For an integral short equation $y^2=x^3+ax+b$, negation acts on $t=-x/y$ by $t\mapsto -t$. Thus the integral formal multiplication series is odd: $[2](t)=2t+t^{3}g(t)$. If $v_2(t)=s\ge 1$, the leading term has valuation $s+1<3s$, so successive doubling never kills a nonzero point. Odd multiplication has unit linear coefficient. Combined with the formal logarithm at odd primes, this proves torsion-freeness of the formal identity neighbourhood at every prime for a short equation. The short-model hypothesis matters at two.

For an elliptic curve over a local field with integral Minimal Weierstrass equation, let $E_0(K)$ consist of points with nonsingular reduction and let $E_1(K)$ be the kernel of $E_0(K)\to {\tilde{E}}_{\mathrm{ns}}(k)$. This definition also applies at a prime of bad reduction of an elliptic curve. The point at infinity is always smooth. With $t=-x/y$ and $w=-1/y$, the Weierstrass equation of an elliptic curve becomesThe unit linear coefficient in $w$ gives a unique integral formal power series $w(t)=t^3+O(t^4)$. Then $x=t/w$ and $y=-1/w$, so the nonidentity points of $E_1(K)$ have $v(x)=-2r$ and $v(y)=-3r$ for $r=v(t)>0$. The local parameter $t$ bijects $E_1(K)$ with ${\mathfrak{m}}_K$ and turns its addition into the formal group of an elliptic curve. At good reduction this agrees with the kernel of reduction of an elliptic curve.

For $h=({\sin }^2\theta +q^{-2}{\cos }^2\theta {)}^{-\gamma /2}$, separation in spherical coordinates makes both stellar potential-energy tensor integrals contain $\int r^{3-\gamma }{\Phi }^{\prime }(r)dr$. Their ratio is the displayed positive finite angular expression. If the radial integrals diverge, identical spherical cutoffs give this same ratio of force moments, but there is no finite global pressure-energy ratio without further hypotheses. A finite-aperture tensor virial theorem retains its surface tensor.

A mean-zero normal distribution prior of variance $s^2$ on the log baseline multiplies the multinomial kernel by the displayed correction, up to a constant independent of coefficients. The factor is not constant in $S$, so the posterior equivalence is approximate. Dominated convergence makes it tend to one as $s^2$ tends to infinity.

For commuting five-term exact sequences in an abelian category, the first vertical map being epic and the second invertible give an invertible induced map on the images entering the middle objects, because these images are the cokernels of the first horizontal maps. The fourth vertical map being invertible and the fifth monic similarly give an invertible map on the images leaving the middle objects, because these are the kernels of the last horizontal maps. The image factorization in an abelian category packages each middle object between these two images in a short exact sequence. The short five lemma then makes the middle vertical map invertible. The proof is categorical and does not require an embedding into a category of modules.

When $d=\sqrt{c^2+2r}$ is real and nonnegative, the discounted drifted Brownian first-passage density is $e^{a(c-d)}{h}_d(t)$. Integrating yields $e^{a(c-d)}\Phi ((dT-a)/\sqrt{T})+e^{a(c+d)}\Phi ((-a-dT)/\sqrt{T})$. In a Black-Scholes model with $c=(r-{\sigma }^2/2)/\sigma$, the identity $c^2+2r=(r+{\sigma }^2/2{)}^2/{\sigma }^2$ ensures this formula covers every real interest rate.

The degree parallelogram identity for endomorphisms of an elliptic curve gives $d_{n+1}+d_{n-1}=2d_n+2$ for $d_n=\mathrm{deg}[n]$, with $d_0=0$ and $d_1=1$. Induction gives $d_n=n^2$ in every characteristic. A zero pullback on differentials therefore does not mean that multiplication is the zero morphism.

This derivative component is carried by the jump set of a BV function. Swapping the two BV traces on a hypersurface and reversing the normal leaves it unchanged. In one dimension it consists of the point measure atoms, each weighted by the difference between the right and left limits.

If $dh$ is not identically zero, translating cotangent fibers by $dh$ preserves the canonical symplectic form but changes the Liouville one-form by ${\pi }^{\ast }dh$. Any positive-dimensional smooth manifold has a suitable nonconstant bump function in a chart. In dimension zero every differential one-form is zero, so this distinction cannot occur.

In a holomorphic local frame, a Hermitian metric on a holomorphic vector bundle has matrix $h$. The trace of its Chern connection curvature is $\bar{\partial }\partial \log \det h$. A frame change multiplies $\det h$ by the squared modulus of a holomorphic unit, whose logarithm has vanishing mixed derivative. For two metrics, the quotient of their determinants is a global positive function, and the trace curvatures differ by the exterior derivative of its logarithmic $\partial$-derivative.

