10.101 Flatness criteria over Artinian rings
We discuss some flatness criteria for modules over Artinian rings. Note that an Artinian local ring has a nilpotent maximal ideal so that the following two lemmas apply to Artinian local rings.
Lemma 10.101.1. Let (R, \mathfrak m) be a local ring with nilpotent maximal ideal \mathfrak m. Let M be a flat R-module. If A is a set and x_\alpha \in M, \alpha \in A is a collection of elements of M, then the following are equivalent:
\{ \overline{x}_\alpha \} _{\alpha \in A} forms a basis for the vector space M/\mathfrak mM over R/\mathfrak m, and
\{ x_\alpha \} _{\alpha \in A} forms a basis for M over R.
Proof.
The implication (2) \Rightarrow (1) is immediate. Assume (1). By Nakayama's Lemma 10.20.1 the elements x_\alpha generate M. Then one gets a short exact sequence
0 \to K \to \bigoplus \nolimits _{\alpha \in A} R \to M \to 0
Tensoring with R/\mathfrak m and using Lemma 10.39.12 we obtain K/\mathfrak mK = 0. By Nakayama's Lemma 10.20.1 we conclude K = 0.
\square
Lemma 10.101.2. Let R be a local ring with nilpotent maximal ideal. Let M be an R-module. The following are equivalent
M is flat over R,
M is a free R-module, and
M is a projective R-module.
Proof.
Since any projective module is flat (as a direct summand of a free module) and every free module is projective, it suffices to prove that a flat module is free. Let M be a flat module. Let A be a set and let x_\alpha \in M, \alpha \in A be elements such that \overline{x_\alpha } \in M/\mathfrak m M forms a basis over the residue field of R. By Lemma 10.101.1 the x_\alpha are a basis for M over R and we win.
\square
Lemma 10.101.3. Let R be a ring. Let I \subset R be an ideal. Let M be an R-module. Let A be a set and let x_\alpha \in M, \alpha \in A be a collection of elements of M. Assume
I is nilpotent,
\{ \overline{x}_\alpha \} _{\alpha \in A} forms a basis for M/IM over R/I, and
\text{Tor}_1^ R(R/I, M) = 0.
Then M is free on \{ x_\alpha \} _{\alpha \in A} over R.
Proof.
Let R, I, M, \{ x_\alpha \} _{\alpha \in A} be as in the lemma and satisfy assumptions (1), (2), and (3). By Nakayama's Lemma 10.20.1 the elements x_\alpha generate M over R. The assumption \text{Tor}_1^ R(R/I, M) = 0 implies that we have a short exact sequence
0 \to I \otimes _ R M \to M \to M/IM \to 0.
Let \sum f_\alpha x_\alpha = 0 be a relation in M. By choice of x_\alpha we see that f_\alpha \in I. Hence we conclude that \sum f_\alpha \otimes x_\alpha = 0 in I \otimes _ R M. The map I \otimes _ R M \to I/I^2 \otimes _{R/I} M/IM and the fact that \{ x_\alpha \} _{\alpha \in A} forms a basis for M/IM implies that f_\alpha \in I^2! Hence we conclude that there are no relations among the images of the x_\alpha in M/I^2M. In other words, we see that M/I^2M is free with basis the images of the x_\alpha . Using the map I \otimes _ R M \to I/I^3 \otimes _{R/I^2} M/I^2M we then conclude that f_\alpha \in I^3! And so on. Since I^ n = 0 for some n by assumption (1) we win.
\square
Lemma 10.101.4. Let \varphi : R \to R' be a ring map. Let I \subset R be an ideal. Let M be an R-module. Assume
M/IM is flat over R/I, and
R' \otimes _ R M is flat over R'.
Set I_2 = \varphi ^{-1}(\varphi (I^2)R'). Then M/I_2M is flat over R/I_2.
Proof.
We may replace R, M, and R' by R/I_2, M/I_2M, and R'/\varphi (I)^2R'. Then I^2 = 0 and \varphi is injective. By Lemma 10.99.8 and the fact that I^2 = 0 it suffices to prove that \text{Tor}^ R_1(R/I, M) = K = \mathop{\mathrm{Ker}}(I \otimes _ R M \to M) is zero. Set M' = M \otimes _ R R' and I' = IR'. By assumption the map I' \otimes _{R'} M' \to M' is injective. Hence K maps to zero in
I' \otimes _{R'} M' = I' \otimes _ R M = I' \otimes _{R/I} M/IM.
Then I \to I' is an injective map of R/I-modules. Since M/IM is flat over R/I the map
I \otimes _{R/I} M/IM \longrightarrow I' \otimes _{R/I} M/IM
is injective. This implies that K is zero in I \otimes _ R M = I \otimes _{R/I} M/IM as desired.
