I am interested in constructing the tangent complex of a commutative ring $A$ as a vector bundle over $\text{Spec}(A)$. Can someone help me with this construction? I am looking for a reference which constructs it explicitly.
I realize there are probably a lot of sources for this, but I can't seem to find one that constructs the entire vector bundle and not just its value at each prime.
The tangent space of an $R$-algebra $A$ at an $R$-point is $\text{Der}_{R}(A, R)$. As a set, the tangent bundle should be $\amalg_{\mathfrak{p}} \ \text{Der}_R (A, A/\mathfrak{p})$. I just need to understand how this assembles into a sheaf of $R$-modules $F$ on $\text{Spec}(A)$ such that $F(U)$ has a $A_U$-module structure.
Maybe we can define $F(U)$ as certain retracts $\text{Spec}(A) \rightarrow \amalg_{\mathfrak{p}} \text{Der}_R(A,A/\mathfrak{p})$ of the projection $\amalg_{\mathfrak{p}} \text{Der}_R(A,A/\mathfrak{p}) \rightarrow \text{Spec}(A)$. But I don't see what condition to put here.