Let $(X, \mathcal{O}_X)$ be a ringed space. Is there such a thing as the sheaf of invertible linear functions $GL_n(\mathcal{O}_X)$? The point is that I cannot see how to define the restriction functions: if I have an $\mathcal{O}_X(U)-$linear function $\mathcal{O}_X(U)^n \to \mathcal{O}_X(U)^n$, how does this restrict to an $\mathcal{O}_X(V)-$linear function $\mathcal{O}_X(V)^n \to \mathcal{O}_X(V)^n$ for $V \subseteq U$ open?
This question came to me as I was considering equivalent definitions of a locally free sheaf $\mathcal{F}$ on $X$. I think that the transition functions between the trivializations $\mathcal{F}(U_i) \cong \mathcal{O}_{U_i}^n$ on open sets $U_i$ and $U_j$ should live in such a $GL_n(\mathcal{O}_{U_i \cap U_j})$, and not just in $GL_n(\mathcal{O}_X(U_i \cap U_j))$, otherwise I cannot see how the data of the transition functions would be enough to determine the sheaf $\mathcal{F}$ up to isomorphism. I wouldn't even know what it means for the transition functions to satisfy the cocycle condition, if I cannot restrict them to a common domain.
Best Answer
Well, in a naive way, if we have a map of rings $A \to B$, then there is an induced map $GL_n(A) \to GL_n(B)$ defined coordinate wise. In this case $A = O_X(U), B = O_X(V).$
Less naively, given a $A$ linear map $f: A^n \to A^n$, we get an induced map by taking tensor products $f: A^n \otimes_A B \to A^n \otimes_A B = B^n \to B^n$.