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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?

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You really will have to use the definitions of the various maps, as stated in the beginning of Chapter 3 in Hartshorne's book.

For $T^1(B/k,B)$, note that it is the cokernel of the map $\text{Der}_k(A,B) \to \text{Hom}_B(I/I^2,B)$. The last module is generated by $dx$ and $dy$ over $B$. The ring $B$ has $k$-dimension $3$, so that $\text{Hom}_B(I/I^2,B)$ has dimension 6.

The image of the map is generated by $2x \ dx, x\ dy+y\ dx$ and $2y\ dy$, so if $char k \neq 2$, the quotient, $T^1$, has $k$-dimension $4$.

For example computations, I recommend the book "Deformations of Algebraic Schemes" by Edoardo Sernesi. There is also a Macaulay2 package for computing $T^i$ for $i=1,2$, written by Nathan Ilten.

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