A Criterion for the Equality of the Little and Big Finitistic Dimensions

Andrew W. Hubery

August 21, 2001

We fix some notation. If R is a ring, then R−Mod will be the category of all (left) R-modules. We identify right R-modules with left R^{op}-modules. Similarly, R−mod will be the full subcategory of R−Mod consisting of the modules of finite length.

If M ∈ R−Mod, then add(M) is the full subcategory of R−Mod consisting of those modules that are direct summands of M^{n} for some n. Similarly Add(M) has objects that are direct summands of M^{(I)} for any set I.

For modules M_{1}, …, M_{n} we define filt(M_{1}, …, M_{n}) to be the set of all modules X which admit a filtration

X = X_{0} ⊃ X_{1} ⊃ ⋯ ⊃ X_{m} = 0

with each successive factor X_{i}/X_{i+1} in the set {M_{1}, …, M_{n}}.

1 Purity

In this section we review some results on pure-exact sequences and pure-injective modules. We shall be working over an arbitrary ring R, and our modules will be in R−Mod.

Theorem 1.1 The following are equivalent for an exact sequence η : 0 → M′ → M → M′′ → 0.

1. For any finitely presented module E we have an exact sequence

0 → Hom(E, M′) → Hom(E, M) → Hom(E, M′′) → 0;

2. For any (finitely presented) right module E we have an exact sequence

0 → E ⊗_{R} M′ → E ⊗_{R} M → E ⊗_{R} M′′ → 0;

3. For any finite matrix (r_{ij}) over R and elements m′_{i} ∈ M′, the system of linear equations

∑_{j} r_{ij}x_{j} = m′_{i}

has a solution over M iff it has a solution over M′;

4. η is a direct limit of split-exact sequences.

Any such sequence will be called pure-exact.

Let k be a commutative ring and R a k-algebra. If I is an injective cogenerator for k−Mod, then ^{∗} := Hom_{k}(−, I) defines a functor from R−Mod → R^{op}−Mod. Corollary 1.2 An exact sequence 0 → K → L → M → 0 of R-modules is pure-exact iff the induced exact sequence 0 → M^{∗} → L^{∗} → K^{∗} → 0 is split-exact. Corollary 1.3 The natural monomorphism ϕ_{M} : M → M^{∗∗}, ϕ_{M}(m)(f) := f(m), is pure.

We now consider those modules that are injective with respect to pure-exact sequences. Theorem 1.4 The following are equivalent for an R-module A:

1. Given any pure-exact sequence 0 → M′ → M → M′′ → 0 we have an exact sequence 0 → Hom(M′′, A) → Hom(M, A) → Hom(M′, A) → 0

2. Any pure-exact sequence 0 → A → M → M′′ → 0 splits. Any such module will be called pure-injective.

We also have the notion of algebraic compactness. We first need a definition.

A positive primitive formula ϕ with one free variable is a formula of the form

∃x_{1}, …, x_{n}  such that  ∑_{j} r_{ij}x_{j} = s_{i}y  for i = 1, …, m

with r_{ij}, s_{i} ∈ R.

Given an R-module M, the set

ϕ(M) := {y ∈ M | ϕ(y)}

is an additive subgroup of M, called a finitely definable subgroup.

The following formulas are immediate:

ϕ(⊕M_{i}) = ⊕ϕ(M_{i}), ϕ( ∏ M_{i}) = ∏ ϕ(M_{i}), ϕ(\underrightarrow{lim} M_{i}) = \underrightarrow{lim} ϕ(M_{i}). Also, it follows from condition 2. of Theorem 1.1 on pure-exact sequences that, for any pure-

submodule N of M,

ϕ(N) = N ∩ ϕ(M).

Clearly a finite intersection of finitely definable subgroups is again finitely definable. We call a subgroup definable if it can be written as an arbitrary intersection of finitely definable subgroups. Theorem 1.5 The following are equivalent for an R-module A:

1. For any row-finite matrix (r_{ij}) over R and elements a_{i} ∈ A, the system of linear equations

∑_{j} r_{ij}x_{j} = a_{i}

has a solution over A iff each finite subsystem has a solution.

