I am trying to solve the Exercise 1.9 from Hartshorne:
Suppose the ideal $\mathfrak a$ in $R$ can be generated by $r$ elements. Show that every irreducible component of $Z(\mathfrak a)$ has dimension $\geq n-r$.
I know it is an immediate corollary from Krull's dimension theorem. However, I try to follow the idea given in this notes:
We know the dimension of any component of $Z(\mathfrak a)$ = transcendence degree of its function field, that is, the quotient field of $A(Z(\mathfrak a))$.
I guess this quotient field is $$K(k[x_1,\cdots,x_n]/(f_1,\cdots,f_r))\cong \, ?.$$ I don't know what the field is. Please verify or deny this statement.
If so, this function field contains $x_1, . . . , x_n$ and the algebraic relations between these are a consequence of the $r$ generators of $a$. Therefore the dimension of any component is at least $n$ minus the number of generators of $\mathfrak a$, that is, greater than $ n − r$.
Note that here we assume it is true that the transcendental degree of $k[x_1,\cdots, x_n]/(f_1,f_2,\cdots,f_r)$ is greater than $n-r$, which is intuitionally correct, but how to prove it?