An odd Fredholm module $(\mathord{\mathcal{H}},F)$ over a $C^{*}$-algebra $A$ is given by an involutive representation $\pi$ of $A$ on a Hilbert space $\mathord{\mathcal{H}}$, together with an operator $F$ on $\mathord{\mathcal{H}}$ such that $F=F^{*}$, $F^{2}=\mathord{\mathrm{1\!\!\!\>I}}$ and $[F,\pi(a)]\in\mathbb{K}(\mathord{\mathcal{H}})$ for all $a\in A$.

An even Fredholm module $(\mathord{\mathcal{H}},F,\Gamma)$ is given by an odd Fredholm module $(\mathord{\mathcal{H}},F)$ together with a $\mathbb{Z}_{2}$-grading $\Gamma$ on $\mathord{\mathcal{H}}$, $\Gamma=\Gamma^{*}$, $\Gamma^{2}=\mathord{\mathrm{1\!\!\!\>I}}$, such that $\Gamma\pi(a)=\pi(a)\Gamma$ and $\Gamma F=-F\Gamma$.

A Fredholm module is called degenerate if $[F,\pi(a)]=0$ for all $a\in A$. Degenerate Fredholm modules are homotopic to the 0-module.