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Let $S$ be some base ring (a commutative ring or even just a field), and $R$ a commutative ring containing $S$ which is finitely generated (as an algebra) over $S$. What conditions guarantee that any two minimal systems of generators of $R$ over $S$ have the same size?

I'm especially interested in a geometric picture to explain the situation, and whether it links to other geometrical ideas such as height or the Cohen-Macaulay property.

I'd also like to know what happens for graded rings - for instance, when are the degrees of two minimal systems of homogeneous generators the same (up to permutation)?

What's the geometry behind this?

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I think this almost never happens without grading. Consider the simple example with $S=\mathbb Q$ and $R=\mathbb Q[X]$. Then $\{ X\}$ and $\{ X^2+X, X^2\}$ are minimal systems of generators of different sizes. Similar constructions should work for any algebra over a field with a transcendental element.

Of course if $R/S$ is a finite extension of prime order, then any minimal system of generators is a singleton. But this fails as soon as we remove the condition of prime order: let $S=\mathbb Q$ and $R=\mathbb Q[\sqrt{2}, \sqrt{3}]$. Then $\{\sqrt{2}, \sqrt{3}\}$ and $\{ \sqrt{2}+\sqrt{3}\}$ are minimal systems of generators of different sizes.

Graded case.

For a homogeneous algebra $R$ over a field $S$, a set of homogeneous elements of $R$ generates $R$ if and only if it contains a set of generators of $R_1$ has vector space. So the minimal systems are exactly the basis of $R_1$ as $S$-vector space.

More generally, if $R$ is a positive graded algebra over a field $S$, we can describe the minimal systems of homogeneous generators as follows. For any $d\ge 1$, denote by $R'_d$ the subvector space of $R_d$ generated by products of homogeneous elements of lower degrees ($R'_1=0$). Then $F\subset R$ is a minimal system of homogeneous generators if and only if for all $d\ge 1$, $F\cap R_d$ is a lifting of a basis of $R_d/R'_d$. In particular, for all $d\ge 1$, any two such systems share the same number of elements of degree $d$.

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Isn't the example $S=\mathbb{Q}$ and $R = \mathbb{Q}[X]$ an example of a homogeneous algebra over a field? I believe if one restricts generators to being inside of $S_1$ your last comment is true, but as your first example shows, this need not be the case. – RghtHndSd Aug 7 '13 at 15:37
@rghthndsd: in the first example, we don't ask the generators to be homogeneous. – user18119 Aug 7 '13 at 15:38
In the sentence "of course if $R/S$ is a finite extension of prime order..." do you mean a finite field extension of prime order? B/c take $S=k$, $R=S[x,y]/(x,y)^2$ for an order 3 extension (i.e. $R$ is a free $S$-module of rank $3$) such that there is no singleton generating (set since the algebra generated by any single nonconstant element is just $2$-dimensional over $k$). – Ben Blum-Smith Dec 11 '15 at 20:51
@user18119 - In my last comment I was assuming that by "finite extension of prime order" you meant a finite extension that is free and of prime rank as a module over the ground ring, but perhaps I misunderstood? – Ben Blum-Smith Dec 11 '15 at 23:07

For a connected, non-negatively graded algebra $A$ over a field, we can compute $Tor_1^A(k,k)$, which is a graded vector space: this is is isomorphic to every minimal space of homogeneous generators of $A$.

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What do you mean by "minimal space of homogeneous generator"? – user26857 Mar 2 '14 at 23:15
What mean is, every graded vector subspace of $A$ which generates $A$ and which is minimal is isomorphic to $Tor_1^A(k,k)$ as a graded vector space. – Mariano Suárez-Alvarez Mar 3 '14 at 8:14

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