Let $K$ be an algebraically closed field, $X\subset K^n$ be an affine algebraic set and $I$ be the ideal generated by all polynomials in $K[x_1,\cdots,x_n]$ that vanish on $X$.

Prove that there exists a linear subspace $L$ of $K^n$ and a linear map of $K^n$ onto $L$ that maps $X$ onto $L$.

I have already proved the hint, which states that if $B/A$ is integral and $K$ is an algebraically closed field then a ring homomorphism from $A$ to $K$ can be extended to $B$ to $K$. My thought of this problem is to use Noether's normalization theorem, so there exists $K[y_1,\cdots,y_m]\subset K[x_1,\cdots,x_n]/I$ where the extension is integral and $y_1,\cdots,y_m$ are linear combinations of $x_1,\cdots,x_n$. Then $L$ should be this $m$-dimensional subspace of $K^n$. But I am not sure how to construct the map. Thanks for any help!


You have almost done the problem. So you have the commutative diagram enter image description here

The bottom arrow is integral and hence induces a surjection from $mSpec(\Gamma V)\rightarrow mSpec(k[Y_1,..Y_m])$ Going to the antiequivalent category of varieties we have the commutative diagram enter image description here

The map $k^n \rightarrow k^m$ is a linear map since each co-ordinate in $k^m$ namely $Y_i$ is a linear combination of the co-ordinates in $k^n$ namely $X_1,...,X_n$ and $V \rightarrow k^m$ is surjective by the discussion above. The argument is now complete.


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