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The absolute Frobenius morphism of a scheme of characteristic p is the identity on the underlying topological space and raises every local function to its $p$th power. On an affine chart $\mathrm{Spec}A$ it is induced contravariantly by the Frobenius endomorphism $A\to A$, $a\mapsto a^p$. It need not be an isomorphism of schemes, even when it acts invertibly on global sections.

The algebraic Frobenius endomorphism induces this scheme map contravariantly on affine charts.

Assume a transitive model $M$ satisfies $2^{{\aleph }_0}={\aleph }_2$. In $M$, encode a bijection $e:{\omega }_2\to \mathcal{P}(\omega )$ by one set $A\subseteq {\omega }_2$, using a fixed pairing of ${\omega }_2\times \omega$ with ${\omega }_2$. Then $L(A)$ decodes $e$ and contains every real of $M$. The two models consequently have the same ${\omega }_1$, and any bijection between ${\omega }_1$ and the real numbers in $L(A)$ would also be one in $M$. Thus $L(A)\models \neg \mathsf{CH}$.

For every $\epsilon >0$ there is a positive integer $K(\epsilon )$ such that every finite bipartite graph has an $\epsilon$-regular pair of partitions with at most $K(\epsilon )$ cells on each side.

To prove this, give $X\times Y$ the uniform probability measure, let $f$ be the indicator function of the edge set, and for partitions $\mathcal{P},\mathcal{Q}$ define their energy byThis conditional expectation is an orthogonal projection, so $0\le \mathcal{E}\le \Vert f{\Vert }_2^2\le 1$. If the partitions are not $\epsilon$-regular, choose witnesses $A_{ij}\subseteq X_i$ and $B_{ij}\subseteq Y_j$ in every irregular cell pair and refine each $X_i$ by all its $A_{ij}$ and each $Y_j$ by all its $B_{ij}$. The refined energy exceeds the old energy byOn $A_{ij}\times B_{ij}$ the absolute value function of the mean of the difference is greater than $\epsilon$, while this rectangle occupies at least an ${\epsilon }^2$ fraction of $X_i\times Y_j$. The Cauchy-Schwarz inequality therefore contributes more than ${\epsilon }^4|X_i||Y_j|/(|X||Y|)$ from that cell pair. Irregular pairs have total weight greater than $\epsilon$, so every refinement raises the energy by more than ${\epsilon }^5$. The process stops after at most $\lceil {\epsilon }^{-5}\rceil$ refinements. If the current cell counts are $r,s$, the construction gives at most $r{2}^s,s{2}^r$ new cells, so a finite iterated bound depending only on $\epsilon$ supplies $K(\epsilon )$.

If $f$ is a continuous probability density on ${\mathbb{R}}^2$ and $B$ is planar Brownian motion, thenalmost surely. The density is bounded below on some disc, recurrence supplies infinitely many visits to a smaller concentric disc, and the Strong Markov property gives infinitely many visits of a fixed positive duration.

Let $B$ be the integral closure of a Dedekind domain $R$ in a finite separable extension $L/K$ of degree $n$. For a nonzero prime ideal $\mathfrak{p}$ of $R$, writeand let $f_{\mathfrak{q}/\mathfrak{p}}=[B/\mathfrak{q}:R/\mathfrak{p}]$. Then

Let $M$ be an indecomposable finite-dimensional $kG$-module and let $P_k$ be the projective cover of the trivial module. Then

If $P$ is an indecomposable projective $kG$-module, then its head and socle are simple and naturally isomorphic:This is the identity Nakayama permutation of the symmetric algebra $kG$.

The homogenization of a polynomial $f(x_1,\dots ,x_n)$ of total degree $d$ is the homogeneous polynomialIts projective zero set is the projective closure of the affine hypersurface defined by $f$.

A principal nilpotent element of a semisimple Lie algebra is a nilpotent element whose centralizer has dimension equal to the rank. For nonzero simple-root vectors $e_i$, the sum $e={\sum }_i{e}_i$ is principal nilpotent and forms the raising element of a Principal sl2 subalgebra.

Every presheaf category is a topos. Finite limits are pointwise; the exponential isand the subobject classifier assigns to $C$ the set of sieves on $C$.

For an oriented codimension-$r$ closed submanifold $Y\subset M$, its Poincare dual is the class in $H^r(M)$ whose cup-product evaluation against any complementary class equals intersection with $Y$.

For a uniformly elliptic operator $Lu=a^{ij}(x)D_{ij}u$ with continuous coefficients and $W⋐U$,Uniform continuity lets one freeze $a^{ij}$ on sufficiently small open balls and use the perturbation of a constant-coefficient elliptic second-derivative estimate. A partition of unity and cutoff functions reduce the global interior estimate to finitely many such balls; derivatives of the cutoffs produce only the displayed lower-order norm.