I'm studying the theory of abelian coverings of varieties.
Let $\mathcal{O}_X$ be the sheaf of regular functions of a varieties $X$ and let $T$ be a closed subset of $X$. What means the symbol $\mathcal{O}_{X,T}$?
At the beginning I thunk that could be the localization of the ring $\mathcal{O}_X$ with respect the multiplicative subset $X\setminus\mathfrak{M}$ where $\mathfrak{M}$ is the prime ideal of $\mathcal{O}_X$ of the regular functions that vanishing on $\mathfrak{M}$, but the problem is that $\mathfrak{M}$ is not always a prime ideal, right?
Moreover, if $T$ is an hypersuface of $X$, then $M$ is a principal ideal generated by an entire regular function $f$, right? So we have that $\mathfrak{M}/\mathfrak{M}^2$ is a $\mathcal{O}_{X,T}/\mathfrak{M}$-vector space?