RESOLUTIONS OVER SYMMETRIC ALGEBRAS WITH RADICAL CUBE ZERO

DAVID J. BENSON

Abstract. Let Λ be a finite dimensional indecomposable symmetric algebra over an algebraically closed field k, satisfying J^{3}(Λ) = 0. Let S_{1}, …, S_{r} be representatives of the isomorphism classes of simple Λ-modules, and let E be the r × r matrix whose (i, j) entry is dim_{k} Ext_{Λ}^{1}(S_{i}, S_{j}). If there exists an eigenvalue λ of E satisfying |λ| > 2 then the minimal resolution of each non-projective finitely generated Λ-module has exponential growth, with radius of convergence \frac{1}{2} (λ−\sqrt{λ^{2} − 4}). On the other hand, if all eigenvalues λ of E satisfy |λ| ≤ 2 then the dimensions of the modules in the minimal projective resolution of each finitely generated Λ-module are either bounded or grow linearly. In this case, we classify the possibilities for the matrix E. The proof is an application of the Perron–Frobenius theorem.

1. Introduction

In the course of investigating the homology of the loop space on the p-completion of the classifying space of a finite group [2], the question of the rate of growth of resolutions over certain finite dimensional symmetric algebras became important. This paper is a small contribution to that subject.

The purpose of this paper is to describe minimal projective resolutions of modules over a finite dimensional symmetric algebra Λ satisfying J^{3}(Λ) = 0.

Let k be an algebraically closed field, and let Λ be a finite dimensional indecomposable symmetric k-algebra satisfying J^{3}(Λ) = 0. Let S_{1}, …, S_{r} a set of representatives for the isomorphism classes of simple Λ-modules. Write E for the Ext matrix

E_{i,j} = dim_{k} Ext_{Λ}^{1}(S_{i}, S_{j}).

Our main theorem is as follows.

Theorem 1.1. The matrix E is symmetric, so its eigenvalues are real. The eigenvalue λ with largest absolute value is positive, and is a simple root of the characteristic equation of E.

(i) If λ > 2 then the minimal projective resolution of each finitely generated nonprojective Λ-module has exponential growth. The Poincaré series

∑_{n=0}^{∞} t^{n} dim_{k} Ω^{n}M

is the power series expansion of a rational function of t whose radius of convergence R is independent of the module. The radius R is determined by λ, and is given by

R = \frac{1}{2}(λ − \sqrt{λ^{2} − 4} ).

(ii) If λ = 2 then the dimensions of the modules in the minimal projective resolution of each finitely generated Λ-module are either bounded or grow linearly. The only modules of linear growth are the modules Ω^{m}(S_{i}) with m ∈ ℤ. The matrix E is the adjacency matrix of a Euclidean diagram Ã_{n} (n ≥ 1), D̃_{n} (n ≥ 4), Ẽ_{6}, Ẽ_{7}, Ẽ_{8}, the graph

Z̃_{n} =  •  •  •  •  (n + 1 nodes, n ≥ 0)

(the adjacency matrix of Z̃_{0} is the 1 × 1 matrix (2)) or the graph

D̃Z_{n} = [figure: •; •; •; •; •; •] (n + 1 nodes, n ≥ 2).

(iii) If λ < 2 then the dimensions of the modules in the minimal projective resolution of each finitely generated non-projective Λ-module are bounded. The matrix E is the adjacency matrix of a Dynkin diagram A_{n} (n ≥ 1), D_{n} (n ≥ 4), E_{6}, E_{7}, E_{8}, or the graph

Z_{n} = •  •  •  •  (n nodes, n ≥ 1)

(the adjacency matrix of Z_{1} is the 1 × 1 matrix (1)).

We remark that if B is a block of the modular group algebra of a finite group, and J^{3}(B) = 0, then Okuyama [6] has determined all the possibilities. Only cases (ii) and (iii) can occur, because in this situation exponential growth is not possible. Case (ii) only occurs for blocks with Klein four group as defect group in characteristic two, in which case the block is Morita equivalent to the group algebra of either the defect group or the alternating group of degree four. Case (iii) only occurs for blocks with cyclic defect group with inertial index p − 1 or (p − 1)/2, for which the Brauer tree is a straight line and the exceptional vertex, if there is one, is at an end.