2. Let B_{i} be (finitely) definable subgroups of A and let a_{i} ∈ A. If the family {a_{i} + B_{i}} has the finite intersection property, then the intersection ⋂_{i}(a_{i} + B_{i}) is non-empty.

Any such module will be called algebraically compact.

Note that any direct summand of an algebraically compact module also satisfies condition 1., and hence is itself algebraically compact.

A compact topological R-module is a module M with a compact Hausdorff topology such that addition and multiplication by any r ∈ R are continuous. Let ℝ be the real numbers with the usual toplogy, and set 𝕋 := ℝ/ℤ with the quotient topology. Then 𝕋 is a compact topological ℤ-module. The set M^{∗} = Hom_{ℤ}(M, 𝕋) is a closed subset of the compact set 𝕋^{M}, and hence is itself a compact ℤ-module. In fact, M^{∗} is compact as an R^{op}-module.

Using Corollary 1.3, we see that we can embed any module as a pure-submodule of a compact module. Since compact implies algebraically compact, this proves half of the following theorem.

Theorem 1.6 A module A is pure-injective iff it is algebraically compact.

We also have

Theorem 1.7 The endomorphism ring of an indecomposable pure-injective module A is local.

A module A is called Σ-pure-injective if, for any set I, the direct sum of I copies of A, A^{(I)}, is pure-injective.

Theorem 1.8 The following are equivalent for an R-module A:

1. A is Σ-pure-injective.

2. 0 → A^{(I)} → A^{I} → A^{I}/A^{(I)} → 0 is split-exact for all I.

3. A satisfies the descending chain condition on finitely definable subgroups.

Since ϕ(N) = N ∩ϕ(M) for any positive primitive ϕ and any pure-submodule N of M, condition 3. implies

Corollary 1.9 If A is Σ-pure-injective, then A^{(I)}, A^{I} and any pure-submodule will also be Σ-pure-injective. In particular, any pure-submodule will be a direct summand.

Corollary 1.10 Let 0 ≠ A be Σ-pure-injective. Then A can be written as a direct sum of indecomposable submodules. Each of these must be Σ-pure-injective, and so has local endomorphism ring. In fact, this condition is equivalent to being Σ-pure-injective.

Finally, an R-module M is called endofinite if it has finite length over its endomorphism ring End_{R}(M). Note that for an Artin algebra Λ and a module M ∈ Λ−mod, each term in the socle series for M is an End_{Λ}(M)-submodule. Using this, we see that any Λ-module of finite length is endofinite.

It is clear from the definition that each finitely definable subgroup of a module M is an End_{R}(M)- submodule. Therefore if M is endofinite, then it has both a.c.c. and d.c.c. on finitely definable submodules. In particular, M must be Σ-pure-injective.

2 Contravariant Finiteness

In this section, we recall the notion of contravariant finiteness for a subcategory X of Λ−mod, where Λ is an Artin algebra. We require that X is a full subcategory, closed under isomorphisms and direct summands.

Let M ∈ Λ−mod, X ∈ X and f : X → M a morphism. Then f is a right X -approximation of M if Hom_{X} (−, X) → Hom_{Λ}(−, M) is surjective. We call f a minimal right X -approximation if f is a right minimal morphism.

Let f : X → M be a right X -approximation and f_{0} : X_{0} → M a right minimal version of f. It follows that f_{0} is also a right X -approximation of M. Thus if M has a right X -approximation, then it has minimal approximations and all such are isomorphic.

The category X is called contravariantly finite in Λ−mod if every module M ∈ Λ−mod has a right X -approximation.

If X is contravariantly finite in Λ−mod, then X contains a (relative) injective cogenerator. For, let I be the minimal injective cogenerator in Λ−mod and let f : J → I be its minimal right X -approximation.

To show that J is a cogenerator, consider X ∈ X. If x ∈ X is non-zero, then there exists a morphism θ : X → I such that θ(x) ≠ 0. By definition, θ = fϕ for some ϕ : X → J, and ϕ(x) ≠ 0.

