A question about Kronecker pairing

Asked Sep 30 '16 at 19:52

Active Sep 30 '16 at 21:33

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Corollary 2.31. If $R$ is a field, $M$ is a vector space over $R$, and $C_*$ is a chain complex over $R$, then $$H^q\left({C_*;M}\right) \cong \operatorname{Hom}\left({H_q\left({C_*}\right),M}\right)$$

Moreover the Kronecker pairing is nondegenerate. $\square$

I am stuck on why the Kronecker pairing here is non-degenerate?

edited Jun 12 '20 at 10:38

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asked Sep 30 '16 at 19:52

Danny

1,897

What is this a corollary of? – Noah Schweber Sep 30 '16 at 21:51
@NoahSchweber universal coefficient theorem for cohomology – Danny Oct 01 '16 at 00:05
The map $h : H^q(C_; M) \to \text{Hom}(H_q(C_), M)$ in UCT takes a cocycle $\varphi$ to the Kronecker pairing, $\langle \varphi, - \rangle$. This is an isomorphism if and only if the pairing is non-degenerate. – feynhat May 22 '22 at 11:40

A question about Kronecker pairing

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