We generalise Okuyama’s theorem to finite dimensional cocommutative Hopf algebras (i.e., finite group schemes) as follows.

Corollary 1.2. If a block Λ of a finite dimensional cocommutative Hopf algebra satisfies J^{3}(Λ) = 0 then all eigenvalues λ of the Ext matrix E are real numbers satisfying |λ| ≤ 2. The conclusion in case (ii) of Theorem 1.1 holds if λ = 2 is an eigenvalue of E, and the conclusion in case (iii) of the theorem holds otherwise.

We also remark that it is possible to generalise the main theorem in various directions. If we allow fields k that are not algebraically closed, then the other Dynkin and Euclidean diagrams come into play. It should also be possible to generalise the main theorem to finite dimensional self injective algebras Λ satisfying J^{3}(Λ) = 0. In this case, E is no longer symmetric, but its transpose is equal to its product with a permutation matrix corresponding to the Nakayama functor, and E commutes with this permutation matrix.

2. Minimal resolutions over Λ

Let Λ be a finite dimensional indecomposable symmetric algebra satisfying J^{3}(Λ) = 0. If J^{2}(Λ) = 0 then E = (1), and every indecomposable module is either projective, or simple and periodic. So we are trivially in case (iii) of the main theorem. Therefore, from now on we assume that J^{2}(Λ) ≠ 0.

Let S_{1}, …, S_{r} be representatives for the isomorphism classes of simple Λ-modules, and let P_{i} be the projective cover of S_{i}. Then P_{i} is a module of Loewy length three with socle Soc(P_{i}) ≅ S_{i}.

If M is a finite dimensional Λ-module, we write \underline{dim} M for the dimension vector of M, namely a column vector of length r whose ith component is the multiplicity of S_{i} as a composition factor of M. If M is indecomposable then there are three possibilities. If M is simple then Rad(M) = 0. If M is projective then Rad^{2}(M) ≅ M/Rad(M). If M is neither simple nor projective then Soc(M) = Rad(M) and Rad^{2}(M) = 0. In this case, we associate to M a column vector d(M) of length 2r, in which the first r components give the dimension vector α(M) = \underline{dim} M/Rad(M), and whose last r components give the dimension vector β(M) = \underline{dim} Rad(M):

d(M) = ( \begin{matrix}α(M) \\ β(M) \end{matrix} ) = ( \begin{matrix}\underline{dim} M/Rad(M) \\ \underline{dim} Rad(M) \end{matrix} ).

The projective cover of M has radical layers with dimension vectors α(M), Eα(M), α(M). So provided ΩM is not a simple module, we have α(ΩM) = Eα(M)−β(M) and β(ΩM) = α(M). So

d(ΩM) = ( \begin{matrix}Eα(M)−β(M) \\ α(M) \end{matrix} ) = ( \begin{matrix}E & −I \\ I & 0 \end{matrix} )( \begin{matrix}α(M) \\ β(M) \end{matrix} ).

Repeating, we have

(2.1)  d(Ω^{n}M) = ( \begin{matrix}E & −I \\ I & 0 \end{matrix} )^{n}d(M)

provided that none of the Ω^{n}M are simple modules.

On the other hand,

( \begin{matrix}E & −I \\ I & 0 \end{matrix} )d(Ω^{−1}S_{i}) = ( \begin{matrix}0 \\ \underline{dim} (S_{i}) \end{matrix} ),

while

d(ΩS_{i}) = ( \begin{matrix}E & −I \\ I & 0 \end{matrix} )( \begin{matrix}\underline{dim} (S_{i}) \\ 0 \end{matrix} ).

Note also that

( \begin{matrix}E & −I \\ I & 0 \end{matrix} )( \begin{matrix}0 \\ \underline{dim} (S_{i}) \end{matrix} ) = ( \begin{matrix}−\underline{dim} (S_{i}) \\ 0 \end{matrix} ),

and that the inverse of the matrix ( \begin{matrix}E & −I \\ I & 0 \end{matrix} ) is the matrix ( \begin{matrix}0 & I \\ −I & E \end{matrix} ).