Similarly, consider a monomorphism g : J → X for some X ∈ X. Since I is injective, we can lift f : J → I to a map h giving a commutative diagram

\xymatrix{0 \ar[r] & J \ar[r]^{g} \ar[d]^{f} & X \ar[dl]^{h} \\  & I}

Again, h = fw for some w : X → J, and since f = hg = fwg, the minimality of f implies that wg is an isomorphism. Therefore J must be a relative injective.

3 The Main Theorem

From now on, we work over an Artin algebra Λ. We denote by P^{∞}(Λ−mod) the set of all modules M ∈ Λ−mod of finite projective dimension. Then the little finitistic dimension of Λ is defined to be

fin.dim.(Λ) := sup{pd(M) | M ∈ P^{∞}(Λ−mod)}.

Similarly, we define P^{∞}(Λ−Mod) and the big finitistic dimension Fin.dim.(Λ).

Note that P^{∞}(Λ−mod) is closed under isomorphisms, extensions, kernels of epimorphisms and contains all the projectives, so is a resolving subcategory. The following theorem was proved in an earlier lecture.

Theorem 3.1 Let P^{∞}(Λ−mod) be contravariantly finite in Λ−mod, and write M_{1}, …, M_{n} for the minimal P^{∞}-approximations of the simple modules. Then

fin.dim.(Λ) = max{pd(M_{i})}  and  P^{∞}(Λ−mod) = add filt(M_{1}, …, M_{n}).

We shall extend this result to

Theorem 3.2 Let P^{∞}(Λ−mod) be contravariantly finite in Λ−mod. Then the little and big finitistic dimensions are equal. Also, for M_{1}, …, M_{n} as above,

P^{∞}(Λ−Mod) = \underrightarrow{lim} filt(M_{1}, …, M_{n}),

Here, for a subcategory X ⊂ Λ−mod, we write \underrightarrow{lim} X for the closure of X under direct limits (in Λ−Mod).

The proof is based on the paper [9] by B. Huisgen-Zimmermann and S.O. Smalø. We shall first need a few preliminary results.

Lemma 3.3 Let R be any ring, and M a Σ-pure-injective R-module. Then the category Add(M) is closed under direct limits.

Proof. Consider a directed family of modules (N_{i}, f_{ij}) with N_{i} ∈ Add(M). Let N denote the direct limit and write g_{i} : N_{i} → ⨁ N_{i} for the canonical morphisms. The exact sequence

η : 0 → K := ∑_{i≤j}(g_{j}f_{ij} − g_{i})(N_{i}) → ⨁ N_{i} → N → 0

is pure-exact, since we can write it as the direct limit of the pull-back sequences

[figure: 0; K; L_{j}; N_{j}; 0; 0; K; ⨁ N_{i}; N; 0]

The map N_{j} → N clearly factors through N_{j} → ⨁ N_{i} and so by the Homotopy Lemma, the pull-back sequences are split-exact.

Now, since M is Σ-pure-injective, so is ⨁ N_{i} and thus η is split-exact. As each Σ-pure-injective is a direct sum of indecomposables with local endomorphism rings, the Krull-Remak-Schmidt-Azumaya Theorem applies, giving N ∈ Add(M). □

Lemma 3.4 Let R be left Noetherian and M a finitely generated Σ-pure-injective R-module. If all monomorphisms in add(M) split, then the same is true in Add(M).

Proof. Let f : K → L be a monomorphism with K and L both in Add(M). We can write K = ⨁ K_{i}, L = ⨁ L_{j} with each K_{i}, L_{j} a direct summand of M, so finitely generated. Thus for each finite sum of the K_{i}, the image is contained in a finite sum of the L_{j}, and so is split by assumption. Since f is the direct limit of all such restrictions, it is a pure-monomorphism. Since L is Σ-pure-injective, K is a direct summand. Also note that the cokernel is in Add(M), using the Krull-Remak-Schmidt-Azumaya Theorem as before. □

In particular, we shall use these lemmas when M = I is a relative injective cogenerator inside P^{∞}(Λ−mod).

