Under the degree-one generation condition for Proj, cover $X=\mathrm{Proj}S$ by degree-one charts. The degree-one localization of a graded module identifies the two local modules in the displayed formula, using localization commutes with tensor products. The maps are natural and agree on overlaps, so they glue to an isomorphism of sheaves of modules. The analogous assertion can fail for general positively graded rings.

The leading scalar spectrum is proportional to $V^3/V^{\prime 2}$. Along a slow-roll trajectory, $d\varphi /d\ln a\simeq -M_{\mathrm{pl}}^2{V}^{\prime }/V$ and $d\ln k/d\ln a=1-{ϵ}_H$. Differentiating at horizon exit and keeping first order gives $n_s-1=-6{ϵ}_V+2{\eta }_V$. The derivative-factor correction contributes at second order.

For an elliptic curve with good reduction over ${\mathbb{Q}}_2$, let $E_1$ be the reduction kernel and $E_2$ its formal subgroup with parameter in $4{\mathbb{Z}}_2$. The deep logarithm subgroup of a formal group makes $E_2$ torsion-free, while $[E_1:E_2]=2$. Thus the torsion of the reduction kernel has order at most $2$. Combined with the Hasse theorem for elliptic curves, rational torsion of order $16$ forces bad reduction at $2$.

A proposed square class in a two-isogeny descent can be excluded by testing its quartic modulo $8$. For $N^2=3U^4+5U^2{V}^2-2V^4$ with $\gcd (U,V)=1$, both variables odd give residue $6$; odd-even gives $3$ or $7$; even-odd gives $6$ or $2$. None is a square modulo $8$, excluding class $3$ and, by the subgroup property of the descent image, its entire coset.

If primitive elements have equal restrictions along $\alpha :P\to m$ and $\beta :P\to n$, then these surjections have equal kernels. To rule out $\alpha (a)=\alpha (b)$ with unequal $\beta$ images, identify those two images by $q:n\to n-1$. The set of pairs equal under $\alpha$ and $q\beta$ maps surjectively both to $P$ by projection and to the kernel pair of $q$ by applying $\beta$. Injective restrictions force the kernel-pair matching condition on the primitive element, contradicting descent along $q$.

Partition the elements of a sheaf by their primitive-ancestor equivalence classes under bijection. Each class gives a sheaf subfunctor, because its membership is preserved and reflected along covering surjections. A representative primitive element names an epic representable map onto its class component. Thus every sheaf is the coproduct of these components, each an atom in a topos.

A smooth real rotation-invariant potential is constant on every positive-radius circle, so its logarithmic-radius function $u$ is smooth on $\mathbb{R}$. Off the origin, $f_{z\bar{z}}=e^{-t}{u}^{\prime \prime }(t)$. Since $f$ is smooth on the entire plane, this coefficient extends smoothly through the origin. A finite Taylor expansion, rather than a convergent series assumption, gives $f(z)=f(0)+b|z{|}^2+O(|z{|}^4)$.

The long roots of a root system $\pm (1,1)$ and $\pm (1,-1)$ give the commuting $SU(2)$ factors of the standard $SO(4)$ subgroup. Choosing $(1,1)$ for the left factor pairs a weight $(x,y)$ with its coroot as $x+y$. The five-dimensional vector then gives two doublets and a singlet, whereas a short-axis root would give a triplet and two singlets.

For distinct closed points over an algebraically closed field, evaluation gives the short exact sequence $0\to {\mathcal{I}}_{\{P,Q\}}\to {\mathcal{O}}_X\to k_P\oplus k_Q\to 0$. If all global regular functions are constant, the long exact sequence in sheaf cohomology embeds $k^2/k(1,1)$ into $H^1(X,{\mathcal{I}}_{\{P,Q\}})$. Thus this ideal detects a concrete obstruction to affineness.

For a target corner $(\alpha ,\beta )$ in the Greene–Nijenhuis–Wilf hook walk, put $A_i=h(i,\beta )-1$ and $B_j=h(\alpha ,j)-1$. The probability of terminating there from $(a,b)$ factors as $F_a{G}_b$, where $F_{\alpha }=G_{\beta }=1$, $F_a=A_a^{-1}{\prod }_{i=a+1}^{\alpha -1}(1+A_i^{-1})$ for $a<\alpha$, and $G_b=B_b^{-1}{\prod }_{j=b+1}^{\beta -1}(1+B_j^{-1})$ for $b<\beta$. It is zero when the start is outside the target's northwest rectangle. The factorization uses $h(i,j)-1=A_i+B_j$ and the first-step recurrence.

Let simple random walk travel from $a\ne z$ to its first visit to $z$, then stop on its subsequent first hit of $a$ in a finite connected unweighted loopless graph. Every directed edge $u\to v$ has expected traversal countThe Strong Markov property splits the count into the two killed legs. By killed-walk occupation voltage, their contributions are $v_{az}(u)$ and $v_{za}(u)$. Their Laplacians cancel, so the harmonic maximum principle on a finite graph makes the sum constant, with value $v_{az}(a)+v_{za}(a)=R_{\mathrm{eff}}(a,z)$. Summing over all directed edges recovers the commute time identity.

