# dimension of a vector space seen as a variety

The dimension of an affine algebraic variety $$V\subset k^n$$ is defined as the Krull dimension of its coordinate ring $$k[V]=k[X_1,\cdots,X_n]/I(V)$$ where $$I(V)$$ is the set of polynomials in $$k[X_1,\cdots,X_n]$$ vanishing on $$V$$.

In the case where $$V$$ is moreover a vector space over the field $$k$$, I don't succeed in showing that the dimension of $$V$$ as vector space is the same as the dimension of $$V$$ as affine algebraic variety. In particular, what can be said about the ideal $$I(V)$$ in this case ?

• What example do you have in mind? Commented Jan 24, 2020 at 18:21
• I have no example in mind. It is just to understand the definition of the dimension of the variety $V$ by the Krull dimension of $k[V]$. Commented Jan 27, 2020 at 13:39

It is true that the dimension of $$V$$ as a vector space is equal to it's dimension as a variety. This is because $$I(V)$$ can be generated by $$n-\dim V$$ linear forms, where we mean dimension as a linear space.

To prove this, select a basis $$v_1,\cdots,v_m$$ for $$V$$ and complete it to a basis of $$k^n$$ with the vectors $$w_{m+1},\cdots,w_n$$. Writing these vectors as the columns of a matrix $$M$$, we see that this matrix is invertible and takes the standard basis vectors $$e_i\mapsto v_i$$ if $$i\leq m$$ and $$e_i\mapsto w_i$$ if $$i>m$$. So $$M$$ and $$M^{-1}$$ define an isomorphism between $$V$$ and the subspace $$V'$$ consisting of vectors with final $$n-m$$ coordinates zero (this is an isomorphism as vector spaces and also as varieties). This means that as the coordinate functions $$x_{m+1},\cdots,x_n$$ generate the ideal of $$V'$$, their pullbacks under $$M^{-1}$$ generate the ideal of $$V$$, and we see the claim.

• Thank you very much for your very clear answer. If I understand well, your change of variables induces a ring isomorphism between $k[V]$ and $k[V']$. Thus this two rings get the same Krull dimension and the Krull dimension of $k[V']$ is cearly dim V. Is it right ? Commented Jan 27, 2020 at 13:37
• Yes, this is correct. Commented Jan 27, 2020 at 19:48