A COURSE IN HOMOLOGICAL ALGEBRA Chapter **: Tor, Flatness, and Purity E. L. Lady (April 7, 1996)

Proposition. If Λ is a K-algebra and M a Λ-module, then the functor M ⊗_{Λ} is left adjoint to Hom_{K}(M, ). Corollary. Tensor products preserve colimits. Definition. Flat modules. Lemma. Projective modules are flat. proof: Free modules are flat. Definition. Tor. Theorem. Tor commutes with (filtered) direct limits and coproducts. Theorem. [Bourbaki, Commutative Algebra I, Lazard, Bull. Soc. Math. France 97(1969,81–128]. The following are equivalent:

(1) M is flat.

(2) X → Y monic implies X ⊗_{Λ} M → Y ⊗_{Λ} M monic.

(3) For every ideal I, the multiplication map I ⊗_{Λ} M → M is monic.

(4) Tor_{1}(Λ/I, M) = 0 for all finitely generated right ideals I.

(5) Whenever a_{j} ∈ M and ℓ_{ij} ∈ Λ are such that ∑_{j} ℓija_{j} = 0 for all i, then there exist y_{k} ∈ M and r_{ij} ∈ Λ such that

m_{j} = ∑_{k} r_{ij}y_{k}
∑_{j} ℓ_{ij}r_{jk} = 0  ∀ i, k.

(6) For every map φ from a finitely generated Λ-module B into M, there exists homomorphism from a finitely generated free Λ-module F into M such that the following diagram commutes:

B ⟶ F ; ↓ ; M  (Insert diagonal arrow for φ.)

(7) M is a direct limit of finitely generated free modules.

Corollary. Finitely presented flat modules are projective. Theorem. Let M be a flat Λ-module.

(1) Every regular element of Λis regular on M.

(2) If I and J are right ideals, then IM ∩ JM = (I ∩ J)M. Theorem. If R → A is a flat morphism of commutative rings and M is a finitely generated R-module, then ann(S ⊗_{R} M) = (ann_{R} M)S. Proposition. If Λ is a flat K-algebra then

Λ ⊗_{K} Tor_{n}^{K}( , ) ≈ Tor_{n}^{Λ}(Λ ⊗_{K} , Λ ⊗_{K} ).

Theorem. If R is a commutative ring, then M is flat if and only if M_{𝔭} is flat for all maximal ideals 𝔭. Theorem. Let Λbe a quasi-local ring with maximal ideal J. Let M be a finitely presented left Λ-module. Then the following are equivalent:

(1) M is free.

(2) Tor_{1}^{Λ}(Λ/J, M) = 0.

(2) The multiplication map J ⊗_{Λ} M → JM is monic.

Semi-Hereditary Rings and Prüfer Domains

Definition. A ring Λis left semi-hereditary if every finitely generated left ideal is projective. Theorem. Λ is left semi-hereditary if and only if all finitely generated submodules of projective modules are projective.

For noetherian rings the concept of “hereditary” and “semi-hereditary” coincide. Theorem. If Λis left semi-hereditary then Tor_{n}^{Λ} = 0 for n ≥ 2. Definition. A Prüfer domain is a semi-hereditary integral domain.

A Prüfer domain is the non-noetherian (or not necessarily noetherian) analog of a dedekind domain. Theorem. An integral domain is a Prüfer domain if and only if all finitely generated torsion free modules are projective. Theorem. A module over a Prüfer domain is flat if and only if it is torsion free.

Theorem. If M and N are modules over a Prüfer domain R, then

Tor_{1}^{R}(M, N) ≈ Tor_{1}^{R}(tM, N),

where tM is the torsion submodule of M. Furthermore, Tor_{1}^{R}(M, N) is a torsion module.

Von Neumann Regular Rings

Definition. A ring is von Neumann regular if for every element a there exists an element x in the ring such that axa = a. In other words,

(∀a ∈ Λ) a ∈ aΛa.

This condition definitely seems to be on the technical side. However it’s actually much more natural than it appears, as the following theorem shows. Recall that a left ideal of Λ is generated by an idempotent if and only if it is a summand of Λ (considered as a left Λ-module).

Theorem. The following conditions for a ring Λ are equivalent:

(1) Λ is von Neumann regular.

(2) Every principal left idea is generated by an idempotent.

(3) Every finitely generated left ideal is generated by an idempotent.

(4) Every right Λ-module is flat.

(5) Every cyclic right Λ-module is flat.

proof: (1) ⇒ (2): If axa = a then xa is an idempotent and the principal left ideal Λa is generated by xa.

