Atiyah-MacDonald, exercise 2.11

Question

Let $A$ be a commutative ring with $1$ not equal to $0$. (The ring A is not necessarily a domain, and is not necessarily Noetherian.) Assume we have an injective map of free $A$-modules $A^m \to A^n$. Must we have $m \le n$?

I believe the answer is yes. For instance, why is there no injective map from $A^2 \to A^1$? Say it's represented by a matrix $(a_1, a_2)$. Then clearly $(a_2, -a_1)$ is in the kernel. In the $A^{n+1} \to A^{n}$ case, we can look at the $n \times (n+1)$ matrix which represents it; call it $M$. Let $M_i$ denote the determinant of the matrix obtained by deleting the $i$-th column. Let $v$ be the vector $(M_1, -M_2, ..., (-1)^nM_{n+1})$. Then $v$ is in the kernel of our map, because the vector $Mv^T$ has $i$-th component the determinant of the $(n+1) \times (n+1)$ matrix attained from $M$ by repeating the $i$-th row twice.

That almost finishes the proof, except it is possible that $v$ is the zero vector. I would like to see either this argument finished, or, even better, a nicer proof.

Thank you!

Answer 1

Let M be the $n\times m$ matrix representing the injection $A^m \to A^n$. Define D_i(M) to be the ideal generated by the determinants of all i-by-i minors of M. Let r be the largest possible integer such that D_r(M) has no annihilator (i.e. there is no nonzero element a∈A such that aD_r(M)=0); I think r is usually called the McCoy rank of M.

Assume that $m>n$. We shall get a contradiction.

Choose a nonzero a∈A such that aD_r+1(M)=0. By assumption, $a$ does not annihilate D_r(M), so there is some r-by-r minor that is not killed by $a$; we may assume it is the upper-left r-by-r minor. Thus, $r \leq n$, so that $r+1 \leq n+1\leq m$. Let M₁, ..., M_r+1 be the cofactors of the upper-left (r+1)-by-(r+1) minor obtained by expanding along the bottom row. (This is well-defined even if the $r+1$-th row of $M$ does not exist, because these cofactors use only the first $r$ rows and the first $r+1$ columns of $A$, and $A$ has both since $r \leq n$ and $r+1 \leq m$.) Note, in particular, we know M_r+1 is the determinant of the upper-left r-by-r minor, so aM_r+1≠0.

The claim is that the vector (aM₁,...,aM_r+1,0,0,...) (which we've already shown is non-zero) is in the kernel of M. To see that, note that when you dot this vector with the k-th row of M, you get $a$ times the determinant of the matrix obtained by replacing the bottom row of the upper-left (r+1)-by-(r+1) minor with the first r+1 entries in the k-th row. If k≤r, this determinant is zero because a row is repeated, and if k>r, this determinant is the determinant of some (r+1)-by-(r+1) minor, so it is annihilated by $a$. But since A is injective, the kernel of M is 0, and we have a contradiction.

Answer 2

Here is another solution using only the Cayley-Hamilton Theorem for finitely generated modules (Proposition 2.4. in Atiyah-Macdonald) which, even though looks quite innocent, is a very powerful statement.

Assume by contradiction that there is an injective map $\phi: A^m \to A^n$ with $m>n$. The first idea is that we regard $A^n$ as a submodule of $A^m$, say the submodule generated by the first $n$ coordinates. Then, by the Cayley-Hamilton Theorem, $\phi$ satisfies some polynomial equation \begin{equation} \phi^k + a_{k-1} \phi^{k-1} + \cdots + a_1 \phi + a_0 = 0. \end{equation} Using the injectivity of $\phi$ it is easy to see that if this polynomial has the minimal possible degree, then $a_0 \ne 0$. But then, applying this polynomial of $\phi$ to $(0,\ldots,0, 1)$, the last coordinate will be $a_0$ which is a contradiction as it should be zero.

Answer 3

Dear CJD, if you are still interested in your problem, already solved three weeks ago by Anton, here is another point of view.

Let $M:A^m \to A^n$ be injective. Let $B=\mathbb Z [\ldots,m_{ij},\ldots]$ be the subring of $A$ generated by all the entries of the matrix $M$ ; this $B$ is a noetherian ring and we have (by restriction) an injective linear map $M:B^m \to B^n$. In other words we may assume that $A$ is noetherian. Now we localize at a prime $\mathfrak p$ of $B$ of height zero and we get an injective map (localization is exact and thus preserves injections) $L:C^m \to C^n$ (the ring $C$ is the ring $B$ localized at $\mathfrak p$).

Ah, but now $C$ is noetherian of dimension zero, hence artinian and we can talk about lengths. Since lengths are additive in exact sequences (Atiyah-MacDonald, Proposition 6.9) we get $m.length(C) + length(coker L)= n.length(C)$, hence $m\leq n$.

Friendly greetings, Georges.

Answer 4

I posted this question on a different site a couple of years ago. Eventually I found that a book of T.Y. Lam has a very nice treatment. Here is the writeup I posted on the other site:

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After paging through several algebra books, I found that T.Y. Lam's GTM Lectures on Rings and Modules has a beautiful treatment of this question.

