# Tensoring over dual numbers- Deformation theory

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.

As a module over $$D$$, $$k = D/(t)$$ so tensoring with it means killing $$t$$. This is exactly the assumption the previous sentence.
In more detail, you can apply the tensor with $$k$$ to the sequence $$I’ \to B’ \to B’/I’$$ and use right exactness and $$B’\otimes k = B$$ and $$I’\otimes k = I$$.