The next result shows that the big finitistic dimension of Λ is attained by looking at the countably generated modules of finite projective dimension.

Proposition 3.5 Let R be left Noetherian. Then each module M is a direct limit of countably generated submodules, each having projective dimension bounded above by that of M.

Proof. Let pd(M) = n. Since M is clearly a direct limit of countably generated submodules, it is enough to show that each countably generated submodule C is contained in one with projective dimension at most n.

Let

\xymatrix{0 \ar[r] & P^{n} \ar[r]^{f^{n}} & P^{n−1} \ar[r]^{f^{n−1}} & ⋯ \ar[r] & P^{1} \ar[r]^{f^{1}} & P^{0} \ar[r]^{f^{0}} & M \ar[r] & 0}

be a projective resolution of M. By Kaplansky’s Theorem, we can write

P^{i} = ⨁_{j∈A^{i}} P_{j}^{i}

with each P_{j}^{i} countably generated.

We shall abuse notation and write f^{i}|_{B^{i}} for f^{i} restricted to ⨁_{j∈B^{i}} P_{j}^{i}, for any subset B^{i} ⊂ A^{i}.

Given C as above, we choose a countable subset B^{0}(1) ⊂ A^{0} such that Im(f^{0}|_{B^{0}(1)}) ⊃ C. Since R is Noetherian, the kernel is again countably generated, so we can take B^{1}(1) ⊂ A^{1} such that Im(f^{1}|_{B^{1}(1)}) ⊃ Ker(f^{0}|_{B^{0}(1)}). Similarly, we pick B^{i}(1) for each i.

Now set B^{n}(2) = B^{n}(1). Since Im f^{n}|_{B^{n}(2)} is countably generated, we can find B^{n−1}(2) ⊃ B^{n−1}(1) such that Im(f^{n}|_{B^{n}(2)}) ⊂ Ker(f^{n−1}|_{B^{n−1}(2)}). Similarly, we define B^{i}(2).

We can now put B^{0}(3) = B^{0}(2) and repeat. In this manner, we form a chain of countable subsets for each i

B^{i}(1) ⊂ B^{i}(2) ⊂ ⋯ ⊂ A^{i}

such that, for odd k,

Ker(f^{i}|_{B^{i}(k)}) ⊂ Im(f^{i+1}|_{B^{i+1}(k)}),

whereas for even k

Ker(f^{i}|_{B^{i}(k)}) ⊃ Im(f^{i+1}|_{B^{i+1}(k)}).

Now set B^{i} := ⋃ B^{i}(k) and define M′ to be the countably generated submodule given by Im(f^{0}|_{B^{0}}). Note that

\xymatrix{0 \ar[r] & ⨁_{j∈B^{n}} P_{j}^{n} \ar[r] & ⋯ \ar[r] & ⨁_{j∈B^{0}} P_{j}^{0} \ar[r] & M′ \ar[r] & 0}

is a projective resolution of M′. □

Corollary 3.6 The big finitistic dimension of Λ equals

sup{pd(M) | M ∈ P^{∞} and M countably generated}.

Proof. Let pd(N) = n. Then we can write N = \underrightarrow{lim} N_{i} with each N_{i} countably generated and pd(N_{i}) ≤ n. Since the projective and weak dimensions are the same, and since Tor commutes with direct limits, we see that pd(N) = sup{pd(N_{i})}. □

4 The Main Proof

We are now in a position to prove Theorem 3.2 , namely that when P^{∞}(Λ−mod) is contravariantly finite in Λ−mod, the little and big finitistic dimensions are equal. From above, it is enough to show this for countably generated modules.

Let M be countably generated with pd(M) = n. We can find a chain

M_{1} ⊂ M_{2} ⊂ ⋯ ⊂ M

of finitely generated submodules such that M = ⋃_{i} M_{i}.

