THE DIFFERENCE BETWEEN THE LITTLE AND THE BIG FINITISTIC DIMENSIONS

KENJI LEFÈVRE-HASEGAWA

Abstract. This note is an expanded version of the article of Sverre O. Smalø “The supremum of the difference between the big and the little finitistic dimensions is infinite” [3].

1. Introduction

Our aim is to show that, in general, the little and the big (left) finitistic dimensions are different, and that their difference can be arbitrarily large. For an overview on the finitistic dimensions conjectures we refer to [6].

All modules, unless otherwise stated, are left modules, ModΛ denotes the category of modules over Λ and modΛ the category of modules of finite type. Recall that the left little finitistic dimension fin.dimΛ of an algebra Λ is defined as

fin.dimΛ = sup{p.dimM|M ∈ modΛ with p.dimM < ∞}

and the big little finitistic dimension Fin.dimΛ as

Fin.dimΛ = sup{p.dimM|M ∈ ModΛ with p.dimM < ∞}.

Put δ(A) = Fin.dimΛ − fin.dimΛ. We will give two different proofs of the fact that this difference can be arbitrarily large:

• The first approach is based on work by Birge Huisgen-Zimmermann: In her paper [5], she showed that for monomial relation algebras Λ, we have δ(Λ) ≤ 1 and exhibited a monomial relation algebra Δ such that δ(Δ) = 1. This was the first counterexample to Bass’ conjecture which claimed that δ(Λ) vanishes for all finite-dimensional algebras Λ.

Rickard has shown that both finitistic dimensions are additive with respect to tensor products of algebras (if the ground field is perfect). Therefore δ is additive as well and we have δ(Δ^{⊗n}) = n. So the difference between the two dimensions can indeed be arbitrarily large.

Date: August 2001.

1991 Mathematics Subject Classification. 16E10, 16G20, 16P10 .

Key words and phrases. Finitistic dimension, Finite dimensional algebra.

• The second approach is direct: We construct a sequence of algebras Λ_{n} such that δ(Λ_{n}) = n − 1. The computation of the big finitistic dimension will be based on the fact that for n ≥ 3, Λ_{n} is a one point extension of Λ_{n−1} by an injective module.

The algebra Λ_{n} has a simpler structure than Δ^{⊗n}: Indeed the cube of the radical of Λ_{n} vanishes and its little finitistic dimension equals 1. The algebra Δ^{⊗n} has little finitistic dimension equal to 2n and non vanishing radical cube even for n = 1. Indeed, it is known that for a monomial relation algebra with vanishing radical cube, the two dimensions coincide.

2. Additivity of the finitistic dimensions with respect to the tensor product of algebras

The algebras Δ_{n} will be constructed as tensor powers of the example of Huisgen-Zimmermann using the following theorem due to Jeremy Rickard.

Theorem 2.1. Let Λ_{1} and Λ_{2} be two finite dimensional algebras over a an algebraically closed field k. We have the following equalities:

(1) fin.dimΛ_{1} ⊗ Λ_{2} = fin.dimΛ_{1} + fin.dimΛ_{2}

(2) Fin.dimΛ_{1} ⊗ Λ_{2} = Fin.dimΛ_{1} + Fin.dimΛ_{2}

Corollary 2.2. The supremum of the difference between the big and little finitistic dimensions is infinite.

Proof of the corollary

The algebra Δ_{n} = Δ^{⊗n} has little finitistic dimension and big finitistic dimension differing by n.

Proof of the theorem

The facts from homological algebra used here can be found in [4].

Denote by p^{Δ}U a projective resolution of a Δ-module U and by Λ the tensor product algebra Λ_{1} ⊗ Λ_{2}.

Recall the following facts :

(1) An arbitrary module M over a finite dimensional algebra A has projective dimension less or equal than l if and only if Tor_{l+1}^{A}(M, S) vanishes for all simple A-modules S.

(2) Let A and B be two finite dimensional algebras over an algebraically closed field. The simple modules over A ⊗ B are isomorphic to S ⊗ T, where S are the simple A-modules and T the simple B-modules.

