I am actually reading the course from Hartshorne about deformation theory. (https://math.berkeley.edu/~robin/math274root.pdf ) After having defined the notion of flatness about a module, the author then defines what the deformation of a closed subscheme of a given one is, and begins with the affine case.
In this case, let $B$ be a $k$-algebra. And let's define $B' := B[t]/t^2$ and $D:=k[t]/t^2$
Let $I$ be an ideal of $B$, and $I' $ be one of $B'$ such that the image of $I' \subset B'$ onto $B'/tB'=B$ is exactly $I$.
My question is the following : why do we have in this case : $$B'/I' \otimes_D k = B/I$$ I tried to figure out on simple exemples but this fact seems weird to me. Thank your for your help.