For each i, we fix the beginning of a projective resolution

\xymatrix{0 \ar[r] & Ω_{i}^{n} \ar[r] & P_{i}^{n−1} \ar[r] & ⋯ \ar[r] & P_{i}^{1} \ar[r] & P_{i}^{0} \ar[r] & M_{i} \ar[r] & 0}

Also, from the inclusions \xymatrix{f_{i,i+1} : M_{i} \ar[r] \ar[r] & M_{i+1}} , we fix chain morphisms

\xymatrix{0 \ar[r] & Ω_{i}^{n} \ar[r] \ar[d]^{f_{i,i+1}^{n}} & P_{i}^{n−1} \ar[r] \ar[d]^{f_{i,i+1}^{n−1}} & ⋯ \ar[r] & P_{i}^{1} \ar[r] \ar[d]^{f_{i,i+1}^{1}} & P_{i}^{0} \ar[r] \ar[d]^{f_{i,i+1}^{0}} & M_{i} \ar[r] \ar[d]^{f_{i,i+1}} & 0 \\ 0 \ar[r] & Ω_{i+1}^{n} \ar[r] & P_{i+1}^{n−1} \ar[r] & ⋯ \ar[r] & P_{i+1}^{1} \ar[r] & P_{i+1}^{0} \ar[r] & M_{i+1} \ar[r] & 0}

For i < j we set f_{i,j} := f_{j−1,j} ⋯ f_{i,i+1}, and similarly f_{i,j}^{k} := f_{j−1,j}^{k} ⋯ f_{i,i+1}^{k}. This gives a directed system indexed by ℕ.

Consider the direct limit

\xymatrix{0 \ar[r] & \underrightarrow{lim} Ω_{i}^{n} \ar[r] & P^{n−1} \ar[r] & ⋯ \ar[r] & P^{1} \ar[r] & P^{0} \ar[r] & M \ar[r] & 0}

Since Λ is an Artin algebra, projective equals flat and so each P^{k} is projective. Since pd(M) = n, the module P^{n} := \underrightarrow{lim} Ω_{i}^{n} is also projective. Clearly each of these modules is countably generated. Thus we can write

P^{n} = ⨁_{j∈ℕ} Q_{j},

where the Q_{j} are finitely generated indecomposable projectives.

We want to approximate the M_{i} by finitely generated modules of finite projective dimension, and we do this by approximating each of the syzygies Ω_{i}^{n}, …, Ω_{i}^{0} = M_{i} in turn.

Let g_{i} : Ω_{i}^{n} → P^{n} denote the canonical morphisms. As ⨁_{j≤m} Q_{j} is finitely generated, it must lie in the image of some g_{i_{m}}. By taking a cofinite subsystem of the M_{i}, we may assume that i_{m} = m for all m.

Now, ⨁_{j≤m} Q_{j} is projective, so the map

g_{m}^{−1}( ⨁_{j≤m} Q_{j}) → ⨁_{j≤m} Q_{j}

must be a split-epimorphism. Therefore we can take a submodule P_{m}^{n} ≅ ⨁_{j≤m} Q_{j} of Ω_{m}^{n}. Also,

g_{m}(P_{m}^{n}) = g_{m+1}f_{m,m+1}^{n}(P_{m}^{n}) = ⨁_{j≤m} Q_{j},

so that f_{m,m+1}^{n} restricted to P_{m}^{n} is a monomorphism. Denote the image by P_{m}^{n} as well. If we consider g_{m+1}^{−1}(Q_{m+1}) → Q_{m+1}, then we can find R_{m+1} ⊂ Ω_{m+1}^{n} isomorphic to Q_{m+1}. Now g_{m+1} restricts to an epimorphism P_{n,m} + R_{m+1} → ⨁_{j≤m+1} Q_{j}, which, by length considerations, must be an isomorphism. Hence the sum on the left is direct. By induction on m, we can therefore choose the P_{m}^{n} = ⨁_{j≤m} P_{m,j}^{n} such that P_{m,j}^{n} ≅ Q_{j} and f_{m,m+1}^{n} induces isomorphisms P_{m,j}^{n} → P_{m+1,j}^{n}. That is, we have

[figure: ⋯; P_{m}^{n}; P_{m+1}^{n}; ⋯; \underrightarrow{lim} P_{m}^{n} = P^{n}; ⋯; Ω_{m}^{n}; Ω_{m+1}^{n}; ⋯; \underrightarrow{lim} Ω_{m}^{n} = P^{n}]

Let I be a relative injective cogenerator for P^{∞}. Note that I is Σ-pure-injective and all monomorphisms in add(I) split. Therefore Add(I) is closed under direct limits and all monomorphisms split.

