= Infinite cotangent space obstructs finite algebra generation
{title2=$\dim_k(I/I^2)=\infty\Longrightarrow A\text{ is not finitely generated}$}
Let $A$ be a commutative algebra over a <field> $k$, with a homomorphism $A\to k$ restricting to the identity on $k$, and let $I$ be its kernel. If $A$ is generated by finitely many elements, subtracting their images in $k$ gives finitely many generators in $I$. Every polynomial in those generators, modulo $I^2$, is a linear combination of their classes. Thus $I/I^2$, the <algebraic cotangent space> at this point, is finite-dimensional. For $A=k+xk[x,y]$ and $I=xk[x,y]$, the classes $xy^j$, $j\ge0$, form an infinite basis modulo $I^2=x^2k[x,y]$. This is a subalgebra of a <polynomial ring> which is not a <finitely generated algebra>.
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