Suppose M is a Λ_{1}-module of projective dimension m and N is a Λ_{2}-module of projective dimension n.

For a Λ_{1}-module X_{1} and a Λ_{2}-module X_{2} , we have the isomorphism

Hom_{Λ}(p^{Λ_{1}}M ⊗p^{Λ_{2}}N, X_{1}⊗X_{2}) ≃ Hom_{Λ_{1}}(p^{Λ_{1}}M, X_{1})⊗Hom_{Λ_{2}}(p^{Λ_{2}}N, X_{2}),

(since it is an isomorphism for M = Λ_{1}, N = Λ_{2}) hence an isomorphism

Ext_{Λ}^{p}(M ⊗ N, X_{1} ⊗ X_{2}) ≃ ⊕_{i+j=p}Ext_{Λ_{1}}^{i}(M, X_{1}) ⊗ Ext_{Λ_{2}}^{j}(N, X_{2}).

This shows that the projective dimension of the Λ-module M ⊗ N is m + n, so we have the inequalities

(1) fin.dimΛ_{1} ⊗ Λ_{2} ≥ fin.dimΛ_{1} + fin.dimΛ_{2},

(2) Fin.dimΛ_{1} ⊗ Λ_{2} ≥ Fin.dimΛ_{1} + Fin.dimΛ_{2}.

Let fin.dimΛ_{1} = m, fin.dimΛ_{2} = n and X be a finite dimensional module over Λ. Since a projective Λ-resolution of p^{Λ}X → X restricts to a projective Λ_{1}-resolution, the Λ_{1}-restriction of the m’th-syzygy Ω = Ω_{m}X in the resolution pX is a projective Λ_{1}-module. Let us show that the Λ-module Ω has Λ-projective dimension n.

Let us compute Tor_{∗}^{Λ}(Ω, S ⊗ T) where S is a simple Λ_{1}-module and T a simple Λ_{2}-module.

Let S be a simple Λ_{1}-module and T be a simple Λ_{2}-module. A complex to compute the functors Tor_{∗}^{Λ}(Ω, S ⊗_{k} T) is given by

p^{Λ}Ω ⊗_{Λ_{1}⊗Λ_{2}} (p^{Λ_{1}}S ⊗_{k} p^{Λ_{2}}T)

which is trivially isomorphic to (p^{Λ}Ω ⊗_{Λ_{1}} p^{Λ_{1}}S) ⊗_{Λ_{2}} p^{Λ_{2}}T. Let us show that the last one is quasi-isomorphic to

(p^{Λ}Ω ⊗_{Λ_{1}} S) ⊗_{Λ_{2}} p^{Λ_{2}}T.

Let V be a Λ_{2}-module and i ≥ 0. We deduce from the isomorphism

((p^{Λ}Ω)_{i} ⊗_{Λ_{1}} S) ⊗_{Λ_{2}} V ≃ (p^{Λ}Ω)_{i} ⊗_{Λ_{1}⊗Λ_{2}} (S ⊗_{k} V)

that the components (p^{Λ}Ω)_{i} ⊗_{Λ_{1}} S of the Λ_{2}-complex p^{Λ}Ω⊗_{Λ_{1}} S are flat Λ_{2}-modules (of finite type) and hence projective. Thus p^{Λ}Ω ⊗_{Λ_{1}} S is a projective Λ_{2}-resolution of Ω ⊗ S and is then homotopically equivalent to the complex p^{Λ}Ω ⊗_{Λ_{1}} p^{Λ_{1}}S as Λ_{2}-modules. Tensorising over Λ_{2} by p^{Λ_{2}}T, this gives us the announced isomorphism above. Since the complex (p^{Λ}Ω⊗_{Λ_{1}} S)⊗_{Λ_{2}} p^{Λ_{2}}T computes the derived functors Tor_{∗}^{Λ_{2}}(Ω⊗_{Λ_{1}} S, T), this shows that Tor_{i}^{Λ_{2}}(Ω⊗_{Λ_{1}} S, T) ≃ Tor_{i}^{Λ}(Ω, S⊗_{k}T) and hence we find that this functor vanishes for i = n. So the m’th syzygy of the Λ-module X has projective dimension at most n and hence p.dim^{Λ}(X) ≤ m + n. The assertion 1. of the theorem is proven.

