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A module over a commutative ring is flat exactly when its character module is injective. For a submodule inclusion $N^{\prime }\hookrightarrow N$, the character adjunction identifies restriction ${\mathrm{Hom}}_A(N,M^{\ast })\to {\mathrm{Hom}}_A(N^{\prime },M^{\ast })$ with the dual of $N^{\prime }{\otimes }_{A}M\to N{\otimes }_{A}M$. The injective cogenerator of abelian groups detects whether the latter map is injective. Thus the two extension/exactness conditions are equivalent.

The arrow is the canonical multiplication map, not an arbitrary abstract module isomorphism. Its inclusion into $M$ is ${\mu }_I:I{\otimes }_{A}M\to M$. Dualizing identifies ${\mu }_I^{\ast }$ with the restriction ${\mathrm{Hom}}_A(A,M^{\ast })\to {\mathrm{Hom}}_A(I,M^{\ast })$. All these maps are surjective precisely when $M^{\ast }$ is injective, by the Baer criterion. The injective cogenerator of abelian groups detects injectivity of ${\mu }_I$, and the Lambek theorem gives flatness. No Noetherian or finite-generation hypothesis is needed.

If $F=P\oplus Q$ is a flat module, then $P$ and $Q$ are flat. For an injection $u:U\hookrightarrow V$, distributivity of the tensor product of modules identifies $u\otimes F$ with $(u\otimes P)\oplus (u\otimes Q)$. This map is injective by flatness of $F$. Each component is therefore injective, proving flatness of both summands.

For a free module $F={⨁}_{i\in J}A{e}_i$, the natural map $U{\otimes }_{A}F\to {⨁}_{i\in J}U$ sends $u\otimes {\sum }_i{a}_i{e}_i$ to $(a_{i}u{)}_i$ and is an isomorphism, with inverse sending the tuple $(u_i)$ to ${\sum }_i{u}_i\otimes e_i$. For any injection $U\hookrightarrow V$, its tensor product of modules with $F$ is therefore the coordinatewise injection ${⨁}_{J}U\hookrightarrow {⨁}_{J}V$. Thus every free module is flat, including those of infinite rank.

For a flat module $M$ over a commutative ring, every finite relation among elements of $M$ is a finite combination of relations among elements of the ring itself, in the displayed sense. To obtain this, tensor the kernel of the linear map $A^r\to A$, $(u_i)\mapsto {\sum }_i{a}_i{u}_i$, with $M$. Flatness says that this tensor maps onto the kernel in $M^r$. Writing an element of that tensor as a finite sum gives the columns $b_{ij}$ and the elements $n_j$. The converse follows by applying this condition to relations in a tensor product to prove that tensoring preserves injections.

For $D=k[\epsilon ]/({\epsilon }^2)$, a module $M$ is a flat module exactly when $\ker (\epsilon :M\to M)=\epsilon M$. Flatness implies this by tensoring the exact sequence $0\to (\epsilon )\to D\to k\to 0$. Conversely lift a $k$-basis of $M/\epsilon M$ to $M$. Every element is a finite combination of the lifts plus $\epsilon$ times another such combination, proving generation over $D$. A relation first reduces to zero coefficients modulo $\epsilon$; the kernel condition then reduces its remaining coefficients to zero as well. The lifts form a free $D$-basis, proving flatness.

In a spherically dominated Newtonian gravitational field, the circular orbit speed satisfies $v_c^2=GM(<r)/r$. A flat galaxy rotation curve therefore implies $M(<r)=v_0^{2}r/G$. Differentiating the enclosed mass gives $\rho =(4\pi r^2{)}^{-1}dM/dr$, yielding the inverse-square density profile. This is a halo-region approximation, not a mass inversion for a flattened stellar disk. A flat curve extending to infinite radius would imply infinite mass, so a real halo profile must eventually change or truncate.

For a connected base and a chosen frame at one point, a flat connection gives a representation of the fundamental group by parallel transport. Conversely, $(\tilde{X}\times G)/{\pi }_1(X)$ with monodromy representation has a flat product connection. Based gauge classes correspond to representations, and unbased classes to conjugacy classes. On a fixed underlying bundle only representations whose associated flat bundle has that bundle type are allowed.

// Destination: geometry-and-topology.bigb

The holonomy of a flat connection is invariant under based homotopy of loops, giving a group homomorphism from the fundamental group into the structure group after choosing a frame at the base point. Changing that frame conjugates the representation. Its associated bundle is constructed from the universal cover and this monodromy. It differs from the natural representation of a Riemannian holonomy group on a tangent fiber.

