3\mu )$, choose velocity $\hat{w}=G{\hat{z}}^2$ and time $\hat{t}=\hat{z}/\hat{w}$. Equivalently $\hat{z}=R/\sqrt{\lambda {\alpha }_0}$, where $\varphi =a^2/R^2$ and $\alpha (\varphi )=2\sqrt{\varphi }/(1-\sqrt{\varphi }{)}^3$ in the leading thin-annulus model. This length describes the vertical range of the basal constraint: a much taller column has a localized basal boundary layer, while a much shorter column responds predominantly through extension.

An alpha-squared dynamo may use a tensorial alpha effect rather than an isotropic coefficient. For constant alpha tensor $\mathbf{\alpha }$ and magnetic diffusivity $\eta >0$, a steady nonzero Fourier mode with wavevector $\mathbf{K}$ satisfies $\eta K^2\mathbf{B}=i\mathbf{K}\times (\mathbf{\alpha }\mathbf{B})$ and $\mathbf{K}\cdot \mathbf{B}=0$. Anisotropy can change both the critical alpha magnitude and the preferred wavevector direction.

A simple ray turning point has outer amplitude proportional to $|Y-Y_s{|}^{-1/4}$. At the Airy scaling at a variable-speed wave turning point, $|Y-Y_s|=O({\epsilon }^{2/3})$, this gives an inner wavefield of size $O({\epsilon }^{-1/6})$ when the incident action normalization is order one. The enhancement is finite for nonzero $\epsilon$ and is resolved by the Airy function.

The local inner solution is a combination of Airy functions $\mathrm{Ai},\mathrm{Bi}$. Decay in the forbidden region removes $\mathrm{Bi}$. The remaining Airy turning-point connection formula is oscillatory on the allowed side and represents an incident/reflected pair, not a single upward branch. Without the forbidden-side condition the local differential equation alone leaves both coefficients free.

An open subset of an algebraic variety equipped with an identification with an affine variety. Such charts make quasi-coherent sheaves and regular maps accessible through their local coordinate rings.