Suppose $A$ is a $k$-algebra and $D$ is the ring of dual numbers. Is there any relation between $\text{Der}_k(A,A)$ and automorphism group of $A\otimes_k D$?
This question comes from the proof of Theorem 5.3 in Hartshorne's deformation theory.
Thanks for the help.
For your specific question, you need to put together several different pieces. At the beginning of section 5, it shows that a deformation of $A$ corresponds to an extension of $k$-algebras as on page 36, say $A'$. As as $k$-vector space, $A'$ is just $A \otimes_k D$. Then use Lemma 4.5 with $R = A'$ to cook up a derivation using the identity and the automorphism as the two maps. Show that all you need is the derivation of $A$ into $A$.