In a holomorphic local frame the Chern matrix is $A=H^{-1}\partial H$. The identity $\partial A+A\wedge A=0$ gives vector-bundle curvature $\Theta =\bar{\partial }A$ of type $(1,1)$. Therefore ${\bar{\partial }}_{\mathrm{End}E}\Theta =0$, allowing the vector-bundle curvature to represent a bundle-valued Dolbeault class.

For the $(1,0)$ component of a Chern connection, $(D_E^{\prime }{)}^{\ast }=-{\ast }_{E^{\ast }}{D}_{E^{\ast }}^{\prime }{\ast }_E$, where the middle operator is the corresponding component of the dual Chern connection. This is the holomorphic-degree counterpart of the bundle-valued Dolbeault adjoint.

The vector-bundle curvature of the unique metric-compatible holomorphic Chern connection is an endomorphism-valued form of type $(1,1)$. It is Dolbeault closed and represents the curvature Dolbeault class. Its trace gives normalized first-Chern forms, while its full endomorphism action enters the Lefschetz-Dolbeault curvature commutator.

This metric pulls back the round spherical metric by $(u,v)\mapsto (\cos u\cos v,\cos u\sin v,\sin u)$. It has unit Gaussian curvature and Ricci curvature equal to its metric, yet the meridian reaches a missing boundary in finite time. Every curve between $(0,0)$ and $(0,L)$ has length at least $|L|/\sqrt{2}$, so its diameter is infinite. It shows that the positive Ricci bound alone does not replace geodesic completeness in the Bonnet-Myers theorem.

If a connected complete Riemannian manifold has $\mathrm{Ric}\ge (n-1)\kappa g$ with $\kappa >0$, its universal cover with the lifted metric is complete and satisfies the same bound. The Bonnet-Myers theorem makes that cover compact. A covering fiber is closed and discrete and hence finite; the number of its elements equals the order of the fundamental group. Applying the diameter theorem only to the base would not prove this conclusion: the positive lower bound must also be used on the cover.

For a scheme over $\mathbb{R}$, its base change of a morphism of schemes to $\mathbb{C}$ has scheme-theoretic fibre over a point $p$ equal to $\mathrm{Spec}(\kappa (p){\otimes }_{\mathbb{R}}\mathbb{C})$. This is the spectrum of $\kappa (p)[s]/(s^2+1)$. It has two points if $-1$ is a square in the residue field, and otherwise one point with a quadratic field extension. In both cases it is reduced, because the quadratic polynomial has no repeated root in characteristic zero. The distinction applies to generic points as well as closed points.

For an integral alternating form $E$ arising as the imaginary part of a Hermitian form, a semicharacter satisfies $\alpha (\lambda +\mu )=e^{\pi iE(\lambda ,\mu )}\alpha (\lambda )\alpha (\mu )$. The Appell–Humbert theorem uses this datum together with the Hermitian form to specify a holomorphic line bundle on a complex torus. The ratio of two semicharacters is an ordinary unitary group homomorphism. On the square period lattice, with $H(z,w)=z\overline{w}$, all semicharacters are $\alpha (m+ni)=(-1{)}^{mn}{u}^m{v}^n$ for $u,v\in U(1)$.

The points of the real affine plane as a scheme are its prime ideals: the zero ideal, the principal ideals of irreducible real polynomials, and its maximal ideals. The latter have residue field $\mathbb{R}$ or $\mathbb{C}$, by the Zariski lemma and the real closed field property. They correspond respectively to real coordinate pairs and conjugate pairs of nonreal complex coordinate pairs. Thus the scheme contains both nonclosed generic points and closed points absent from the real locus. Its topology is the Zariski topology, and its structure sheaf has sections $\mathbb{R}[x,y{]}_h$ on each principal open subscheme $D(h)$.

For a smooth complex submanifold $V\subset M$, the tangent exact sequence gives $\det TM{|}_V=\det TV\otimes \det N_V$. Dualizing yields $K_V=K_M{|}_V\otimes \det N_V$. This form of adjunction formula applies in any codimension.

For regular homogeneous equations of degrees $d_1,\dots ,d_r$ in ${\mathbb{P}}^n$, the normal bundle of a regular zero locus is ${⨁}_i\mathcal{O}(d_i){|}_V$. The Euler sequence gives $K_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$, hence $K_V=\mathcal{O}({\sum }_i{d}_i-n-1){|}_V$. Equations with unreduced multiplicities do not have this conclusion for the underlying reduced submanifold.