Let $f: X\to Y$ be a morphism of schemes and $y\in Y$ be a point. Then we can define the fibre of $f$ at $y$ by $X_y:=X\times_{Y}k(y)$. My question is the following: let $x\in X_y$ be a point. Then what can we say about $\mathcal{O}_{X_y,x}$, the stalk of $X_y$ at $x$?
Thank you!
Let $\mathfrak m_y\subset \mathcal O_{Y,y}$ be the maximal ideal. Then the local ring you are looking for is $$\mathcal O_{X_y,x}=\mathcal O_{X,x}\otimes_{\mathcal O_{Y,y}}k(y)=\frac {\mathcal O_{X,x}}{\mathfrak m_y \mathcal O_{X,x}} $$ The proof consists in reducing to the affine case.
The affine case is then handled in Matsumura, Commutative Ring Theory, pages 47-48.