The main ingredient for the above proof is that projective and flat finite dimensional modules coincide. For infinite dimensional modules, we know that, for artinian algebras, flatness is equivalent to projectivity, see for example [1]. Since finite dimensional algebras are artinian, we argue as above to prove the assertion 2.

3. The example

For each n > 1 we construct an algebra Λ_{n} whose little and big finitistic dimensions differ at least by n − 1.

The algebra Λ_{n} will be given as the path algebra of a quiver modulo relations. We will obtain a lower bound for the big finitistic dimension using the fact that Λ_{n} is an iterated one-point-extension by injective modules.

Let k be any field and Γ_{n} be the quiver

[figure: n; ρ_{n}; σ_{n}; τ_{n}; n − 1; ρ_{n−1}; σ_{n−1}; τ_{n−1}; n − 2; …; 3; ρ_{3}; σ_{3}; τ_{3}; 2; ρ_{2}; σ_{2}; τ_{2}; 1; ρ_{1}; σ_{1}; τ_{1}; 0 .; α; β]

Let Λ_{n} be the path algebra of Γ_{n} over the field k modulo the ideal generated by the following relations : α^{2}, β^{2}, αβ, βα, αρ_{1}, ασ_{1}, βτ_{1}, x_{i}y_{i+1} for i = 1, …, n−1 and x = y and x_{i}x_{i+1} −y_{i}y_{i+1}, where x, y ∈ {σ, τ, ρ} and i = 1, …, n − 1. Let C be the path category, i.e. the category obtained from the quiver Γ_{n} the following way: A path from i to j in Γ is a formal composition of arrows starting in i and ending in j. If i = j, we also admit the zero length path. The category C̃ has the vertices of Γ as its objects; the set of morphisms from i to j is the vector space whose basis is the set of paths from i to j; the composition is induced by the composition of paths. The category C is the quotient of C̃ by the ideal generated by the relations. The category of left modules over the path algebra Λ_{n} is equivalent to the category of functors C → Vect k where Vectk is the category of k-vector spaces.

Theorem 3.1. For each natural number n ≥ 1, the algebra Λ_{n} given above has

(1) left little finitistic dimension equal to 1,

(2) left big finitistic dimension equal to n.

Reminder on projective modules and minimal resolutions(for details see [2])

Each vertex i of the quiver yields a minimal idempotent e_{i} in the path algebra. This idempotent corresponds to the identity of i in C. By the Yoneda lemma, the functor i ↦ P_{i} = Λ_{n}e_{i} ≃ Hom_{C}(i,?) is an equivalence from the category C^{op} to the category of indecomposable projective left modules. The simple left module S_{i} = P_{i}/𝔯P_{i} ≃ (where 𝔯 denotes the radical of the path algebra Λ_{n} whose generators are the arrows of the quiver Γ_{n}) has projective cover P_{i}. The socle socM of a module M is the biggest semi-simple submodule of M. The Loewy length of a module M is the minimal number of extensions of semisimple modules needed to obtain M. Since the biggest semi-simple quotient of 𝔯^{i}M is 𝔯^{i}M/𝔯^{i+1}M, the Loewy length can also be viewed as the first exponent n for which 𝔯^{n}M equals zero. Note that 𝔯^{n−1}M is contained in the socle of M. A minimal projective resolution of a module M is an exact sequence

0 → Q_{m} → Q_{m−1} → … → Q_{1} → Q_{0} → M → 0

where the Q_{j} are projective modules and such that any other projective resolution admits a epimorphism onto it. Hence minimal resolution are unique up to isomorphisms. By minimality, the module Q_{i−1} is a projective cover of kerd_{i−2} (i.e. the morphism Q_{i−1} ↠ kerd_{i−2} restricts to an isomorphism Q_{i−1}/𝔯Q_{i−1} ≃ kerd_{i−2}/𝔯kerd_{i−2}) and, thus, the composition Q_{i} → Q_{i−1} → Q_{i−1}/𝔯Q_{i−1} equals zero because the composition

