Cup and cap products with local coefficients

Let $R$ be a commutative unital ring and let $\mathcal{L},\mathcal{K}$ be left $R$-module local systems on $X$. Their objectwise tensor product is $(\mathcal{L}{\otimes }_R\mathcal{K}{)}_x={\mathcal{L}}_x{\otimes }_R{\mathcal{K}}_x,T_{\gamma }(\mathrm{\ell }\otimes k)=T_{\gamma }\mathrm{\ell }\otimes T_{\gamma }k\mathrm{.}$ The tensor relations make the displayed transport well defined; identity and composition hold on elementary tensors, and $T_{\bar{\gamma }}\otimes T_{\bar{\gamma }}$ is its inverse. Thus this is again a local system.

A local-coefficient pairing is a natural transformation $b:\mathcal{L}{\otimes }_R\mathcal{K}\to \mathcal{N}$, equivalently bilinear maps $b_x:{\mathcal{L}}_x\times {\mathcal{K}}_x\to {\mathcal{N}}_x$ satisfying $T_{\gamma }{b}_x(\mathrm{\ell },k)=b_y(T_{\gamma }\mathrm{\ell },T_{\gamma }k)$ for every path class $\gamma :x\to y$.

For a simplex $\sigma :[v_0,\dots ,v_{p+q}]\to X$, let ${\lambda }_{0p}^{\sigma }$ be its affine edge path from $v_0$ to $v_p$. For $\phi \in C^p(X;\mathcal{L})$ and $\psi \in C^q(X;\mathcal{K})$ define $(\phi {⌣}_b\psi )(\sigma )=b_{v_0}(\phi (\sigma [0,\dots ,p]),T_{\overline{{\lambda }_{0p}^{\sigma }}}^{\mathcal{K}}\psi (\sigma [p,\dots ,p+q]))\mathrm{.}$ The reverse transport is necessary because the back-face value lies over $v_p$ while the output cochain value must lie over $v_0$. The Alexander--Whitney face calculation, with naturality of $b$ on each triangular transport comparison, gives $\delta (\phi {⌣}_b\psi )=\delta \phi {⌣}_b\psi +(-1{)}^p\phi {⌣}_b\delta \psi \mathrm{.}$ Hence cocycles give cup products in cohomology. The same vanishing-on-front-face argument as for ordinary relative cups gives $H^p(X,A;\mathcal{L}){\otimes }_R{H}^q(X,B;\mathcal{K})\to H^{p+q}(X,A\cup B;\mathcal{N})$ under the usual excisive-triad comparison.

For $\phi \in C^p(X;\mathcal{L})$ and a local chain generator $m\sigma \in C_n(X;\mathcal{K})$, define the cohomology-first cap product by zero when $p>n$ and otherwise by $\phi {\cap }_b(m\sigma )=T_{{\lambda }_{0p}^{\sigma }}^{\mathcal{N}}(b_{v_0}(\phi (\sigma [0,\dots ,p]),m))\sigma [p,\dots ,n]\mathrm{.}$ Here forward transport is necessary because the retained back face begins at $v_p$. The published front/back face cancellation, with the same naturality squares, yields $\mathrm{\partial }(\phi {\cap }_{b}c)=(-1{)}^p(\phi {\cap }_b\mathrm{\partial }c-\delta \phi {\cap }_{b}c)\mathrm{.}$ Consequently cap descends to the same relative quotient patterns as Relative cap products with quotient domains displayed, with the chain coefficient system changed from $\mathcal{K}$ to $\mathcal{N}$ in the target. In particular, $H^p(X,A;\mathcal{L}){\otimes }_R{H}_n(X,A;\mathcal{K})⟶H_{n-p}(X;\mathcal{N}),$ and $H^p(X;\mathcal{L}){\otimes }_R{H}_n(X,A;\mathcal{K})⟶H_{n-p}(X,A;\mathcal{N})\mathrm{.}$

If a cohomology class is represented with compact support $K$, cup with any ordinary class remains supported in $K$, while the cap of a class in $H^p(X,X\setminus K;\mathcal{L})$ with a class in $H_n(X,X\setminus K;\mathcal{K})$ is an absolute class. Enlargement of $K$ commutes with the formulas, giving the compact-support cup and supportwise cap operations used in duality. Constant systems and multiplication recover the published operations. Empty spaces, zero systems or rings, $p=0$, $p=n$, and $p>n$ follow from the displayed formulas. Only finite faces of a supplied simplex occur, so no AC is used.