Let $F$ be a field and $C$ be a chain complex (of free abelian groups). I am confused about what the isomorphism $\phi : H^p(C; F) \to \hom_F(H_p(C;F),F)$ is exactly.

I know that there is a natural map $H^p(C; F) \to \hom_\mathbb{Z}(H_p(C;\mathbb{Z}),F)$ (the "Kronecker map") which is the one appearing in the universal coefficients theorem. I have also seen an isomorphism of chain complexes of vector spaces \begin{align*} \tau: \hom_\mathbb{Z}(C_p, F) &\to \hom_F(C_p \otimes F, F) \\ f &\mapsto c \otimes \alpha \mapsto \alpha f(c) \end{align*} where this first chain complex is the chain complex for $H^p(C; F)$.

Is the universal coefficients isomorphism $\phi$ given by $\phi([f])([c]) = \tau(f)(c)$?


1 Answer 1


Here are the details. For all $p$, you have a natural map: $$\tau^p:\operatorname{Hom}_{\mathbb{Z}}(C_p,F)\to\operatorname{Hom}_{F}(C_p\otimes F,F),\quad (\tau^p(f))(x\otimes a) = af(x)$$ (the tensor-hom adjunction in disguise) which, when applied to the differential $d_p:C_p\to C_{p-1}$ gives rise to a commutative square: $$ \require{AMScd} \begin{CD} \operatorname{Hom}_{\mathbb{Z}}(C_{p-1},F) @>{d^p}>> \operatorname{Hom}_{\mathbb{Z}}(C_{p-1},F)\\ @V{\tau^{p-1}}VV @VV{\tau^p}V\\ (C_{p-1}\otimes F)^* @>>{(d_p\otimes 1)^*}> (C_{p-1}\otimes F)^* \end{CD} $$ where $(C_p\otimes F)^*=\operatorname{Hom}_{F}(C_p\otimes F,F)$ is the usual dual space notation. That is, the $\tau^p$ constitute a cochain map between the complexes $\operatorname{Hom}_{\mathbb{Z}}(C,F)$ and $(C\otimes F)^*$. Since the $\tau^p$ are isomorphisms, the induced map on cohomology: $$\tau^*:H^p(C,F)\to H^p((C\otimes F)^*),\quad \tau^*[f] = [\tau^p(f)]$$ is an isomorphism. However, since $F$ is an injective $F$-module, the (contravariant) functor $\operatorname{Hom}_F(-,F)$ is exact, and therefore we have an isomorphism $$H^p(\operatorname{Hom}_F(C\otimes F,F)) \cong \operatorname{Hom}_F(H_p(C;F),F)$$ which sends a an element $[g:C_p\otimes F\to F]$ to $\bar{g}:H_n(C;F)\to F$, where $\bar{g}[x\otimes a] = g(x\otimes a)$. Composing these two isomorphisms gives: $$[f]\mapsto [\tau^p(f)]\mapsto \overline{\tau^p(f)},$$ where $\overline{\tau^p(f)}[x\otimes a]=\tau^p(f)(x\otimes a)$. So the map does act as you expected.

The isomorphism between $H^n(C;F)$ and $H_n(C;F)^*$ follows from similar reasoning found here.

  • $\begingroup$ What you have just described is the Kronecker map which is not what I am asking about. But thank you for the response! $\endgroup$
    – user101010
    Feb 28, 2017 at 23:38
  • $\begingroup$ @user101010 Not sure how I missed that! I've updated the answer. $\endgroup$
    – user171308
    Mar 1, 2017 at 3:53

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