Basic question on the dimension of affine varieties

Question

Suppose I have an affine variety defined by some polynomials $f_1(x_1, ..., x_n), ..., f_r(x_1, ..., x_n)$ in $\mathbb{C}^n$, and suppose this has dimension $L$. We can also view the polynomials as sitting inside $\mathbb{C}[x_1, ..., x_n, y_1, ..., y_r]$ so it produces an affine variety inside $\mathbb{C}^{n + r}$. I mean they are really the ``same", so does the two (the one sitting inside $\mathbb{C}^n$ and $\mathbb{C}^{n + r}$) have the same dimension?

Answer 1

No. The variety corresponding to the ideal $I = (f_1, \dots, f_r) $ in $\mathbb C^{n+r}$ will be $X \times \mathbb C^r$, where $X$ is the zero set of $I$ in $\mathbb C^n$.