---- > [!definition] Definition. ([[(co)homology with coefficients]]) > Assume $M$ is a $\mathbb{Z}$-[[module]] (an [[abelian group]]). Any (say, [[commutative ring|commutative]]) [[ring]] $R$ is naturally an $(R,\mathbb{Z})$-[[bimodule]]. [[Extension of scalars]] gives an *$R$*-[[module]] $M \otimes_{\mathbb{Z}} R$. > > Now let $X$ be a [[topological space]]. In the case where $M=C_{k}(X)$ is a [[singular homology|singular chain group]], basic [[tensor product of modules|tensor product]] properties give a [[chain complex of modules|chain complex of]] $R$-modules $\begin{align} > C_{k}(X; R)&:=\mathbb{Z}^{\oplus\{ \text{maps } \Delta^{k} \to X \}} \otimes R \cong R^{\oplus \{ \text{maps } \Delta^{k} \to X \}} > \end{align}$ > with differentials $d_{k} \otimes \id_{R}$.[^1] Elements of $C_{k}(X; R)$ look like $R$-linear (rather than $\mathbb{Z}$-linear) [[linear combination|combinations]] of $k$-[[singular simplex|singular simplicies]]. > > The [[(co)homology of a complex|homology]] $H_k(X; R):= H_{k}\big( C_{\bullet}(X; R), d \otimes \id_{R} \big)$ of this complex is called the **singular homology of $X$ with coefficients in $R$**. > > To get our cohomology to 'take coefficients in $R, we let $C^{k}(X; R):=\text{Hom}\big(C_{k}(X), R\big)$, leaving the codifferential untouched. The cohomology $H^{k}(X; R)=H^{k}\big( C_{\bullet}(X; R) \big)$ > of this complex is called the **singular cohomology of $X$ with coefficients in $R$**. > > > Can do similar definitions for [[cellular homology]] and [[cellular cohomology]]. In fact, this is just an [[extension of scalars]], so can really do it for any kind of [[(co)homology of a complex|(co)homology]] theory. > [!generalization] > In [[extension of scalars]] we allow for tensoring with any $R$-module $N$, not just with $R$. Here we can do the same thing. This is relevant e.g. in the [[universal coefficients theorem for homology]]. ^generalization > [!basicexample] > > In the case of $C_{\bullet}(\mathbb{R}P^{n})$, [[cellular homology|recall]] that the cochain groups are all $\mathbb{Z}$ and the differentials are all $0$ or $2$. So we have $C_{i}(\mathbb{R}P^{n}; \mathbb{Z}/2)=\mathbb{Z} / 2$ for all $i \leq n$ ($0$ above $n$), and the differentials are all zero. Hence $H_{i}(\mathbb{R}P^{n}; \mathbb{Z} / 2)=\mathbb{Z} / 2$ > for all $i \leq n$ ($0$ above). On the other hand, if we take coefficients to be $\mathbb{Q}$, then $C_\bullet(\mathbb{R}P^{n}; \mathbb{Q}) \equiv \mathbb{Q}$ > for all $i \leq n$ ($0$ above) and multiplication by $2$ is now an [[isomorphism]]. So $H_{i}(\mathbb{R}P^{n}; \mathbb{Q})=\begin{cases} > \mathbb{Q} & i=0 \\ > 0 & 1 \leq i \leq n-1 \\ > \begin{cases} > \mathbb{Q} & n \text{ odd} \\ > 0 & n\text{ even} > \end{cases}& i=n. > \end{cases}$ > > > ---- #### [^1]: This also coming from the [[extension of scalars]] [[covariant functor|functor]] definition. ---- #### 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 > ```