(2) ⇒ (3): Consider a left ideal I generated by a_{1}, …, a_{n}. By induction on the number of generators, we may suppose that there exists an idempotent f such that Λa_{1} + ⋯ + Λa_{n−1} = Λf. Then by the modular law we get

Λa_{1} + ⋯ + Λa_{n} = Λf ⊕ (I ∩ Λ(1 − f))

and the rest is easy.

(3) ⇒ (4): If suffices to see that Tor_{1}^{Λ}(M, Λ/I) = 0 for all finitely generated left ideals I. But (3) implies that such an I is a summand of the ring, so Λ/I is projective.

(4) ⇒ (5): Clear.

(5) ⇒ (1): Let a be an element of Λ. Then by assumption Λ/Λa is flat. This yields an exact sequence

Tor_{1}^{Λ}(aΛ, Λ/aΛ) = 0 ⟶ aΛ ⊗_{Λ} Λa ⟶ aΛ ⟶^{β} aΛ ⊗_{Λ} Λ/Λa → 0 .

Since Ker β = aΛa, we see that aΛ ⊗_{Λ} Λa ≈ aΛa. The inclusion aΛ ↪ Λ induces an exact diagram

0 ⟶ aΛa ⟶ aΛ ⟶ aΛ ⊗_{Λ} (Λ/Λa) ⟶ 0
↓  ↓  ^{γ}↓
0 ⟶ Λa ⟶ Λ ⟶  Λ/Λa  ⟶ 0 .

Since Λ/Λa is flat, γ is monic and the left-hand square must be a pull back, in other words aΛa = Λa ∩ aΛ. Thus a ∈ aΛa, as required. ✓

Theorem. A commutative ring is von Neumann regular if and only if it has Krull dimension one and has no non-trivial nilpotent elements.

proof: ( ⇒ ): Easy.

( ⇐ ): It suffices to prove that R_{𝔭} is von Neumann regular for every prime ideal 𝔭. But since R_{𝔭} is zero-dimensional, 𝔭R_{𝔭} = nilrad(R_{𝔭}) = 0, so R_{𝔭} is a field. ✓

Faithfully Flat Modules

Definitions. A functor T: C → D is faithful if it preserves the distinctness of morphisms, i. e. T (φ) ≠ T (ψ) whenever φ and ψ are distinct morphisms from an object X to an object Y in C.

Generator. Cogenerator.

Proposition. An exact functor T from one abelian category to another is faithful if and only if T A is non-zero whenever A is non-zero.

proof: ( ⇒ ): Clear.

( ⇐ ): Let α: X → Y be a non-zero morphism and let A be the image, so that α factors as X ↠ A ↣ Y . Since T is exact, T α factors as T X ↠ T A ↣ T Y . Since A ≠ 0, by hypothesis T A ≠ 0. Thus T α is not zero. ✓

Proposition. An additive functor between abelian categories is faithful if and only if it reflects exactness, i. e. if X → Y → Z is not exact in the source category, then T X → T Y → T Z will not be exact.

proof: ( ⇐ ): Let 0 ≠ β: B → C be a non-zero morphism in the source category. Then

the sequence B ⟶^{1_{B}} B ⟶^{β} C is not exact. Thus

T B ⟶^{1_{T B}} T B ⟶^{T β} C

is not exact, so T β ≠ 0.

( ⇒ ): Let T be faithful and consider a sequence A ⟶^{α} B ⟶^{β} C in the source category.

Now if βα ≠ 0, then T (βα) ≠ 0 since T is faithful, so clearly T A ⟶^{T α} T B ⟶^{T β} T C is not exact.

Suppose therefore that βα = 0 but that A ⟶^{α} B ⟶^{β} C is not exact. Thus (thinking in terms of modules), α(A) ⊊ Ker β.

In more categorial terms, let K ⟶^{σ} B be the kernel of β and B ⟶^{τ} L be the cokernel of α. Intuitively, L = B/α(A) and the fact that α(A) ⊊ Ker β can be expressed in categorical terms by saying that τσ ≠ 0. Then since T is faithful, T (τσ) ≠ 0. Now βσ = 0 so T (β)T (σ) = 0 (since T is additive), which says that the image of T (σ) is contained in Ker T β. Also, T (τα) = 0, so Image (T α) ⊆ Ker(T τ). **** Thus to show

that T A ⟶^{T α} T B ⟶^{T β} T C is not exact, it will certainly suffice to show that

Image(T α) ⊊ Image(T σ) ⊆ Ker T β.

