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Theorem: For any compact abelian group $G$, the homogeneous component $% H^{2}\left( B_{G};% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion \right) $ of degree $2$ is naturally isomorphic to the character group $% \widehat{G}$. If $G$ is connected, then the entire cohomology algebra $% H^{\ast }\left( B_{G};% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion \right) $ is the symmetric $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $-algebra $P\left( \widehat{G}\right) $ generated by this component of homogeneous degree $2$.

Observation: If $G$ is a compact connected abelian group and $N$ any closed normal subgroup, then the quotient morphism $q:G\longrightarrow G/N$ induces an injection $\widehat{q}:\widehat{G/N}\cong N^{\bot }\longrightarrow \widehat{G}$ by Pontryagin Duality with annihilator $N^{\bot }$ of $N$ in $% \widehat{G}$.

Someone says that the symmetric algebra functor $P$ preserves inclusions, we have $P\left( N^{\bot }\right) \subset P\left( \widehat{G}\right) $, and so $% H^{\ast }\left( q;% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion \right) :H^{\ast }\left( B_{G/N};% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion \right) \longrightarrow H^{\ast }\left( B_{G};% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion \right) $ is injective.

Another person says that is not true that the symmetric algebra functor $P$ preserves inclusions.

Which one of these is correct? That is does the symmetric algebra functor $P $ preserve inclusions?

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    $\begingroup$ It is classical that the symmetric algebra functor, as a functor from $R$-modules to commutative $R$-algebras, does not preserve inclusions. But it seems that this is not the symmetric algebra functor that you have in mind. Instead, I think you intend $P$ to be a functor from compact topological abelian groups to commutative rings, sending a group $G$ to the symmetric algebra on the character group of $G$. Is that right? More generally, can you clarify what you intend the domain of your functor $P$ to be? $\endgroup$
    – user509184
    Commented Jan 6 at 20:44
  • $\begingroup$ @user509184 I saw this symmetric algebra in K. H. Hofmann, P. S. Moster's ''Cohomology Theories for Compact Abelian Groups''. I didn't see any information about this in this book. There is a statement somewhere like this: $P \widehat{G}$ is the symmetric algebra generated by $\widehat{G}$ in degree 2. $\endgroup$
    – Mehmet Onat
    Commented Jan 6 at 21:05
  • $\begingroup$ What do you intend the domain of the functor $P$ to be? $\endgroup$
    – user509184
    Commented Jan 6 at 21:08
  • $\begingroup$ @user509184 books.google.com.tr/… $\endgroup$
    – Mehmet Onat
    Commented Jan 6 at 21:08
  • $\begingroup$ @user509184 Is the character groups of compact abelian groups? $\endgroup$
    – Mehmet Onat
    Commented Jan 6 at 21:10

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