# Classification of local Artin (commutative) rings which are finite over an algebraically closed field.

A result in deformation theory states that if every morphism $$Y=Spec(A)→XY=\operatorname{Spec}(A)\rightarrow X$$ where $$AA$$ is a local Artin ring finite over $$kk$$ can be extended to every $$Y′⊃YY'\supset Y$$ where $$Y′Y'$$ is an infinitesimal thickening of $$YY$$, then $$XX$$ is non-singular.

My question is:

if $$kk$$ is algebraically closed, can we say explicitly what every local Artin ring finite over $$kk$$ is?

$$AA$$ can be of the form $$k[t]/(tn)k[t]/(t^n)$$ or for instance if its maximal ideal isn’t principally generated $$k[t2,t3]/(t4)k[t^2,t^3]/(t^4)$$. Are there $$AA$$ which we cannot write in this “adjoin various powers of $$tt$$ and mod out by some power of $$tt$$” form? If there are more exotic $$AA$$ can we say anything non-tautological about the structure of such an $$AA$$?

Thanks.