# Minimal number of generators for $A^n$

Let $$A$$ be a commutative ring and $$n \in \mathbb{N}$$. What is the minimal number $$e_A(n)$$ of generators of the $$A$$-algebra $$A^n$$? Here is what I already know (I can add proofs if necessary) from a little joint work:

1. We have $$e_A(n)=0$$ for $$n=0,1$$, therefore let's exclude these trivial cases. We have $$e_A(2) \leq 1$$ since $$A^2$$ is generated by $$(1,0)$$.

2. We have $$e_A(mn) \leq e_A(m) + e_A(n)$$ since $$A^{mn} \simeq A^m \otimes_A A^n$$ as $$A$$-algebras.

3. The function $$e_A : \mathbb{N} \to \mathbb{N}$$ is non-decreasing.

4. When there is a homomorphism $$A \to B$$, then $$e_B \leq e_A$$ holds pointwise. In particular we have $$e_A = e_B$$ when $$B$$ is an $$A$$-algebra with a section. This happens, for example, when $$B$$ is a polynomial ring over $$A$$.

5. We have $$e_A(n) \leq \lceil \log_2(n) \rceil$$ (use 1,2,3).

6. If $$A$$ has elements $$\alpha_1,\dotsc,\alpha_n$$ such that $$\alpha_i - \alpha_j \in A^*$$ for $$i \neq j$$, then $$e_A(n)=1$$ (use Vandermonde). In particular: If $$K$$ is a field with $$\geq n$$ elements, then $$e_K(n)=1$$. If $$K$$ is an infinite field, we therefore have $$e_K=1$$.

7. For a finite field $$\mathbb{F}_q$$ we have $$e_{\mathbb{F}_q}(n)=\lceil \log_q(n) \rceil$$.

In particular, it follows (use 4,5,7) that $$e_A = \lceil \log_2 \rceil$$ when there is a homomorphism $$A \to \mathbb{F}_2$$ (for example for $$A=\mathbb{Z}$$).

8. We have $$e_{A \otimes B} \leq \min(e_A,e_B)$$ with equality for $$A=B$$. But this is not always an equality.

9. We have $$e_{A \times B} = \max(e_A,e_B)$$.

10. If $$I \subseteq A$$ is a nil ideal, then $$e_A = e_{A/I}$$. In particular, we may assume always that $$A$$ is reduced.

11. If $$A$$ is a local ring with residue field $$k$$, then $$e_A = e_k$$. More generally, if $$A$$ has only finitely many maximal ideals $$\mathfrak{m}_i$$, then $$e_A = \max_i e_{A/\mathfrak{m}_i}$$.

12. If $$A=\mathrm{colim}_i A_i$$ is a directed colimit, then $$e_A = \min_i e_{A_i}$$.

Questions. Is there any (geometric) intuition behind the number $$e_A(n)$$? How can we compute $$e_A$$ for other examples, or even for an arbitrary commutative ring? Is there always some $$q \in \mathbb{N}$$ such that $$e_A=\lceil \log_q \rceil$$?

• A related number is, given $m\ge 1$, how many disjoint sections you can put in the affine space $\mathbb A^m$ over $A$. The maximal number is the greastest $n$ such that $e_A(n)\le m$. For exemple, in $\mathbb A^1_{\mathbb Z}$, you can't put three disjoint sections. So $e_{\mathbb Z}(3)\ge 2$. Nov 17, 2012 at 11:15
• The number $e_A(n)$ makes sense for any scheme $S$ : it is the smallest integer $e$ such that there is a closed immersion from $S \sqcup \ldots \sqcup S$ ($n$ times) into the affine space ${\bf A}^e_S$. Nov 17, 2012 at 23:33
• @François, I guess you mean $e_S(n)$. Nov 18, 2012 at 0:04
• @Qing Liu : Yes, I should have written $e_S(n)$. There is also the related number $n_S(e)$ which one can define as in your first comment (fixing the dimension of the affine space and trying to put the maximal number of disjoint sections). Nov 18, 2012 at 0:31
• Write $A$ as an inductive limit of finitely generated algebras $A_0$ over ${\bf Z}$. Then $e_A = \min{A_0} e_{A_0}$ pointwise. So in some sense one is reduced to the case of f.g. algebras. Don't know if this helps : it is not clear how to deduce a result purely in terms of $A$. Nov 19, 2012 at 14:01

You already completely solved the question over fields, noetherian artinian rings and $\mathbb Z$.

Let $A$ be any commutative unitary ring. Then the maximum $f_A(n)$ of all $e_k(n)$ when $k$ runs the residue fields of $A$ (at maximal ideals) satisfies clearly $e_A(n)\ge f_A(n)$ by your (4).

Suppose $A$ is noetherian of dimension $d$, then $$f_A(n) \le e_A(n) \le \max \{ d+1, f_A(n)\}.$$

Proof. Let $m=\max \{ d+1, f_A(n)\}$. We want to show that the affine space $\mathbb A^m$ over $A$ contains $n$ disjoint sections. Let $r\le n-1$ be such that $\mathbb A^m$ contains $r$ disjoint sections. We are going to show that $\mathbb A^m$ contains one more section disjoint from the previous one. This will prove the claim.

Let $T$ be the union of $r$ sections. For every residue field $k$ of $A$, $\mathbb A^m_k$ contains at least $r+1$ rational points. In particular, $T$ doesn't contain $\mathbb A^m_k(k)$. By hypothesis, we also have $\dim T=\dim A< m$. By Proposition 1.10 of this preprint, there is a section in $\mathbb A^m$ disjoint from $T$ and we are done.

Edit (Remove generalization to non-noetherian rings).

Remark. Let $A$ be any finite dimensional noetherian ring. If $A$ has a finite residue field, there exists $q$ such that $e_A(n)$ coincides asymptotically with $\lceil \log_q n \rceil$. It is enough to take for $q$ the smallest cardinality of the finite residue fields of $A$. If all residue fields of $A$ are infinite, then $e_A(n)$ is bounded hence asymptotically constant (because it is increasing). It would be interesting to decide whether these properties hold without noetherian and finite-dimensional hypothesis.

• There is gap in my proof in this part. Enlarging $A_0$ can introduce maximal ideals that I can't control. So I will delete the part concerning non-necessarily noetherian rings. Nov 18, 2012 at 22:45
• What can be said about the eventual value $\lim_{n \to \infty} e_A(n)$ when $A$ has only infinite residue fields? Nov 19, 2012 at 11:15
• I don't know, even when $A$ is noetherian (of infinite dimension). Nov 19, 2012 at 12:48
• Let's assume that $A$ is noetherian of finite dimension $d$. Then your result just says that $e_A(n)$ is eventually constant with some value $< d$. How can we determine this, say, when $A$ is given by generators and relations over a field? And can you give an example where this value is $>1$? Nov 19, 2012 at 15:30
• If $A$ has only infinite residue fields and contains a field $k$, then probably $A$ contains also an infinite field ? Anyway I think this is not the hardest case. Staying with the hypothesis on the residue fields, if we want $e_A(n)=1$ for all $n\ge 2$, then this means that for any $n$, there are $n$ units in $A$ whose differences are all units. This is true for example if $A$ is the ring of all algebraic integers thanks to cyclotomic units $(\xi^n-1)/(\xi -1)$. So far I don't have an example with $e_A(n)>1$. But this is an interesting question. Nov 20, 2012 at 14:07

The paper http://msp.org/ant/2012/6-2/p03.xhtml and the references in it should be of interest to you. Please do not hesitate to contact me with any questions.