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For finite-dimensional complex linear operators with $C=[A,B]$ commuting with both $A$ and $B$, differentiating $e^{-vB}A{e}^{vB}$ gives the constant derivative $C$. Hence $e^{-B}A{e}^B=A+C$, and conjugating the matrix exponential gives $e^{-B}{e}^{-A}{e}^B=e^{-A-C}$. Since $[A,C]=0$, multiplication by $e^A$ proves the identity. The commutator order determines the minus sign.

The free commutative monoid on $d$ generators consists of the nonnegative integral sums of those generators, with unique coefficients. Its semigroup algebra over a commutative ring is the polynomial ring in $d$ variables. In a smooth toric chart, the dual-cone lattice points form such a monoid, yielding affine-space coordinates.

For a height-one CM elliptic curve formal group and compatible ${\mathfrak{p}}^n$-division parameters $t_n$, an integral Coleman series interpolates a norm-compatible sequence of local units $u_n$. Its norm relation is multiplication of its translates by the $\mathfrak{p}$-torsion, with coefficient Frobenius where required by the unramified base. Coates–Wiles homomorphisms and the Frobenius-corrected logarithm turn this local series into critical values and measures.

On $R={\mathbb{Z}}_p[[T]]$, this injective substitution reduces modulo $p$ to $f(T)\mapsto f(T^p)$. The ring is a free module of rank $p$ over $\phi (R)$, with basis $1,(1+T),\dots ,(1+T{)}^{p-1}$. Prove this first modulo $p$, where the change of basis from $1,T,\dots ,T^{p-1}$ is triangular and invertible, and then lift coefficients successively modulo powers of $p$.

The determinant norm for the finite free extension $R/\phi (R)$ defines $\mathcal{N}$ on units. After adjoining the p-th roots of unity, the norm is the displayed product of conjugates. A Coleman power series of a norm-compatible unit sequence satisfies $\mathcal{N}f=f$. For symmetric cyclotomic units, the product relation follows directly from ${\prod }_{{\xi }^p=1}(1-{\xi }^a{X}^a)=1-X^{ap}$ when $p\nmid a$.

For a norm-compatible sequence of local units, its Coleman power series $f_u\in {\mathbb{Z}}_p[[T]{]}^{\times }$ defines these integral additive homomorphisms. In the formal coordinate $z=\log (1+T)$ they are the derivatives of $\log f_u(e^z-1)$; the constant term of the logarithm is irrelevant. A Galois element with cyclotomic value $a$ multiplies the kth derivative by $a^k$. This is the multiplicative-formal-group version of the Coates–Wiles homomorphism.

Such a sequence $u=(u_n)$ satisfies $N_{n/n-1}{u}_n=u_{n-1}$ at every successive layer. The inverse limit uses the field norm, rather than inclusion of units from the lower field. In a p-power cyclotomic tower a Coleman power series interpolates the sequence, and its Coleman norm operator fixes the interpolating series.

Let $V_{\infty }$ be the norm limit of local principal units in the $p$-power cyclotomic field tower, for odd $p$. The Coleman power series $f_u$ gives the Frobenius-corrected logarithm $p^{-1}\log (f_u^p/\varphi f_u)$, whose inverse Amice transform is supported on ${\mathbb{Z}}_p^{\times }$. This is the middle map in the displayed exact sequence; the last map is the first moment $\mu \mapsto \int xd\mu$. Its kernel consists of norm-compatible $p$-power roots of unity. The logarithmic difference equation has one resonant degree, namely degree one in $\log (1+T)$, accounting for the last term. Complex conjugation acts by $-1$ on both copies of ${\mathbb{Z}}_p(1)$. Taking its even part therefore identifies the local principal-unit limit in the real tower with ${\mathbb{Z}}_p[[{\mathbb{Z}}_p^{\times }/\{\pm 1\}]]$.

For odd $p$ and a norm-compatible sequence of local units in ${\mathbb{Q}}_p({\mu }_{p^{n+1}})$, there is a unique $f_u\in {\mathbb{Z}}_p[[T]{]}^{\times }$ interpolating all its entries, and the Coleman norm operator fixes it. To construct it, choose integral unit polynomials $F_j$ interpolating entry $u_j$, and take a convergent subsequence of $N^j{F}_{2j}$ in the compact $(p,T)$-adic unit space. The congruence $NF\equiv F(\mathrm{mod}p)$ and Coleman norm contraction show that its evaluation at any fixed level tends to the desired entry. Uniqueness follows from the Weierstrass preparation theorem: a nonzero integral power series has only finitely many zeros in the open unit disk. The norm relation on evaluations then proves norm fixation. Principal units correspond to $f_u(0)\equiv 1(\mathrm{mod}p)$.

With $⟨{\mathrm{ad}}_{\xi }^{\ast }\ell ,\eta ⟩=-⟨\ell ,[\xi ,\eta ]⟩$, the displayed form on a coadjoint orbit is well-defined: changing a generator by an element of the stabilizer changes no pairing. It is nondegenerate on orbit tangents, and its closedness follows from the Jacobi identity. This sign agrees with the positive Lie-Poisson bracket when Hamiltonian vector fields act by $X_f(g)=\{g,f\}$. A negative Lie-Poisson convention reverses the orbit form as well.