For a nondegenerate isometry direction $y$ and initially zero two-form, the local T-dual fields in alpha-prime-one units are $g_{yy}^{\prime }=1/g_{yy}$, $g_{yi}^{\prime }=0$, $g_{ij}^{\prime }=g_{ij}-g_{yi}{g}_{yj}/g_{yy}$, $B_{yi}^{\prime }=g_{yi}/g_{yy}$ and $B_{ij}^{\prime }=0$. Gauging the isometry and eliminating its auxiliary connection derives these rules. The spectator metric is a Schur complement. The sign of the mixed two-form follows the multiplier and dual-coordinate orientation convention.

The degree-one Bockstein homomorphism makes modulo-$n$ cohomology into a cochain complex. Its cohomology is the Bockstein cohomology. For prime $n$ this is a graded vector space; for composite $n$ it is a graded $\mathbb{Z}/n$-module. For ${\mathbb{RP}}^3$ at modulus two, its only nonzero groups are ${\mathbb{F}}_2$ in degrees zero and three. This is the cohomological counterpart of Bockstein homology.

Start with the highest weight and process weights by increasing height below it. For each simple root, compute the upper endpoint from the already known higher weights, use the weight string endpoint relation, and append all downward steps. Repeat until the finite set stabilizes. This recovers the weights of a finite-dimensional Irreducible Lie algebra representation without determining weight multiplicities.

A diagram placing the weights of a representation in the real span of its weight lattice, recording each weight multiplicity. Coincident weights must retain their multiplicity: for an Adjoint representation the zero-weight multiplicity is the dimension of its Cartan subalgebra.

For a finite-dimensional Irreducible Lie algebra representation of a complex semisimple Lie algebra with highest weight $\lambda$, this recursion computes its weight multiplicities from $m_{\lambda }(\lambda )=1$. Here $\rho$ is the half-sum of positive roots and the inner product is induced by the Killing form. Write the Casimir operator on the weight-$\mu$ space as $(\mu ,\mu +2\rho )I+2{\sum }_{\alpha >0}{F}_{\alpha }{E}_{\alpha }$. The cyclic trace identity between adjacent weight spaces gives $\mathrm{tr}(F_{\alpha }{E}_{\alpha }{|}_{V_{\mu }})={\sum }_{j\ge 1}(\mu +j\alpha ,\alpha )m_{\lambda }(\mu +j\alpha )$. Taking the trace and using the Casimir eigenvalue proves the formula. Only finitely many terms are nonzero.

For linear maps $E:U\to W$ and $F:W\to U$ between finite-dimensional vector spaces, ${\mathrm{tr}}_U(FE)={\mathrm{tr}}_W(EF)$ even when their dimensions differ; expanding rectangular matrices proves it. For root vectors normalized by $[E_{\alpha },F_{\alpha }]=t_{\alpha }$, with $B(t_{\alpha },h)=\alpha (h)$, this gives $T_{\alpha }(\mu )=T_{\alpha }(\mu +\alpha )+(\mu +\alpha ,\alpha )\dim V_{\mu +\alpha }$. Iteration along a finite weight string is the trace step in Freudenthal multiplicity formula.

For $E:y^2=x(x^2+ax+b)$, set $b^{\prime }=a^2-4b$ and $E^{\prime }:Y^2=X(X^2-2aX+b^{\prime })$. The isogeny of elliptic curves with kernel $\{O,(0,0)\}$ is $\varphi (x,y)=(x+a+b/x,y(1-b/x^2))$. Its dual is $\hat{\varphi }(X,Y)=((X-2a+b^{\prime }/X)/4,Y(1-b^{\prime }/X^2)/8)$. The maps extend over their exceptional affine points and satisfy $\hat{\varphi }\varphi =[2]$.

For the two-isogeny formula $\varphi :E\to E^{\prime }$ over a number field, the two-torsion square-class homomorphisms have kernels $\hat{\varphi }{E}^{\prime }(K)$ and $\varphi E(K)$. ConsequentlyIndeed $2E(K)=\hat{\varphi }\varphi E(K)$, and the surjection from $E^{\prime }(K)/\varphi E(K)$ to $\hat{\varphi }{E}^{\prime }(K)/2E(K)$ has kernel $\ker \hat{\varphi }/(\ker \hat{\varphi }\cap \varphi E(K))$. Combined with the Mordell-Weil theorem, the index is $2^r\#E(K)[2]$, where $r$ is the rank of an abelian group. This formula prevents an erroneous extra factor of two in an isogeny descent.

Suppose the ring of integers of a number field is a principal ideal domain, $a,b$ are integral, and $b$ is a unit. On $y^2=x(x^2+ax+b)$ every nonzero $x$-coordinate has even valuation at every finite prime: positive valuation gives $2v(y)=v(x)$, and negative valuation gives $2v(y)=3v(x)$. Thus the two-torsion square-class homomorphism takes values among unit square classes. For the Gaussian integers, their unit group modulo squares has two elements, represented by $1,i$. This can reduce a two-isogeny descent to a very small calculation even though the field contains no ordering.