Lemma 4.1 Suppose that we are given the exact commutative diagram

[figure: 0; X; β; P; α; Y; 0; Q; g; 0; X′; β′; f_{X}; P′; α′; f_{P}; Y′; f_{Y}; 0; Q′; g′; h_{Q}]

with Q, Q′, P, P′ ∈ P^{∞}. Then we can complete it to an an exact commutative diagram

[figure: 0; X; P; Y; 0; 0; Q; P ⊕ I_{0}; Z; 0; 0; X′; P′; Y′; 0; 0; Q′; P′ ⊕ I′_{0}; Z′; 0]

with I_{0}, I′_{0} ∈ add(I). Also note that Z, Z′ ∈ P^{∞} and the maps Z → Y and Z′ → Y′ are onto.

Proof. We have monomorphisms

γ : Q → I_{0}, γ′ : Q′ → I′_{0}

for some I_{0}, I′_{0} ∈ add(I). Thus the diagram below is commutative:

\xymatrix{0 \ar[r] & Q \ar[r]^{^{( \begin{matrix}βg \\ γ \end{matrix} )}} \ar[d]^{g} & P ⊕ I_{0} \ar[r] \ar[d]^{_{(_{1 0} )}} & Z \ar[r] \ar[d] & 0 \\ 0 \ar[r] & X \ar[r]^{β} & P \ar[r]^{α} & Y \ar[r] & 0}

Similarly for the bottom plane.

Since I_{0} is a relative injective module, we can find a map h making the following diagram commute:

\xymatrix{0 \ar[r] & Q′ \ar[r]^{γ′} \ar[d]_{γh_{Q}} & I′_{0} \ar@{-->}[dl]^{h} \\  & I_{0}}

The map

(\begin{matrix}f_{P} \\ & h \end{matrix}) : P′ ⊕ I′_{0} → P ⊕ I_{0}

makes the front left square commute and so, by diagram chasing, the whole diagram is commutative. □

We now apply the lemma, first to the diagrams

[figure: 0; P_{i+1}^{n−1}/Ω_{i+1}^{n}; P_{i+1}^{n−2}; Ω_{i+1}^{n−2}; 0; P_{i+1}^{n−1}/P_{i+1}^{n}; 0; P_{i}^{n−1}/Ω_{i}^{n}; f_{i,i+1}^{n−1}; P_{i}^{n−2}; f_{i,i+1}^{n−2}; Ω_{i}^{n−2}; 0; P_{i}^{n−1}/P_{i}^{n}]

obtaining the cokernels Z_{i+1}^{n−2}, Z_{i}^{n−2}, and then inductively to the diagrams

[figure: 0; Ω_{i+1}^{k}; P_{i+1}^{k−1}; Ω_{i+1}^{k−1}; 0; Z_{i+1}^{k}; 0; Ω_{i}^{k}; P_{i}^{k−1}; Ω_{i}^{k−1}; 0; Z_{i}^{k}]

for n − 2 ≥ k ≥ 1.

In this way, we construct a directed system of commutative diagrams indexed by ℕ:

[figure: 0; 0; 0; 0; 0; 0; Ω_{i}^{n}; P_{i}^{n−1}; P_{i}^{n−2}; ⋯; P_{i}^{1}; P_{i}^{0}; M_{i}; 0; 0; P_{i}^{n}; P_{i}^{n−1}; P_{i}^{n−2} ⊕ I_{i}^{n−2}; _{(_{1 0} )}; ⋯; P_{i}^{1} ⊕ I_{i}^{1}; _{(_{1 0} )}; P_{i}^{0} ⊕ I_{i}^{0}; _{(_{1 0} )}; N_{i}; 0; 0; I_{i}^{n−2}; _{( \begin{matrix}0 \\ 1 \end{matrix} )}; ⋯; I_{i}^{1}; _{( \begin{matrix}0 \\ 1 \end{matrix} )}; I_{i}^{0}; _{( \begin{matrix}0 \\ 1 \end{matrix} )}; K_{i}; 0; 0; 0; 0; 0]

Note that the top two rows are exact, as are all the columns. The third row is not exact, although it is a complex.