****

Theorem. Let T be a faithful additive functor from Λ-modules to Λ′-modules.

(1) Suppose that T preserves direct limits. Let me be M a Λ-module such that T M is finitely generated. Then M is finitely generated.

(2) Suppose that T preserves all colimits and that T Λ is a finitely presented Λ′-module. Then M is finitely presented.

Definition. A left Λ-module M is faithfully flat if ⊗_{Λ} M is a faithful and exact functor.

Theorem. The following conditions are equivalent for a left Λ-module M.

(1) M is faithfully flat.

(2) A sequence 0 → A → B → C → 0 of right Λ-modules is exact if and only if the induced sequence

0 → A ⊗_{Λ} M ⟶ B ⊗_{Λ} M ⟶ C ⊗_{Λ} M → 0

is exact.

(3) M is flat and for every non-trivial right Λ-module A, A ⊗_{Λ} M ≠ 0.

(4) M is flat and for all [maximal] right ideals I, IM ≠ 0.

Faithfully Flat Ring Extensions

Definitions. Faithfully flat algebra. Faithfully flat morphism of rings.

Theorem. Let γ: K → Λ be a K-algebra. The following conditions are equivalent.

(1) Λ is a faithfully flat K-algebra.

(2) Λ is flat, γ is monic, and Λ/γ(K) is a flat K-module.

(3) Λ is flat and for every K-module G, the standard map G → Λ ⊗_{K} G is monic.

(4) Λ is flat and every ideal 𝔞 of K is the contraction of its extension to Λ, i. e. 𝔞 = γ^{−1}(𝔞Λ) (or, more informally, 𝔞 = K ∩ 𝔞Λ).

(5) Λ is flat and every maximal ideal 𝔪 in K is the contraction of an ideal in Λ, i. e. there exists an ideal I in Λ such that 𝔪 = γ^{−1}(I).

Lemma. Let Λ be a flat K-algebra. Then the standard map Λ_{K} Hom_{K}(A, B) → Hom_{Λ}(Λ ⊗_{K} A, Λ ⊗_{K} B) is a monomorphism when A is finitely generated and an isomorphicm when A if finitely presented.

Theorem. Let Λ be a faithfully flat K-algebra and let G be a K-module.

(1) G is flat if and only if Λ ⊗_{K} G is flat.

(2) G is finitely generated [presented] if and only if Λ ⊗_{K} G is finitely generated [presented].

(3) G is a finitely generated projective K-module if and only if Λ ⊗_{K} G is a finitely generated projective Λ-module.

Proposition. Let Λ be a K-algebra. If there exists a faithfully flat Λ-module M which is flat as a K-module, then Λ is a flat K-algebra.

Pure Submodules

We were led to the idea of flat modules by looking for those modules M such that whenever A is a submodule of a right Λ-module B, the induced map A ⊗_{Λ} M → B ⊗_{Λ} M is monic.

Another natural line of investigation is to look for pairs A ⊆ B such that for every possible left Λ-module M, the induced map A ⊗_{Λ} M → B ⊗_{Λ} M is monic. When this is the case, we say that A is a pure submodule of B.

Proposition. ℚ/ℤ is an injective cogenerator for the category of abelian groups.

Proposition. Let T = Hom_{ℤ}( , ℚ/ℤ) taking right Λ-modules to left Λ-modules. A right Λ-module A is flat if and only if T (A) is an injective Λ-module.

Definition. A sequence 0 → A → B → C → 0 of right Λ-modules is called pure exact if for every left Λ-module M the induced sequence

0 → A ⊗_{Λ} M ⟶ B ⊗_{Λ} M ⟶ C ⊗_{Λ} M → 0

is exact. (Since the tensor product is right exact, if the original sequence is exact, the only issue in doubt is that A ⊗_{Λ} M → B ⊗_{Λ} M be monic.)

A is called a pure submodule of B if the natural sequence

0 → A ⟶^{⊆} B ⟶ B/A → 0

is a pure exact sequence.

Clearly every split exact sequence is pure exact. Also, since the tensor product preserves direct limits, it is not hard to see that a direct limit of pure exact sequences is pure exact.

The following theorem says that all pure exact sequences can be obtained as direct limits of split exact sequences.

Theorem. Let 0 ⟶^{α} A ⟶^{β} B → C → 0 be a sequence of right Λ-modules. The following are equivalent.

(1) The sequence is pure exact.

(2) The sequence is exact and the morphism T (A) ⊗_{Λ} α is monic.