3. Eigenvalues

The Perron–Frobenius theorem is a very useful theorem in the context of Ext matrices of indecomposable algebras. We state it in its general form, even though we do not use the full force of it in this paper.

Definition 3.1. We say that an r × r matrix A of non-negative real numbers is connected if there is no partition of {1, …, r} into non-empty subsets I and J in such a way that A_{i,j} = A_{j,i} = 0 for all i ∈ I, j ∈ J. This is equivalent to the condition that every entry in the matrix I + A + A^{2} + ⋯ + A^{r−1} is strictly positive. The Ext matrix E of an indecomposable finite dimensional algebra is connected; see Proposition II.5.2 of [1].

Theorem 3.2 (Perron–Frobenius). Let A be a connected r ×r matrix of non-negative real numbers. Enumerate the eigenvalues λ_{1}, …, λ_{r} so that

|λ_{1}| = ⋯ = |λ_{h}| > |λ_{h+1}| ≥ ⋯ ≥ |λ_{r}| (h ≥ 1).

Then

(i) The positive real number λ = |λ_{1}| is a simple root of the characteristic equation of A. In other words, the corresponding algebraic eigenspace is one dimensional.

(ii) There exists an eigenvector v of A with all entries positive, with eigenvalue λ. (iii) After reordering if necessary, the numbers λ_{1}, …, λ_{h} are equal to λ, ωλ, …, ω^{h−1}λ

where ω = e^{2πi/h}. (iv) The product of any eigenvalue of A with ω is another eigenvalue of A.

(v) If h > 1 then after conjugating by a permutation matrix if necessary, A has the form

( \begin{matrix}0 & A_{1} & 0 & ... & 0 \\ 0 & 0 & A_{2} & ... & 0 \\ ... & ... & ... & ... & ... \\ 0 & 0 & 0 & ... & A_{h−1} \\ A_{h} & 0 & 0 & ... & 0 \end{matrix} ) ,

where each A_{i} is an r/h × r/h matrix.

Proof. Perron [7] proved parts (i) and (ii) for matrices with all entries strictly positive. Frobenius [4] proved the full form of the theorem. □

Remark 3.3. If the matrix A in the theorem happens also to be symmetric, then we are forced to have h = 1 or h = 2. In other words, there is a positive real eigenvalue λ_{1} of A such that for all i > 1 we have λ_{1} > |λ_{i}|, with the possible exception of a single eigenvalue λ_{2} = −λ_{1} if h = 2. We call this eigenvalue λ_{1} the dominant eigenvalue.

We now return to the situation of the last section. We apply the Perron–Frobenius theorem to the Ext matrix E to obtain a dominant eigenvalue λ. If v is an eigenvector of E with eigenvalue λ, and μ satisfies μ^{2} − λμ + 1 = 0, then ( \begin{matrix}μv \\ v \end{matrix} ) is an eigenvector of ( \begin{matrix}E & −I \\ I & 0 \end{matrix} ) with eigenvalue μ. So we investigate the solutions μ of this quadratic equation.

Lemma 3.4. Let λ be a positive real number. If μ is one solution of μ^{2} − λμ + 1 = 0 then the other solution is μ^{−1}, and we have λ = μ + μ^{−1}.

(i) If λ > 2 then μ and μ^{−1} are distinct real numbers.

(ii) If λ = 2 then μ = 1 (with multiplicity two).

(iii) If λ < 2 then μ and μ^{−1} are non-real complex numbers with absolute value one.

