I just start to study Hartshorne's 'Deformation theory'.
I have a stupid question about $T^i$-functors. It may be an easy algebra.
Exercise 1.3.3. Let $B=k[x,y]/(x^2,xy,y^2)$. Show that $T^0(B/k,B)=k^4, T^1(B/k,B)=k^4$, and $T^2(B/k,B)=k$.
I tried to calculate this. Let $A=k[x,y], I=(x,y)$. Then by Proposition 3.10. there is an exact sequence $0\to Der_k(B,B)=T^0(B/k,B)\to Der_k(A,B)\to Hom(I/I^2,B)\to T^1(B/k,B)\to 0$ and an isomorphism $T^2(B/A,B)\to T^2(B/k,B)$. But I can't calculate anything.
Can you help me? Any hint or partial solution?