Q_{i} → Q_{i−1} → kerd_{i−2} → kerd_{i−2}/𝔯kerd_{i−2}

equals also zero. Now suppose that, in a resolution, each morphism d_{i} : Q_{i} → Q_{i−1} factors through 𝔯Q_{i−1}. The projective Q_{i−1} and the cokernel of d_{i} must then have isomorphic biggest semi-simple quotient, hence Q_{i−1} is the projective cover of cokerd_{i}. This gives us a minimality caracterisation : namely, each morphism d_{i} : Q_{i} → Q_{i−1} factors through 𝔯Q_{i−1}.

Computation of the left little finitistic dimension

All modules considered during this computation are of finite type. We will show in the next paragraph that the syzygy of a module can not have projective dimension one. Suppose that this has been proven and that we have a minimal resolution of a module M

0 → Q_{m} → Q_{m−1} → … → Q_{1} → Q_{0} → M → 0

with m > 1. For each n > 1, the cokernel of d_{n} is the syzygy of coker d_{n−1}. Since the resolution above is minimal, the sequence Q_{m} → Q_{m−1} → cokerd_{m} is a minimal resolution of cokerd_{m}. This contradicts the fact that it can not have projective dimension 1. Hence a minimal resolution of a non projective M of finite projective dimension should be of the form Q_{1} → Q_{0} → M. This proves the first statement of the theorem.

The Loewy length of the projective module P_{0} equals 2 : the socle of P_{0} is S_{0} ⊕ S_{0} generated as k-vector space by α and β. The quotient of P_{0} by its socle is the simple module S_{0} = P_{0}/𝔯P_{0}. The Loewy length of P_{1} is 3 :

P_{1} ≃ ( \begin{matrix}P_{1}/𝔯P_{1} \\ 𝔯P_{1}/𝔯^{2}P_{1} \\ 𝔯^{2}P_{1} \end{matrix} ) ≃ ( \begin{matrix}S_{1} \\ S_{0} ⊕ S_{0} ⊕ S_{0} \\ S_{0} ⊕ S_{0} ⊕ S_{0} \end{matrix} ) ≃ ( \begin{matrix}ke_{1} \\ kτ_{1} ⊕ kσ_{1} ⊕ kρ_{1} \\ kατ_{1} ⊕ kβρ_{1} ⊕ kβσ_{1} \end{matrix} ) .

The Loewy length of P_{i} for i > 1 equals 3 :

P_{i} ≃ ( \begin{matrix}S_{i} \\ S_{i−1} ⊕ S_{i−1} ⊕ S_{i−1} \\ S_{i−2} \end{matrix} ) ≃ ( \begin{matrix}ke_{i} \\ kτ_{i} ⊕ kσ_{i} ⊕ kρ_{i} \\ kσ_{i−1}σ_{i} \end{matrix} ) .

This shows that the Loewy length of any projective is at most 3. Suppose we have an inclusion of a projective P into the radical of another projective Q. Since the Loewy length of 𝔯Q is 2, and that if M ⊂ N then the Loewy length of N is at least that of M, the projective P needs must have Loewy length 2 and is then the direct sum of copies of the projective P_{0}. However, the only non zero morphisms of source P_{0} have targets P_{0}, P_{1} and P_{2}. So the inclusion P → Q factors through an inclusion

f = [ \begin{matrix}f_{0} \\ f_{1} \\ f_{2} \end{matrix} ] : P = P_{0}^{r} ↪ P_{0}^{u} ⊕ P_{1}^{v} ⊕ P_{2}^{w}