A surjective ring homomorphism $A\to B=A/I$ identifies the spectrum of a ring $B$ with the closed subset $V(I)$ and induces a surjective map of structure sheaves. Conversely, the direct image sheaf of ${\mathcal{O}}_{\mathrm{Spec}B}$ under an affine morphism is the sheaf associated with the $A$-module $B$: its sections on $D(a)$ are $B_{\varphi (a)}$. The cokernel of the structure-sheaf map is therefore associated with the module cokernel of $A\to B$. If the sheaf map is surjective, all localizations of that cokernel vanish and the module cokernel is zero. This proves the converse without assuming global sections preserve arbitrary sheaf surjections.

A nonempty flat closed immersion of finite presentation into a connected scheme is an isomorphism. On a chart it is $A\to A/I$ with $I$ finitely generated. At a prime containing $I$, the quotient is a finite flat module over the local ring, hence free of rank one, so $I$ vanishes there. Finite generation extends this vanishing to a neighbourhood. Thus the image is open as well as closed; connectedness makes it all of the target, and the ideal is everywhere zero. Without finite presentation the conclusion can fail.

For finitely many orthonormal local trivializations $t_i$ and a subordinate partition of unity ${\rho }_i$, the displayed map embeds a rank-k complex vector bundle isometrically in $X\times {\mathbb{C}}^{kr}$. Outside the support of each ${\rho }_i$ its component is zero. Its continuous image planes give a finite-dimensional Grassmannian map and identify the original bundle with a tautological pullback.

Let P,Q be continuous orthogonal projection matrices of equal rank on a common trivial complex vector bundle, with the displayed bound at every base point. Restricting Q to the image of P is injective: if Qv=0, then $v=(P-Q)v$ contradicts the norm bound unless v=0. Equal dimensions give a continuously invertible vector bundle isomorphism. Compactness applies this observation uniformly to a homotopy of image planes.

The contractible universal frame bundle gives ${\pi }_4(B{U}_2)\cong {\pi }_3(U_2)$ and simple connectedness of $B{U}_2$. The determinant fibration $S{U}_2\to U_2\to S^1$ identifies ${\pi }_3(U_2)$ with ${\pi }_3(S{U}_2)$. Since $S{U}_2\cong S^3$, this is $\mathbb{Z}$. Hence the classification of complex vector bundles by a Grassmannian gives the displayed bijection of isomorphism classes; fixed-rank classes do not acquire a group operation from Whitney sum.

A symplectomorphism of a closed symplectic surface carries the complementary components of a separating simple closed curve to those of its image, preserving their unordered symplectic areas. On the unit sphere a latitude at height $h$ has complementary areas $2\pi (1-h)$ and $2\pi (1+h)$. Thus the equator cannot be sent to any nonequatorial latitude.

For symmetric nonnegative $D$ with zero diagonal, a centered Euclidean realization implies $D_{ij}^2=G_{ii}+G_{jj}-2G_{ij}$ and hence the displayed double-centering identity. Conversely, if the double-centered matrix is positive semidefinite, factor it as $X{X}^T$. Its reconstructed squared distances equal the original squared distances: their difference has zero double-centering, so is of the form $1a^T+a{1}^T$; the zero diagonal forces $a=0$. The minimum exact embedding dimension is $\mathrm{rank}G$.

With $D=\sum a_i{F}_i$ and $\sum q_i{a}_i=1$, the dual toric lattice is $M=\{m\in {\mathbb{Z}}^{n+1}:\sum q_i{m}_i=0\}$. A torus character ${\chi }^m$ is a section of $rD$ exactly when $b_i=m_i+r{a}_i\ge 0$. The vector $b$ has weighted degree $\sum q_i{b}_i=r$, and the correspondence ${\chi }^m\mapsto X^b$ is bijective on bases in every degree and respects multiplication. This gives the displayed graded-ring isomorphism, even when $D$ is a non-Cartier Weil divisor.

For smooth projective varieties $X,Y$, pull back Chow classes from the two factors and take their intersection product on $X\times Y$. This gives the external product and a graded map ${\mathrm{CH}}^{\ast }(X)\otimes {\mathrm{CH}}^{\ast }(Y)\to {\mathrm{CH}}^{\ast }(X\times Y)$. Unlike the cohomological Künneth theorem, it need not generate every class of the product.

The probabilities that a random chordal filling avoids each admissible compact H-hull away from zero determine its law as an unparameterized closed set. Each point of its complement can be joined to the relevant real boundary arc through the complement and lies in a thin rational polygonal tube attached to that arc. These countably many hull tests recover the complement. Their joint avoidance events are tested by filled finite unions; if a union separates zero from infinity, the joint avoidance event is empty. Otherwise connectedness makes avoidance of the union equivalent to avoidance of its filled hull. Thus individual hull probabilities give every finite joint indicator distribution and determine the random complement.