\square
Lemma 10.101.5. Let \varphi : R \to R' be a ring map. Let I \subset R be an ideal. Let M be an R-module. Assume
I is nilpotent,
R \to R' is injective,
M/IM is flat over R/I, and
R' \otimes _ R M is flat over R'.
Then M is flat over R.
Proof.
Define inductively I_1 = I and I_{n + 1} = \varphi ^{-1}(\varphi (I_ n)^2R') for n \geq 1. Note that by Lemma 10.101.4 we find that M/I_ nM is flat over R/I_ n for each n \geq 1. It is clear that \varphi (I_ n) \subset \varphi (I)^{2^ n}R'. Since I is nilpotent we see that \varphi (I_ n) = 0 for some n. As \varphi is injective we conclude that I_ n = 0 for some n and we win.
\square
Here is the local Artinian version of the local criterion for flatness.
Lemma 10.101.6. Let R be an Artinian local ring. Let M be an R-module. Let I \subset R be a proper ideal. The following are equivalent
M is flat over R, and
M/IM is flat over R/I and \text{Tor}_1^ R(R/I, M) = 0.
Proof.
The implication (1) \Rightarrow (2) follows immediately from the definitions. Assume M/IM is flat over R/I and \text{Tor}_1^ R(R/I, M) = 0. By Lemma 10.101.2 this implies that M/IM is free over R/I. Pick a set A and elements x_\alpha \in M such that the images in M/IM form a basis. By Lemma 10.101.3 we conclude that M is free and in particular flat.
\square
It turns out that flatness descends along injective homomorphism whose source is an Artinian ring.
Lemma 10.101.7. Let R \to S be a ring map. Let M be an R-module. Assume
R is Artinian
R \to S is injective, and
M \otimes _ R S is a flat S-module.
Then M is a flat R-module.
Proof.
First proof: Let I \subset R be the Jacobson radical of R. Then I is nilpotent and M/IM is flat over R/I as R/I is a product of fields, see Section 10.53. Hence M is flat by an application of Lemma 10.101.5.
Second proof: By Lemma 10.53.6 we may write R = \prod R_ i as a finite product of local Artinian rings. This induces similar product decompositions for both R and S. Hence we reduce to the case where R is local Artinian (details omitted).
Assume that R \to S, M are as in the lemma satisfying (1), (2), and (3) and in addition that R is local with maximal ideal \mathfrak m. Let A be a set and x_\alpha \in A be elements such that \overline{x}_\alpha forms a basis for M/\mathfrak mM over R/\mathfrak m. By Nakayama's Lemma 10.20.1 we see that the elements x_\alpha generate M as an R-module. Set N = S \otimes _ R M and I = \mathfrak mS. Then \{ 1 \otimes x_\alpha \} _{\alpha \in A} is a family of elements of N which form a basis for N/IN. Moreover, since N is flat over S we have \text{Tor}_1^ S(S/I, N) = 0. Thus we conclude from Lemma 10.101.3 that N is free on \{ 1 \otimes x_\alpha \} _{\alpha \in A}. The injectivity of R \to S then guarantees that there cannot be a nontrivial relation among the x_\alpha with coefficients in R.
\square
Please compare the lemma below to Lemma 10.99.15 (the case of Noetherian local rings), Lemma 10.128.8 (the case of finitely presented algebras), and Lemma 10.128.10 (the case of locally nilpotent ideals).
Lemma 10.101.8 (Critère de platitude par fibres: Nilpotent case). Let
\xymatrix{ S \ar[rr] & & S' \\ & R \ar[lu] \ar[ru] }
be a commutative diagram in the category of rings. Let I \subset R be a nilpotent ideal and M an S'-module. Assume
The module M/IM is a flat S/IS-module.
The module M is a flat R-module.
Then M is a flat S-module and S_{\mathfrak q} is flat over R for every \mathfrak q \subset S such that M \otimes _ S \kappa (\mathfrak q) is nonzero.
Proof.
As M is flat over R tensoring with the short exact sequence 0 \to I \to R \to R/I \to 0 gives a short exact sequence
0 \to I \otimes _ R M \to M \to M/IM \to 0.
Note that I \otimes _ R M \to IS \otimes _ S M is surjective. Combined with the above this means both maps in
I \otimes _ R M \to IS \otimes _ S M \to M
are injective. Hence \text{Tor}_1^ S(IS, M) = 0 (see Remark 10.75.9) and we conclude that M is a flat S-module by Lemma 10.99.8. To finish we need to show that S_{\mathfrak q} is flat over R for any prime \mathfrak q \subset S such that M \otimes _ S \kappa (\mathfrak q) is nonzero. This follows from Lemma 10.39.15 and 10.39.10.
\square
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