For $D_i={\partial }_i+A_i$, compatibility for arbitrary initial vectors is equivalent to zero gauge curvature. An invertible parallel bundle frame $S$ obeys $D_{i}S=0$, so $[D_i,D_j]S=F_{ij}S=0$ forces $F_{ij}=0$. Conversely, solve the two ordinary transport equations along one coordinate axis and then along parallel lines. The unused transport residual solves a homogeneous transport equation with zero initial value when the gauge curvature is zero; it therefore vanishes. Thus $A_i=-({\partial }_{i}S)S^{-1}=g^{-1}{\partial }_{i}g$ with $g=S^{-1}$. One particular parallel vector requires only $F_{ij}v=0$, a weaker condition.

If a thermal front has speed $Y^{\prime }=\Gamma U$ with $U$ the Darcy velocity, while a material front has speed $X^{\prime }=U/\varphi$, their position ratio from coincident initial locations is $Y/X=\Gamma \varphi$. Calling a retardation factor the ratio to pore velocity instead would give $Y/X=\Gamma$; these conventions must not be mixed. With cold viscosity $\mu$ and heated viscosity $b\mu$, the displacement's effective viscosity is ${\mu }_{\mathrm{eff}}=\mu [b-(b-1)\Gamma \varphi ]$. In the pressure resistance the cold layer has length $Y$, the hot injected layer has length $X-Y$, and the displaced layer has length $L-X$.

The Complex projective plane is simply connected. If it were the total space of a covering of degree at least two, it would be the universal cover and have a nonidentity deck transformation. Such a transformation acts by $u\mapsto \pm u$ on $H^{\ast }({\mathbb{CP}}^2)=\mathbb{Z}[u]/(u^3)$, giving Lefschetz number 1 or 3. It therefore has a fixed point, impossible for a nonidentity deck transformation.

Every Fatou component of a rational map of degree at least two is eventually periodic. Thus the periodic-component classification accounts for the entire Fatou set.

The largest normal p-subgroup of a finite group. Products of normal p-subgroups are again normal p-subgroups, since $|AB|=|A||B|/|A\cap B|$. Thus the subgroup generated by all such subgroups is itself a p-subgroup. It is a characteristic subgroup.

A normal subgroup which is a nilpotent group. In a finite group its unique Sylow subgroups are characteristic subgroups and hence normal in the ambient group. Products of nilpotent normal subgroups are nilpotent groups: their normal Sylow subgroups of different prime characteristics commute.

For a finite group, $F(G)$ centralizes its group socle. A nonabelian minimal normal subgroup meets $F(G)$ trivially: otherwise it is nilpotent, and its nontrivial characteristic center makes it abelian. Their group commutators therefore vanish. An abelian minimal normal subgroup $M$ is an elementary abelian p-group; $M\le O_p(G)$ and $M\cap Z(O_p(G))\ne 1$ by the fixed-point counting of a finite p-group on $M$. Minimal normality makes $M\le Z(O_p(G))$, and other p-cores commute with $M$.

In a finite soluble group, $C_G(F(G))\le F(G)$. Put $C=C_G(F)$ and $D=C\cap F=Z(F)$. If $C>D$, choose an elementary abelian minimal normal subgroup of $G/D$ inside $C/D$, and let $K$ be its preimage. Then $K^{\prime }\le D\le Z(K)$, so $K$ is a nilpotent normal subgroup, contradicting $K\nleqq F$.

A connected correlation function with critical anomalous dimension $\eta$ scales as $G(r)={\xi }^{-(D-2+\eta )}\mathcal{G}(r/\xi )$. If its large-distance shape is $s^{-(D-1)/2}{e}^{-s}$, substitution of $s=r/\xi$ produces the displayed amplitude. The extra ${\xi }^{-\eta }$ relative to the massive Gaussian field correlation tail is necessary for matching at $r$ of order $\xi$. Integrating over space gives magnetic susceptibility proportional to ${\xi }^{2-\eta }$; omitting that factor would instead make the long-distance contribution scale as ${\xi }^2$.

Congelation ice grows by freezing onto the underside of an existing sheet. Competitive growth selects grains whose fast basal-plane growth can advance downward, yielding columnar sea-ice crystal fabric with brine-bearing intergranular regions.

// Target: geophysics.bigb

Tennenbaum's theorem says that no countable nonstandard model of Peano arithmetic can have both addition and multiplication computable under a presentation of its domain by the natural numbers.