whose image is contained in the radical. Let us show that f_{1} is a monomorphism. The morphism f_{0} (resp. f_{2}) is a linear combination of the morphisms α^{∗} and β^{∗} (resp. of (σ_{1}σ_{2})^{∗}), so its image contained in 𝔯P_{0}^{u} (resp. in 𝔯^{2}P_{2}^{w}) is semi-simple. Hence f_{0} and f_{2} factor through the quotient P → P/𝔯P and the morphism f through P → P/𝔯P ⊕ P_{1}^{v}. Since it is an inclusion, the kernel of f_{1} : P_{0}^{r} → P_{1}^{v} has a zero intersection with 𝔯P which also happens to be the socle of P, so it is also zero and f_{1} is an inclusion. Now notice that dimαP_{0}^{r} is r and that the dimension of α𝔯P_{1}^{v} (which contains αImf_{1}) is v. So in order to have an inclusion f_{1}, we must have the inequality v ≥ r. Now, the socle of P_{1}^{v}, whose dimension is 3v, can not be contained in the image of socP_{0}^{r} (of dimension 2r). This implies that the Loewy length of the cokernel of f_{1} (and hence of f) is 3. Now recall that the radical of any projective has Loewy length equal to 2. This implies that no submodule of the radical of a projective (e.g. a syzygy) can have projective dimension 1.

Computation of the left big finitistic dimension

We denote the set of strictly positive integers by N. Let e_{0,i} be the element of P_{0}^{(N)} with e_{0} in the i’th place and 0 elsewhere. These elements form a set of Λ_{n}-generators of P_{0}^{(N)}. In the same way, define the elements e_{1,i} of P_{1}^{(N)}. Let ϕ_{1} : P_{0}^{(N)} → P_{1}^{(N)} be the morphism of Λ_{n}-modules defined by ϕ_{1}(e_{0,2i−1}) = τ_{1}e_{1,2i−1} + σ_{1}e_{1,i} and ϕ_{1}(e_{0,2i}) = τ_{1}e_{1,2i} + ρ_{1}e_{1,i}. The map ϕ_{1} sends the elements

αe_{0,2i−1} to ατ_{1}e_{1,2i−1},

αe_{0,2i} to ατ_{1}e_{1,2i},
βe_{0,2i−1} to βσ_{1}e_{1,i}
and βe_{0,2i} to βρ_{1}e_{1,i}.

The matrix of ϕ_{1} with respect to the generators e_{0,j} and e_{1,i} is :

[ \begin{matrix}τ_{1} + σ_{1} & ρ_{1} & 0 & 0 & 0 & 0 & 0 & … \\ 0 & τ_{1} & σ_{1} & ρ_{1} & 0 & 0 & 0 & … \\ 0 & 0 & τ_{1} & 0 & σ_{1} & ρ_{1} & 0 & … \\ 0 & 0 & 0 & τ_{1} & 0 & 0 & σ_{1} & … \\ 0 & 0 & 0 & 0 & τ_{1} & 0 & 0 & … \\ 0 & 0 & 0 & 0 & 0 & τ_{1} & 0 & … \\ 0 & 0 & 0 & 0 & 0 & 0 & τ_{1} & … \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

This shows that the morphism ϕ_{1} is injective. Its cokernel, denoted by X_{1}, is the module P_{1}^{(N)} divided by the relations :

ατ_{1}e_{1,i}, βσ_{1}e_{1,i}, βρ_{1}e_{1,i}, τ_{1}e_{1,2i−1} + σ_{1}e_{1,i}, τ_{1}e_{1,2i} + ρ_{1}e_{1,i}.

Hence the residue classes of e_{1,i} and τ_{1}e_{1,i} form a k-basis of the cokernel X_{1}. Clearly it is annihilated by the residue classes of α and β and its socle is the submodule generated by the τ_{1}e_{1,i} and is isomorphic to S_{0}^{(N)}. The quotient of X_{1} by its socle is isomorphic to S_{1}^{(N)}.