(3) The sequence of Λ-modules 0 → T C → T B → T A → 0 is split exact.

(4) The sequence 0 → T C → T B → T A → 0 is pure exact.

(5) For every finitely presented right Λ-module F, the induced map Hom_{Λ}(F, B) ⟶ Hom_{Λ}(F, C) is surjective.

(6) The sequence 0 → A → B → C → 0 can be obtained as a direct limit of split exact sequences.

proof: 1) ⇒ 2) ⇒ 3) ⇒ 4) ⇒ 6) ⇒ 1).

To prove 4) ⇒ 5) consider the map σ: T A ⊗_{Λ} F → T (Hom_{Λ}(F, A)) given by σ(θ ⊗ f)(φ) = θ(φ(f)). This map σ is an isomorphism if F is finitely presented. ✓

Corollary [P. M. Cohn]. Let N be a submodule of the left Λ-module M. Then N is a pure submodule of M if and only if the following condition is satisfied.

Whenever there exist elements n_{1}, …, n_{r} ∈ N, m_{1}, …, m_{s} ∈ M, and ℓ_{ij} ∈ Λ such that for all i, ∑_{j} ℓ_{ij}m_{j} = n_{i}, then there exist m′_{1}, …, m′_{s} ∈ N satisfying the same equations, i. e.

(∀i) ∑_{j} ℓ_{ij}m′_{j} = n_{i}.

Corollary. If N is a pure submodule of M and I if a right ideal in Λ, then IM = M ∩ IN.

Corollary. A pure subring of a noetherian [artinian] ring is noetherian [artinian].

Theorem. If K is a pure subring of the center of Λ, then Λ ⊗_{K} is a faithful functor from K-modules to Λ-modules.

Lemma. If K is a pure subring of the center of Λ and 𝔞 is an ideal in K, then 𝔞 is a pure K-submodule of 𝔞Λ. proof: Suppose that there exist a_{i} ∈ 𝔞, b_{j} ∈ 𝔞Λ and k_{ij} ∈ K such that for all i,

∑_{j} b_{j}k_{ij} = a_{i},

Since b_{j} ∈ 𝔞Λ, we can write b_{j} = ∑_{m} c_{m}ℓ_{jm}, for c_{m} ∈ 𝔞. Then

∑ ∑_{j,m} c_{m}ℓ_{jm}k_{ij} = a_{i}.

Since K is pure in Λ, there exist ℓ′_{jm} ∈ K such that ∑ ∑_{j,m} c_{m}ℓ′_{jm}k_{ij} = a_{i}. Now let

b′_{j} = ∑_{m} c_{m}ℓ′_{jm} ∈ 𝔞 .

Theorem. Let K be a pure subring of the center of Λ and let G be a K-module. (1) G is flat if and only if Λ ⊗_{K} G is a flat Λ-module.

(2) G is finitely generated [finitely presented] if and only if Λ ⊗_{K} G is a finitely generated [finitely presented] Λ-module.

(3) G is a finitely generated projective module if and only if Λ ⊗_{K} G is a finitely generated projective Λ-module. proof: (1)( ⇐ ): Suppose that Λ ⊗_{K} G is a flat Λ-module. Let 𝔞 be an ideal in K and consider the diagram

𝔞 ⊗_{K} G  ⟶  G
↓^{monic}  ^{monic}↓
𝔞Λ ⊗_{K} G ⟶^{≈} 𝔞Λ ⊗_{Λ} Λ ⊗_{K} G ⟶^{monic} Λ ⊗_{K} M .

( Insert diagonal arrow. )

Theorem. Let N be a submodule of a flat Λ-module M. The following are equivalent. (1) N is a pure submodule of M.

(2) IN = N ∩ IM for all finitely generated right ideals I of Λ.

(3) M/N is flat.

Corollary. A pure submodule of a flat module is flat.

Theorem. If Λ is left noetherian, then pure submodules of injective left Λ-modules are injective.

Theorem. A push-out or pull-back of a pure exact sequence is pure exact, as is the Baer sum of two pure exact sequences. Thus there is a subfunctor Pext_{Λ}^{1}( , ) of Ext_{Λ}^{1}( , ) corresponding to the pure exact sequences.

Theorem. If R is a commutative ring, then a sequence 0 → A → B → C → 0 of R-modules is pure exact if and only if for every maximal ideal 𝔭, the induced sequence

0 ⟶ A_{𝔭} ⟶ B_{𝔭} ⟶ C_{𝔭} ⟶ 0

is pure exact.