---- > [!theorem] Theorem. ([[universal coefficients theorem for cohomology]]) > Let $X$ be a [[topological space]]. Let $R$ be a [[PID]] (e.g., $\mathbb{Z}$ or a [[field]]). And $M$ an $R$-[[module]]. There is a natural map on [[singular cohomology]] $\begin{align} H^{*}(X; M) &\to \text{Hom}_{R\text{-}\mathsf{Mod}}\big( H_{*}(X ; R), M \big) \\ [\varphi: C_{n}(X;R) \to M] & \mapsto \big([c] \mapsto \varphi(c) \big) \end{align}$ If $H_{n}(X; R)$ is a [[free module|free]] $R$-[[module]] for all $n$, then this map is an [[isomorphism]]. > [!proof]- Proof. ([[universal coefficients theorem for cohomology]]) > (bring over) ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```