Proof. This is straightforward. □

Since E is a symmetric matrix, the algebraic and geometric eigenspaces of E coincide. If v_{i} is an eigenvector of E with eigenvalue λ_{i}, and μ_{i} satisfies the quadratic equation μ_{i}^{2} − λ_{i}μ_{i} + 1 = 0, then ( \begin{matrix}μ_{i}v_{i} \\ v_{i} \end{matrix} ) is an eigenvector of ( \begin{matrix}E & −I \\ I & 0 \end{matrix} ) with eigenvalue μ_{i}. Provided λ_{i} ≠ 2, the quadratic equation gives two different values of μ_{i}. If λ_{i} = 2 then μ_{i} = 1 is the only solution, but the eigenspace contains not only the geometric eigenvector ( \begin{matrix}v_{i} \\ v_{i} \end{matrix} ) but also the algebraic eigenvector ( \begin{matrix}v_{i} \\ 0 \end{matrix} ). In either case, we get two dimensions of algebraic eigenvectors of ( \begin{matrix}E & −I \\ I & 0 \end{matrix} ) for each dimension of eigenvectors of E, but it is only for λ_{i} = 2, μ_{i} = 1 that the algebraic and geometric eigenspaces of ( \begin{matrix}E & −I \\ I & 0 \end{matrix} ) differ.

If |λ_{i}| ≤ 2 then both values of μ_{i} satisfy |μ_{i}| = 1, while if |λ_{i}| > 2 then the two values

of μ_{i} are real, and given by μ_{i} = \frac{1}{2}(λ_{i} ± \sqrt{λ_{i}^{2} − 4}). The larger of these has the plus sign; since this gives a monotonic increasing function of λ_{i} for λ_{i} > 2, it follows that the largest positive eigenvalue is μ = \frac{1}{2}(λ + \sqrt{λ^{2} − 4}), with multiplicity one, and all other eigenvalues of ( \begin{matrix}E & −I \\ I & 0 \end{matrix} ) have strictly smaller absolute value. All the entries in the eigenvector ( \begin{matrix}μv \\ v \end{matrix} ) are strictly positive. The corresponding row eigenvector is (−μv^{T}, v^{T}).

Proposition 3.5. Suppose that λ > 2 and ( \begin{matrix}x \\ y \end{matrix} ) is any non-zero column vector such that for all n ≥ 0, ( \begin{matrix}E & −I \\ I & 0 \end{matrix} )^{n}( \begin{matrix}x \\ y \end{matrix} ) has non-negative integer entries. If (δ, δ′) is any row vector of length 2r with strictly positive integer entries then the power series

∑_{n=0}^{∞} t^{n}(δ, δ′)( \begin{matrix}E & −I \\ I & 0 \end{matrix} )^{n}( \begin{matrix}x \\ y \end{matrix} )

is a rational function of t with radius of convergence R = \frac{1}{2}(λ − \sqrt{λ^{2} − 4}). In particular, the growth rate of the coefficients of this power series is exponential.

Proof. Let v be a positive eigenvector of E with eigenvalue λ, as above. Write

( \begin{matrix}x \\ y \end{matrix} ) = a( \begin{matrix}μv \\ v \end{matrix} ) + b( \begin{matrix}v \\ μv \end{matrix} ) + ( \begin{matrix}x′ \\ y′ \end{matrix} )

where v^{T}x′ = v^{T}y′ = 0. This is possible since μ and μ^{−1} are simple roots of the characteristic equation of ( \begin{matrix}E & −I \\ I & 0 \end{matrix} ). We claim that a ≠ 0. For if a = 0 then

( \begin{matrix}E & −I \\ I & 0 \end{matrix} )^{n}( \begin{matrix}x \\ y \end{matrix} ) = μ^{−n}b( \begin{matrix}v \\ μv \end{matrix} ) + ( \begin{matrix}E & −I \\ I & 0 \end{matrix} )^{n}( \begin{matrix}x′ \\ y′ \end{matrix} ).

The first term on the right hand side tends to zero as n → ∞. So the entries in the second term get closer and closer to non-negative integers. For any particular value of n, not all these integers can be zero. This contradicts the fact that

(v^{T}, v^{T})( \begin{matrix}E & −I \\ I & 0 \end{matrix} )^{n}( \begin{matrix}x′ \\ y′ \end{matrix} ) = 0,

since v has positive entries.

