]> No Title

a a a a above, above. also and and and and and and and and and and are asbasic be be by

clear conclude Consequently, contrary cotorsion

defined direct divisible divisible, divisible.

ET

lact follows follows follows follows For from from

G. GRÄBE group group group, group,

bave have hence hence

if in is is is is is is is is It it it it it

latter Lemma Let Let let

Moreover,

$\mathit{H}\mathrm{=}\mathit{Z}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathrm{\oplus }\mathit{H}\mathrm{\prime }\mathrm{,}$ $\mathit{X}\mathrm{\right)}\mathrm{\cong }\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{\left(}\mathit{t}\mathit{G}\mathrm{,}\mathrm{}\mathit{X}\mathrm{/}{⨁}_{\mathit{r}\mathrm{\in }\mathit{P}}\mathit{Z}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathit{X}\mathrm{=}{\prod }_{\mathit{p}\mathrm{\in }\mathit{P}}\mathit{Z}\mathrm{\left(}\mathit{p}\mathrm{\right)}$. $\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{\left(}\mathit{Z}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathrm{,}\mathrm{}\mathit{X}\mathrm{\right)}\mathrm{,}{\prod }_{\mathit{p}\mathrm{\in }\mathit{P}}\mathit{Z}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathrm{/}\mathit{X}\mathrm{\cong }{\mathit{Q}}^{\mathrm{\left(}\mathfrak{n}\mathrm{\right)}}$.$\mathrm{X}\mathrm{\otimes }\mathrm{\left(}\mathit{Q}\mathrm{/}\mathit{Z}\mathrm{\right)}\mathrm{\cong }\mathit{Q}\int \mathit{Z}\mathrm{\left(}\mathrm{\left[}\mathrm{2}\mathrm{\right]}$,

$\mathit{G}\mathit{H}\mathit{H}\mathit{X}\mathit{H}\mathrm{,}$ ${\mathfrak{m}}_{\mathit{p}}{\mathfrak{m}}_{\mathit{\rho }}{\mathfrak{m}}_{\mathit{P}\mathit{i}}\mathfrak{B}\mathrm{\subseteq }\mathfrak{U}$. $\mathit{G}\mathrm{\in }\mathfrak{B}\mathrm{,}$ $\mathit{G}\mathrm{\in }\mathfrak{B}$. $\mathrm{\left(}\mathit{t}\mathit{G}{\mathrm{\right)}}_{\mathit{p}}\mathrm{,}$ $\mathrm{\left(}\mathit{t}\mathit{G}{\mathrm{\right)}}_{\mathit{p}}$. $\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{\left(}\mathit{t}\mathit{G}$,

$\mathit{G}\mathrm{\cong }\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{\left(}\mathit{G}\mathrm{,}\mathrm{}\mathit{X}\mathrm{\right)}\mathrm{\cong }\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{\left(}\mathit{t}\mathit{G}\mathrm{,}\mathrm{}\mathit{X}\mathrm{\right)}\mathrm{,}$ $\mathit{t}\mathit{G}\mathrm{\cong }{⨁}_{\mathit{P}\mathrm{\in }\mathit{P}}\mathit{t}\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{\left(}\mathrm{\left(}\mathit{t}\mathit{G}{\mathrm{\right)}}_{\mathit{p}}\mathrm{,}\mathrm{}\mathit{Q}\mathrm{/}\mathit{Z}\mathrm{\right)}$ $\mathrm{\left(}\mathit{t}\mathit{G}{\mathrm{\right)}}_{\mathit{p}}\mathrm{\cong }\mathit{t}\prod _{\mathit{i}\mathit{r}}^{\mathrm{\infty }}\mathrm{\left(}\mathrm{\left(}\mathit{C}\mathrm{\left(}{\mathit{p}}^{\mathit{i}}\mathrm{\right)}{\mathrm{\right)}}^{{\mathfrak{m}}_{\mathit{P}\mathit{i}\mathrm{\right)}}}$.

${\mathit{B}}^{\mathrm{\left(}\mathit{\rho }\mathrm{\right)}}\mathrm{=}{\mathit{B}}_{\mathrm{1}}\mathrm{\oplus }\mathrm{\text{...}}\mathrm{\oplus }{\mathit{B}}_{\mathit{i}}\mathrm{\oplus }\mathrm{\text{...}}$ ; ${\mathit{B}}_{\mathit{i}}\mathrm{=}\mathrm{\left(}\mathit{C}\mathrm{\left(}{\mathit{p}}^{\mathit{i}}\mathrm{\right)}{\mathrm{\right)}}^{\mathrm{\left(}{\mathfrak{m}}_{\mathit{P}\mathit{i}}\mathrm{\right)}}\mathrm{,}$ $\mathit{G}\mathrm{\cong }\mathrm{I}{\mathrm{|}}_{\mathit{\rho }}\mathit{Z}\mathrm{\left(}\mathit{p}{\mathrm{\right)}}^{{\mathrm{m}}_{\mathit{\rho }}}\mathrm{\oplus }\prod _{\mathit{i}\mathit{p}\mathrm{\in }\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\mathrm{\left(}\mathrm{\left(}\mathit{C}\mathrm{\left(}{\mathit{p}}^{\mathit{i}}\mathrm{\right)}{\mathrm{\right)}}^{{\mathfrak{m}}_{\mathit{P}\mathit{i}\mathrm{\right)}}}$,