The above property of a (possibly noncommutative) ring is called the "strong rank condition." It is indeed stronger than the corresponding statement for surjections ("the rank condition") which is stronger than the isomorphism version "Invariant basis number property". However, in fact it is the case that all commutative rings satisfy the strong rank condition. Lam gives two proofs [pp. 12-16], and I will now sketch both of them.

First proof:

Step 1: The result holds for (left-) Noetherian rings. For this we establish:

Lemma: Let $M$ and $N$ be (left-) $A$-modules, with $N$ nonzero. If the direct sum $M \oplus N$ can be embedded in $M$, then $M$ is not a Noetherian $A$-module.

Proof: By hypothesis $M$ has a submodule $M_1 \oplus N_1$, with $M_1 \cong M$ and $N_1 \cong N$. But we can also embed $M \oplus N$ in $M_1$, meaning that $M_1$ contains a submodule $M_2 \oplus N_2$ with $M_2 \cong M$ and $N_2 \cong N$. Continuing in this way we construct an ascending chain of submodules $N_1$, $N_1 \oplus N_2$,..., contradiction.

So if A is (left-) Noetherian, apply the Lemma with $M = A^n$ and $N = A^{m-n}$. $M$ is a Noetherian $A$-module, and we conclude that $A^m$ cannot be embedded in $A^n$.

Step 2: We do the case of a commutative, but not necessarily Noetherian, ring. First observe that, defining linear independent subsets in the usual way, the strong rank condition precisely asserts that any set of more than $n$ elements in $A^n$ is linearly dependent. Thus a ring $A$ satisfies the strong rank condition iff: for all $m > n$, any homogeneous linear system of $n$ linear equations and $m$ unknowns has a nonzero solution in $A$.

So, let $MX = 0$ be any homogeneous linear system with coefficient matrix $M = (m_{ij}), \ 1 \leq i \leq n, 1 \leq j \leq m$. We want to show that it has a nonzero solution in $A$. But the subring $A' = \mathbb{Z}[a_{ij}]$, being a quotient of a polynomial ring in finitely many variables over a Noetherian ring, is Noetherian (by the Hilbert basis theorem), so by Step 1 there is (even) a nonzero solution $(x_1,...,x_m) \in (A')^m$.

This makes one wonder if it is necessary to consider the Noetherian case separately, and it is not. Lam's second proof comes from Bourbaki's Algebra, Chapter III, §7.9, Prop. 12, page 519. [Thanks to Georges Elencwajg for tracking down the reference.] It uses the following elegant characterization of linear independence in free modules:

Theorem: A subset $\{u_1,...,u_m\}$ in $M = A^n$ is linearly independent iff: if $a \in A$ is such that $a \cdot (u_1 \wedge \ldots \wedge u_m) = 0$, then $a = 0$.

Here $u_1\wedge \ldots \wedge u_m$ is an element of the exterior power $\Lambda^m(M)$.

(I will omit the proof here; the relevant passage is reproduced on Google books.)

This gives the result right away: if $m > n$, $\Lambda^m(A^n) = 0$.

Answer 5

Every few years this question re-appears in my life (as it just did this week) and I have to rediscover the Euler Characteristic proof which I half-remember. Let me just write my thoughts down here so I don't ever have to do this again.

Definition. Let $A$ be a non-zero commutative ring. A finite free resolution (or FFR) of an $A$-module $M$ is an exact sequence $$0\to F_n \to F_{n-1} \to \cdots \to F_1 \to F_0 \to M\to 0$$ where each $F_i$ is a free $A$-module, of rank $r_i$.

If $M$ has an FFR then we define the Euler Characteristic $\chi(M)$ of $M$ to be $\sum_i(-1)^ir_i$. This turns out to be a well-defined invariant of $M$ (i.e. independent of the resolution) because a relatively straightforward induction on $n$ shows that if $G_i$ is another resolution then $F_0\oplus G_1\oplus F_2\oplus\cdots\oplus (F\ \mbox{or}\ G)_n$ is isomorphic to $G_0\oplus F_1\oplus G_2\oplus\cdots$. For more details on this part of the argument see Lemma 4 in section 19 of Matsumura's book "Commutative Ring Theory".

The fundamental fact is:

Theorem. If $M$ has an FFR then $\chi(M)\geq0$.

As a consequence, not only does an injection $A^m \to A^n$ imply $m\leq n$ (apply the theorem to the cokernel), but if $0\to A^m\to A^n \to A^r$ is exact then $m+r\geq n$ and so on and so on. Do the other solutions also generalise to prove this?

The theorem is Theorem 19.7 of Matsumura's "commutative ring theory". Here's a sketch. Localising at a minimal prime one can assume that $A$ is local with nilpotent maximal ideal. It suffices to prove that in this case $M$ is free, as then the exact sequence splits and the Euler Characteristic is just the rank of $M$. It suffices to prove that the cokernel of $F_n\to F_{n-1}$ is free, because then one is done by induction on $n$. The trick now is to replace $F_n$ and $F_{n-1}$ by a minimal free resolution, look at the image of $F_n$ and see that its coordinates must lie in the maximal ideal, and it's then not difficult to concoct a non-zero element of $A$ which kills this finite set of coordinates, and hence $F_n$ is a free module annihilated by a non-zero element of $A$ and must hence be zero.

This last part of the argument feels very similar to Georges Elencwajg's argument above, which reduces to the Noetherian case and uses lengths, but Matsumura explicitly avoids this reduction.