“Algebra is but written geometry and geometry is but figured algebra.”
Sophie Germain
Recall that for a topological space \(X\), a fibre bundle of fibres \(F\), where \(F\) is a topological space, is a topological space \(E\), with a surjective map \(\pi: E\rightarrow X\) such that for all \(x\in X\), exists an open nbhd \(x\in U\subseteq X\) such that there is a homeomorphism \(\varphi: \pi^{-1}(U)\rightarrow U\times F\) such that \(\mathrm{pr}_U\circ \varphi=\pi|_{\pi^{-1}(U)}\) where \(\mathrm{pr}_U:U\times F\rightarrow U\) is the projection onto \(U\). Fibre bundles generalize vector bundles and covering spaces. In this post, we interpret this algebraically and generalize this notion to schemes.