Let us analyze symmetry and repetition occurring in the quiver Λ_{n}. One can show that for n ≥ 3 the monomial relation algebra Λ_{n} is obtained from Λ_{n−1} as a one point extension by the injective envelope E of S_{n−2} considered as a Λ_{n−1}-module. This means that the algebra Λ_{n} is isomorphic to the 2 × 2-matrix algebra

Λ̃_{n} = [ \begin{matrix}Λ_{n−1} & E \\ 0 & k \end{matrix} ] .

The injective envelope E is the dual of the projective cover Hom(?, n−2) of the right module S_{n−2}^{∗}. We denote its k-generators by ρ_{n−1}^{o}, σ_{n−1}^{o}, τ_{n−1}^{o} and e_{n−1}^{o}. The isomorphism Λ_{n} → Λ̃_{n} associates the generator of the vector space k with the idempotent e_{n} and maps

ρ_{n} to ρ_{n−1}^{o},

σ_{n} to σ_{n−1}^{o},
τ_{n} to τ_{n−1}^{o}
and ρ_{n}ρ_{n−1} to e_{n−1}^{o}.

This permits us to make the following construction : suppose we have a morphism of Λ_{n−1}-modules P → C where P is projective and where the socle of C is S_{n−2}^{(N)}. Since E^{(N)} is also the injective envelope of S_{n−2}^{(N)}, the inclusion of S_{n−2}^{(N)} into E^{(N)} extends to another inclusion C ↪ E^{(N)}. In this way, one obtains morphisms of Λ_{n−1}-modules

P → C → E^{(N)}

and by composition

P → E^{(N)}.

Now we can construct a morphism of projective Λ_{n}-modules by replacing the module P with the column module

[ \begin{matrix}P \\ 0 \end{matrix} ]

(which is again projective over Λ_{n} !) and by replacing E with the projective Λ_{n}-module (isomorphic to P_{n})

[ \begin{matrix}E \\ k \end{matrix} ] .

One can apply the same construction to a morphism of Λ_{1}-modules P → C where P is projective and where the socle of C is S_{0}^{(N)} noting that the algebra Λ_{2} is the one point extension of Λ_{1} by the injective Λ_{1}/(α, β)-envelope of S_{0} considered as a Λ_{1}-module (here α and β denote their residue classes in Λ_{1}).

Now we apply the above construction to the morphism P_{1} → X_{1}. We get a monomorphism X_{1} → P_{2}. To see what is happening precisely let us analyze P_{2} in terms of extensions of simple modules. The socle of P_{2} is the k-vector space with basis σ_{1}σ_{2} and is isomorphic to S_{0}. The quotient of P_{2} by its socle has socle S_{1} ⊕ S_{1} ⊕ S_{1} generated as a vector space by the residue classes of σ_{2}, τ_{2}, ρ_{2}. The embedding of X_{1} into P_{2}^{(N)} sends the element

e_{1,i} to τ_{2}e_{2,i} − σ_{2}e_{2,2i−1} − ρ_{2}e_{2,2i}.

Its matrix is

[ \begin{matrix}τ_{2} − σ_{2} & 0 & 0 & 0 & 0 & 0 & 0 & … \\ −ρ_{2} & τ_{2} & 0 & 0 & 0 & 0 & 0 & … \\ 0 & −σ_{2} & τ_{2} & 0 & 0 & 0 & 0 & … \\ 0 & −ρ_{2} & 0 & τ_{2} & 0 & 0 & 0 & … \\ 0 & 0 & −σ_{2} & 0 & τ_{2} & 0 & 0 & … \\ 0 & 0 & −ρ_{2} & 0 & 0 & τ_{2} & 0 & … \\ 0 & 0 & 0 & −σ_{2} & 0 & 0 & τ_{2} & … \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

By the same kind of arguments as for ϕ_{1}, one can show that the cokernel of ϕ_{2} (denoted by X_{2}) has socle S_{1}^{(N)}. At the level of composition factors, one has the diagram

0 → X_{1} ≃ ( \begin{matrix}S_{1}^{(N)} \\ S_{0}^{(N)} \end{matrix} ) ⟶^{ϕ_{2}} P_{2}^{N)} ≃ ( \begin{matrix}S_{2} \\ S_{1}^{(3)} \\ S_{0} \end{matrix} )^{(N)} ↠ X_{2} ≃ ( \begin{matrix}S_{2}^{(N)} \\ S_{1}^{(N)} \end{matrix} ) → 0.