If we now pass to the direct limit, we have the commutative diagram

[figure: 0; 0; 0; 0; 0; 0; P^{n}; P^{n−1}; P^{n−2}; ⋯; P^{1}; P^{0}; M; 0; 0; P^{n}; P^{n−1}; P^{n−2} ⊕ I^{n−2}; ⋯; P^{1} ⊕ I^{1}; P^{0} ⊕ I^{0}; N; 0; 0; I^{n−2}; ⋯; I^{1}; I^{0}; K; 0; 0; 0; 0; 0]

Again, the top two rows and all the columns are exact.

Using this, consider

\xymatrix{0 \ar[r] & I^{n−2} \ar[r] \ar[d] & P^{n−2} ⊕ I^{n−2} \ar[r] \ar[d] & P^{n−2} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I^{n−3} \ar[r] & P^{n−3} ⊕ I^{n−3} \ar[r] & P^{n−3} \ar[r] & 0}

Then the Snake Lemma says that the kernels and cokernels of the vertical maps fit into an exact sequence

\xymatrix{0 \ar[r] & U^{n−2} \ar[r] & P^{n−1}/P^{n} \ar[r] & P^{n−1}/P^{n} \ar[r] & V^{n−3} \ar[r] & W^{n−3} \ar[r] & Ω^{n−3} \ar[r] & 0}

where

U^{k} := Ker(I^{k} → I^{k−1}), V^{k−1} := Coker(I^{k} → I^{k−1}), W^{k} := Ker(P^{k} ⊕ I^{k} → P^{k−1} ⊕ I^{k−1}).

Thus U^{n−2} = 0 and 0 → V^{n−3} → W^{n−3} → Ω^{n−3} is a short exact sequence.

Similarly, for the (n − 3)-rd and (n − 4)-th rows, we have an exact sequence

\xymatrix{0 \ar[r] & U^{n−3} \ar[r] & W^{n−3} \ar[r] & Ω^{n−3} \ar[r] & V^{n−4} \ar[r] & W^{n−4} \ar[r] & Ω^{n−4} \ar[r] & 0}

Thus U^{n−3} = V^{n−3} and 0 → V^{n−4} → W^{n−4} → Ω^{n−4} is short exact.

In this way, we see that the third row is also exact.

Now, as each I_{i}^{k} ∈ add(I), we have that I^{k} ∈ Add(I). Since all monomorphisms in Add(I) split, K ∈ Add(I) also. Also, N_{i} ∈ P^{∞}, so we can find monomorphisms N_{i} → J_{i} for some J_{i} ∈ add(I). Using the relative injectivity of I, we obtain commutative squares

\xymatrix{N_{i+1} \ar@{^{(}->}[r] & J_{i+1} \\ N_{i} \ar[u] \ar@{^{(}->}[r] & J_{i} \ar[u]}

Thus N := \underrightarrow{lim} N_{i} embeds in J := \underrightarrow{lim} J_{i} ∈ Add(I).

Since 0 → K → J must split, so does 0 → K → N. Therefore M is a direct summand of N and hence

pd(M) ≤ pd(N) ≤ sup{pd(N_{i})} ≤ fin.dim.(Λ).

Since P^{∞}(Λ−mod) = add filt(M_{1}, …, M_{n}), where the M_{i} are the minimal approximations of the simples, we see that P^{∞}(Λ−Mod) = add \underrightarrow{lim} filt(M_{1}, …, M_{n}). However, since any category closed under direct limits must also be closed under direct summands, we get

P^{∞}(Λ−Mod) = \underrightarrow{lim} filt(M_{1}, …, M_{n}).

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