$\mathrm{X}\mathrm{\otimes }\mathrm{\left(}\mathit{Q}\mathrm{/}\mathit{Z}\mathrm{⟩}\mathrm{\right)}\mathrm{\cong }{\prod }_{\mathit{p}\mathrm{\in }\mathit{P}}\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{\left(}\mathrm{\left(}\mathit{t}\mathit{G}{\mathrm{\right)}}_{\mathit{\rho }}\mathrm{,}\mathrm{}\mathrm{x}\mathrm{\otimes }\mathrm{\left(}\mathit{Q}\mathrm{/}\mathit{Z}\mathrm{\right)}{\mathrm{\right)}}^{\mathrm{\infty }}\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{\left(}\mathrm{\left(}\mathit{t}\mathit{G}{\mathrm{\right)}}_{\mathit{p}}\mathrm{,}\mathrm{}\mathit{C}\mathrm{\left(}{\mathit{p}}^{\mathrm{\infty }}\mathrm{\right)}\mathrm{\right)}\mathrm{\cong }\mathit{Z}\mathrm{\left(}\mathit{p}{\mathrm{\right)}}^{{\mathfrak{m}}_{\mathit{p}}}\mathrm{\oplus }\mathrm{I}{\mathrm{I}}_{\mathrm{1}}^{\mathrm{\left(}\mathrm{\left(}\mathit{C}\mathrm{\left(}{\mathit{p}}^{\mathit{i}}{\mathrm{\right)}}^{{\mathfrak{m}}_{\mathit{P}\mathit{i}}}\mathrm{\right)}}\mathit{i}\mathrm{=}$

$\mathit{U}\mathrm{\in }\mathrm{\Gamma }{\mathrm{I}}_{\mathit{P}}^{\mathit{Z}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathrm{/}\mathit{X}\mathrm{\cong }}\mathrm{\left(}{\prod }_{\mathit{p}\mathrm{\in }\mathit{P}}\mathit{Z}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathrm{/}{⨁}_{\mathit{p}\mathrm{\in }\mathit{P}}\mathit{Z}\mathrm{\left(}\mathit{p}\mathrm{\right)}\mathrm{\right)}\mathrm{/}{\mathrm{\left(}}^{\mathrm{x}\mathrm{/}{⨁}_{\mathit{p}\mathrm{\in }\mathit{P}}\mathit{Z}\mathrm{\left(}\mathit{p}\mathrm{\right)}}\mathrm{\right)}$

$\mathrm{G}\mathrm{\cong }\mathrm{E}\mathrm{x}\mathrm{t}\mathrm{\left(}\mathit{t}\mathit{G}\mathrm{,}\mathrm{}\mathit{X}\mathrm{\right)}\mathrm{\cong }{\prod }_{\mathit{p}\mathrm{\in }\mathit{P}}\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{\left(}\mathrm{\left(}\mathit{t}\mathit{G}{\mathrm{\right)}}_{\mathit{p}}\mathrm{,}\mathrm{}\mathit{Q}\mathrm{/}\mathit{Z}\mathrm{\right)}\mathrm{\cong }\mathrm{\Gamma }\mathrm{\left[}\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{\left(}\mathrm{\left(}\mathit{t}\mathit{G}{\mathrm{\right)}}_{\mathit{\rho }}\mathrm{\in }\mathit{P}\mathit{C}\mathrm{\left(}\mathrm{\left(}{\mathit{p}}^{\mathrm{\infty }}\mathrm{\right)}\mathrm{\right)}$.ion-trivial now

$\mathrm{D}\mathrm{f}$ of of

133), 134), 178 2@2 255), 2.6 final $\mathrm{\left(}\mathrm{I}\mathrm{o}\mathrm{\right)}\mathrm{\left(}\mathrm{\left[}\mathrm{3}\mathrm{\right]}\mathrm{,}$ $\mathrm{\left(}\mathrm{\left[}\mathrm{5}\mathrm{\right]}\mathrm{,}$ $\mathrm{\left(}\mathrm{\left[}\mathrm{5}\mathrm{\right]}$,

$\mathrm{p}$. $\mathrm{p}$. $\mathrm{p}$. $\mathrm{p}$. $\mathrm{1}\mathrm{3}\mathrm{7}$) P. $\mathrm{J}$. prove

rank recapitulate, reduced. reduced.

${\mathrm{5}}^{\mathrm{⌝}}\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{1}$ shows Since since subgroup summand

that that that that that that that that that that the the the the the then then To totorsion-free torsion-free

$\mathit{V}\mathrm{I}\mathrm{L}\int \mathrm{O}\mathrm{E}\mathrm{N}$.

We We We we where which