We have

∑_{n=0}^{∞} t^{n}(δ, δ′)( \begin{matrix}E & −I \\ I & 0 \end{matrix} )^{n}( \begin{matrix}x \\ y \end{matrix} ) = a ∑_{n=0}^{∞} t^{n}(δ, δ′)( \begin{matrix}E & −I \\ I & 0 \end{matrix} )^{n}( \begin{matrix}μv \\ v \end{matrix} ) + ∑_{n=0}^{∞} t^{n}(δ, δ′)( \begin{matrix}E & −I \\ I & 0 \end{matrix} )^{n}( \begin{matrix}bv+x′ \\ bμv+y′ \end{matrix} )

The first sum on the right hand side is equal to

a ∑_{n=0}^{∞} t^{n}(δ, δ′)( \begin{matrix}μ^{n+1}v \\ μ^{n}v \end{matrix} ) = a(μδ + δ′)v ∑_{n=0}^{∞} μ^{n}t^{n} = \frac{a(μδ + δ′)v}{1 − μt} .

This is a rational function of t with radius of convergence μ^{−1} = \frac{1}{2}(λ − \sqrt{λ^{2} − 4}). The remaining sum breaks up into terms corresponding to smaller eigenvalues, which therefore give rational functions of the same form, but with larger radii of convergence. □

We can now prove the case of Theorem 1.1 for λ > 2. If M is a non-projective indecomposable Λ-module, then at most a finite number of the modules Ω^{n}M can be simple. So by replacing M by Ω^{n}M for n large enough, we may suppose that Ω^{n}M is not simple for n ≥ 0. Let δ be the row vector whose ith entry is dim_{k} S_{i}. Then using equation (2.1), we have

dim_{k} Ω^{n}M = (δ, δ)d(Ω^{n}M) = (δ, δ)( \begin{matrix}E & −I \\ I & 0 \end{matrix} )^{n}d(M).

So we can apply Proposition 3.5 to the Poincaré series

∑_{n=0}^{∞} t^{n} dim_{k} Ω^{n}M = ∑_{n=0}^{∞} t^{n}(δ, δ)( \begin{matrix}E & −I \\ I & 0 \end{matrix} )^{n}d(M)

to deduce the case λ > 2 of Theorem 1.1.

4. Eigenvalues at most two

Now suppose that λ ≤ 2. We need the following theorem.

Theorem 4.1. The only connected graphs with no loops, having maximum eigenvalue equal to 2 are Ã_{n}, D̃_{n}, Ẽ_{6}, Ẽ_{7} and Ẽ_{8}. The only connected graphs with no loops, having maximum eigenvalue strictly less than 2 are A_{n}, D_{n}, E_{6}, E_{7} and E_{8}.

Proof. See J. H. Smith [8], or Theorem 5.1 of Lemmens and Seidel [5]. □

We need the following generalisation of Theorem 4.1.

Theorem 4.2. The only connected graphs having maximum eigenvalue equal to 2 are Ã_{n}, D̃_{n}, Ẽ_{6}, Ẽ_{7}, Ẽ_{8}, Z̃_{n} and D̃Z_{n}. The only connected graphs having maximum eigenvalue strictly less than 2 are A_{n}, D_{n}, E_{6}, E_{7}, E_{8} and Z_{n}.

Proof. The proof is exactly the same as the proof of Theorem 4.1. The main steps are to find positive eigenvectors for Ã_{n}, D̃_{n}, Ẽ_{6}, Ẽ_{7}, Ẽ_{8}, Z̃_{n} or D̃Z_{n} with eigenvalue equal to 2, and to show that if a graph does not contain a subgraph isomorphic to one of these, then it is isomorphic to a proper subgraph of one of them. In the cases of Z̃_{n} and D̃Z_{n}, the eigenvectors are as follows:

Z̃_{n}:  _{1}  _{1}  _{1}  _{1} • • • •

D̃Z_{n}:  [figure: 1; •; 2; 2; 2; 2; •; •; •; •; 1; •]

So for example the adjacency matrix for Z̃_{3} is ( \begin{matrix}1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{matrix} ) and the eigenvector is ( \begin{matrix}1 \\ 1 \\ 1 \\ 1 \end{matrix} ); the

adjacency for D̃Z_{3} is ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{matrix} ) and the eigenvector is ( \begin{matrix}1 \\ 1 \\ 2 \\ 2 \end{matrix} ).