Now we can embed X_{2} into P_{3} using the same arguments as we did for the embedding of X_{1} into P_{2}. We get a morphism ϕ_{3} whose cokernel X_{3} has socle S_{1}^{(N)} and the quotient of X_{3} by its socle is isomorphic to S_{2}^{(N)}. This process can be repeated and then leads to an exact sequence of modules

0 → P_{0}^{(N)} → P_{1}^{(N)} → … → P_{n−1}^{(N)} → P_{n}^{(N)} → X_{n} → 0.

This sequence gives rise to a minimal projective resolution of X_{n} since for each i we have isomorphisms (P_{i})^{(N)}/𝔯(P_{i})^{(N)} ≃ X_{i}/𝔯X_{i} ≃ S_{i}^{(N)}. This shows that the big finitistic dimension of Λ_{n} is at least n.

To prove the other inequality, let us look at the algebra Λ̃_{n} corresponding to the paths algebra of the subquiver of Γ_{n}

\xymatrix{n \ar[r]^{ρ_{n}} \ar[r]^{σ_{n}} \ar[r]_{τ_{n}} & n − 1 \ar[r]^{ρ_{n−1}} \ar[r]^{σ_{n−1}} \ar[r]_{τ_{n−1}} & n − 2 & … & 3 \ar[r]^{ρ_{3}} \ar[r]^{σ_{3}} \ar[r]_{τ_{3}} & 2 \ar[r]^{ρ_{2}} \ar[r]^{σ_{2}} \ar[r]_{τ_{2}} & 1}

divided by the same relations (when they make sense). The restriction ˜ : ModΛ_{n} → ModΛ̃_{n} is exact, sends projectives to projectives and radical morphisms to radical morphisms. The restriction of a minimal projective resolution of a Λ_{n}-module M

pM_{∗} = (… → Q_{n+1} → Q_{n} ⟶^{δ_{n}} Q_{n−1} → … Q_{0} → M → 0)

is thus a minimal projective resolution of the restriction of M. Suppose we know that the big finitistic dimension of Λ̃_{n} is less or equal than n − 1. This means that the restriction of pM_{∗} must be such that Q̃_{l} = 0 for l ≥ n. Hence the projective module Q_{l} (for l ≥ n) in a minimal Λ_{n}-resolution is a direct sum of copies of P_{0}. Now suppose the morphism δ_{n} is not an injection. This means that its kernel will contain a copy of S_{0} in its biggest semi-simple quotient. Since S_{0} is of infinite projective dimension, so is the module M. We have shown that, for a module M of finite projective dimension, the differential δ_{n} must be injective, i.e. p.dimM ≤ n.

The proof of the theorem will be complete when we will have proven that Fin.dimΛ̃_{n} ≤ n − 1. By definition, a minimal projective resolution of a Λ̃_{n}-module M has projective components and radical differential. Any radical morphism between projectives is given by a linear combination of paths of non-zero length. Thus, the coefficients of the matrix (with respect to the projective components of the Q_{i}’s) of a differential δ_{i} : Q_{i} → Q_{i−1} in a minimal resolution are of the form P_{j} → P_{k} with j < k. Hence the maximal length of a minimal projective resolution can not exceed n − 1 and there no module of infinite projective dimension.

References

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[3] Smalø Sverre O., The supremum of the difference between the big and little finitistic dimensions is infinite, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2619–2622. MR 98k:16012

[4] Charles A. Weibel, An introduction to homological algebra, Cambridge University Press, Cambridge, 1994. MR 95f:18001

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UFR de Mathématiques, Université Paris 7, 2, place Jussieu, 75251 Paris Cedex 05, France

E-mail address: lefevre@math.jussieu.fr www.math.jussieu.fr/˜ lefevre