Let $V$ be an affine real algebraic set. That is, $V$ is the zero set of some polynomials in $\mathbb{R}^n$. I would like to show that there is not a proper algebraic subset $W\subset V$ which admits a surjective polynomial map $W\twoheadrightarrow V$. The plan to do this is to take the ordered list of dimensions of irreducible components of $V = V\_1\cup \cdots \cup V\_p$. For each irreducible component $V\_i$, $W\cap V\_i$ will equal $V\_i$ or will have smaller dimension. Thus, the ordered list of dimensions of irreducible components of $W$ will be less than or equal to that of $V$ with respect to lexicographic order, with equality only if $W=V$. The same holds for images of polynomial maps which are algebraic sets, and hence the statement.


Thus, I am wondering if there is a name for the ordered list of dimensions of the irreducible components of an algebraic set? I would also be happy to know if this terminology exists for algebraic sets over algebraically closed fields (which is the usual setting for algebraic geometry). Maybe there is appropriate terminology in commutative algebra?


Also, if the above result is written down somewhere, that would be a helpful reference to have.



Reference request: ordered list of dimensions of components of a variety?
You can prove this without needing to keep track of dimensions.


Suppose that $W$ and $V$ are real algebraic sets with $W \subsetneq V$, and that there exists a surjective polynomial map
$\phi: W \twoheadrightarrow V$. Then define a sequence of algebraic sets $W\_i$
with $W\_0 = W$ and $W\_{i+1} = \phi^{-1}(W\_i)$. Since $W \subsetneq V$ and $\phi$ is surjective, $W\_1 = \phi^{-1}(W)$ must be a proper subset of $W\_0$. Then $\phi|\_{W\_1}$ gives a surjection $W\_1 \twoheadrightarrow W\_0$, so by the same logic $W\_2 \subsetneq W\_1$.


By induction, the $W\_i$ form a sequence of nested of algebraic sets $W\_0\supsetneq W\_1 \supsetneq \ldots$, which contradicts the Zariski topology on $\mathbb{R}^n$
being a Noetherian topological space.