Finally, every proper subgraph of one of these graphs is of one of the Dynkin types A_{n}, D_{n}, E_{6}, E_{7}, E_{8}, or the graph Z_{n}. □

Now we begin the proof of Theorem 1.1 in the cases where λ ≤ 2. Theorem 4.2 shows that the matrix E has the type stated. All the eigenvalues μ of ( \begin{matrix}E & −I \\ I & 0 \end{matrix} ) satisfy |μ| = 1, by Lemma 3.4. Moreover, it follows from the discussion preceding Proposition 3.5 that the algebraic and geometric eigenspaces coincide, with one possible exception. The exception is that if λ = 2 then μ = 1 occurs with algebraic multiplicity two and geometric multiplicity one. Thus if λ < 2 then the dimensions of the modules in the minimal projective resolution of each finitely generated Λ-module are bounded.

Now suppose that λ = 2, and that v is a positive eigenvector of E with eigenvalue 2. Suppose that M is an indecomposable module with an unbounded resolution. Choose m large enough so that for n ≥ m, Ω^{n}M is not simple. Then for all n ≥ m, d(Ω^{n}M) is an algebraic eigenvector of ( \begin{matrix}E & −I \\ I & 0 \end{matrix} ) but not a geometric eigenvector. Thus d(Ω^{m}M) is of the form a( \begin{matrix}v \\ v \end{matrix} ) + b( \begin{matrix}v \\ 0 \end{matrix} ), where a and b are real numbers and b ≠ 0. Then for n ≥ m, equation (2.1) gives

(4.3)  d(Ω^{n}M) = (a + (n − m)b)( \begin{matrix}v \\ v \end{matrix} ) + b( \begin{matrix}v \\ 0 \end{matrix} ).

Since the entries in d(Ω^{n}M) have to be non-negative integers for all n ≥ m, it follows that b > 0. Furthermore, equation (4.3) cannot hold for all n ∈ ℤ, again by non-negativity. Using equation (2.1) again, the only possibility then is that for some n ∈ ℤ, the module Ω^{n}M is simple.

Finally, we prove Corollary 1.2. By a theorem of Friedlander and Suslin [3], if Γ is a finite dimensional cocommutative Hopf algebra over a field k then the cohomology ring H^{∗}(Γ, k) a finitely generated graded commutative k-algebra. Furthermore, if M is a finitely generated Γ-module then for each simple module S_{i}, Ext_{Γ}^{∗}(S_{i}, M) is a finitely generated H^{∗}(Γ, k)-module. It follows that the minimal projective resolution of M has polynomial growth. If Λ is a block of Γ with J^{3}(Λ) = 0 then Λ is also a symmetric algebra, and so it satisfies the hypotheses of Theorem 1.1. It follows that only cases (ii) and (iii) of the theorem are possible.

References

1. M. Auslander, I. Reiten, and Sverre O. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, 1995, reprinted in paperback, 1997.

2. D. J. Benson, An algebraic model for chains on Ω(BGp^{∧}), preprint, 2007.

3. E. M. Friedlander and A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), 209–270.

4. G. Frobenius, Über Matrizen aus nicht negativen Elementen, Sitzungsber. Königl. Preuss. Acad. Wiss. Berlin (1912), 456–477.

5. P. W. Lemmens and J. J. Seidel, Equiangular lines, J. Algebra 24 (1973), 494–512.

6. T. Okuyama, On blocks of finite groups with radical cube zero, Osaka J. Math. 23 (1986), 461–465.

7. O. Perron, Zur Theorie der Matrizen, Math. Ann. 64 (1907), 248–263.

8. J. H. Smith, Some properties of the spectrum of a graph, Combinatorial Structures and their Applications (R. Guy, ed.), Gordon and Breach, New York, 1970, pp. 403–406.

Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE