PIECEWISE HEREDITARY SKEW GROUP ALGEBRAS

LIPING LI

Abstract. Let Λ be a finite dimensional algebra and G be a finite group whose elements act on Λ as algebra automorphisms. Under the assumption that Λ has a complete set E of primitive orthogonal idempotents, closed under the action of a Sylow p-subgroup S ⩽ G, we show that the skew group algebra ΛG and Λ have the same finitistic dimension and the same strong global dimension if the action of S on E is free. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce that ΛG is piecewise hereditary if and only if S acts freely on E and Λ is piecewise hereditary as well.

1. Introduction

Throughout this note let Λ be a basic finite dimensional k-algebra, where k is an algebraically closed field with characteristic p ⩾ 0, and let G be a finite group whose elements act on Λ as algebra automorphisms. The skew group algebra ΛG is the vector space Λ ⊗_{k} kG equipped with a bilinear product · determined by the following rule: for λ, μ ∈ Λ, g, h ∈ G, (λ ⊗ g) · (μ ⊗ h) = λg(μ) ⊗ gh, where g(μ) is the image of μ under the action of g. We write λg rather than λ ⊗ g to simplify the notation. Correspondingly, the product can be written as λg · μh = λg(μ)gh. Denote the identity of Λ and the identity of G by 1_{Λ} and 1_{G} respectively.

It has been observed that when |G|, the order of G, is invertible in k, ΛG and Λ share many common properties [4, 11, 13]. We wonder for arbitrary groups G, under what conditions this phenomena still happens. This problem is considered in [9], where under the hypothesis that Λ has a complete set E of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S ⩽ G, we show that Λ and ΛG share certain properties as finite global dimension, finite representation type, etc., if and only if the action of S on E is free. Clearly, this answer generalizes results in [13] since if |G| is invertible in k, the only Sylow p-subgroup of G is the trivial group.

In this note we continue to study representations and homological properties of modular skew group algebras. Using the ideas and techniques described in [9], we show that Λ and ΛG share more common properties under the same hypothesis and condition. Explicitly, we have:

Theorem 1.1. Let Λ and G be as above, and let S ⩽ G be a Sylow p-subgroup. If S acts freely on a complete set E of primitive orthogonal idempotents in Λ, then ΛG and Λ have the same finitistic dimension and the same strong global dimension.

In [4] it has been proved that if Λ is a piecewise hereditary algebra (defined in Section 3) and |G| is invertible in k, then ΛG is piecewise hereditary as well. Applying the homological characterization of piecewise hereditary algebras by Happel and Zacharia in [6], we get the following immediate corollary, generalizing the result of [4] and characterizing piecewise hereditary skew group algebras.

Corollary 1.2. Let Λ, G and S be as above, and suppose that Λ has a complete set E of primitive orthogonal idempotents closed under the action of S. Then ΛG is a piecewise hereditary algebra if and only if Λ is piecewise hereditary and S acts freely on E.

We introduce some notations and conventions here. Throughout this note all modules are finitely generated left modules. Composition of maps and morphisms is from right to left. For an algebra A, gl. dim A, fin. dim A, and sgl. dim A are the global dimension, finitistic dimension, and strong global dimension (defined in Section 4) of A respectively. For an A-module M, pd_{A} M is the projective dimension of M. We use A-mod to denote the category of finitely generated A-modules. Its bounded homotopy category and bounded derived category are denoted by K^{b}(A) and D^{b}(A) respectively.

2. projective dimensions and finitistic dimensions

We first describe some background knowledge and elementary results. Most of them can be found in literature. We suggest the reader to refer to [1, 2, 3, 9, 10, 11, 12, 13, 14] for more details.

For every subgroup H ⩽ G, elements in H also act on Λ as algebra automorphism, so we can define a skew group algebra ΛH, which is a subalgebra of ΛG. The induction functor and the restriction functor are defined in the usual way. For a ΛH module V, the induced module is V ↑_{H}^{G}= ΛG ⊗_{ΛH} V, where ΛG acts on the left side. Every ΛG-module M can be viewed as a ΛH-module, denoted by M ↓_{H}^{G}. Observe that ΛG is a free (both left and right) ΛH-module. Therefore, these two functors are exact, and perverse projective modules.

Proposition 2.1. Let H ⩽ G be a subgroup. Then:

(1) Every ΛH-module V is isomorphic to a summand of V ↑_{H}^{G}↓_{H}^{G}.

(2) If |G : H| is invertible in k, then every ΛG-module M is isomorphic to a summand of M ↓_{H}^{G}↑_{H}^{G}.

Proof. This is Proposition 2.1 in [9]. □

The above proposition immediately implies:

Corollary 2.2. Let H ⩽ G be a subgroup. For every M ∈ ΛG-mod, pd_{ΛH} M ↓_{H}^{G}⩽ pd_{ΛG} M. If |G : H| is invertible in k, then the equality holds.

Proof. Take a minimal projective resolution P^{•} → M of ΛG-modules and apply the restriction functor termwise. Since this functor is exact and preserves projective modules, we get a projective resolution P^{•} ↓_{H}^{G}→ M ↓_{H}^{G} of ΛH-modules, which might not be minimal. Thus pd_{ΛH} M ↓_{H}^{G}⩽ pd_{ΛG} M, which is true even for the case that either pd_{ΛH} M ↓_{H}^{G}= ∞ or pd_{ΛG} M = ∞.

Now suppose that |G : H| is invertible in k. Take a minimal projective resolution Q^{•} → M ↓_{H}^{G} of ΛH-modules and apply the induction functor termwise. Since it is exact and preserves projective modules, we get a projective resolution Q^{•} ↑_{H}^{G}→ M ↓_{H}^{G}↑_{H}^{G} of ΛG-modules, so pd_{ΛG} M ↓_{H}^{G}↑_{H}^{G}⩽ pd_{ΛH} M ↓_{H}^{G}. But M is isomorphic to a summand of M ↓_{H}^{G}↑_{H}^{G}, so pd_{ΛG} M ⩽ pd_{ΛH} M ↓_{H}^{G}. Putting two inequalities together, we get the equality. □

Let E = {e_{i}}_{i∈[n]} be a set of primitive orthogonal idempotents in Λ. We say it is complete if ∑_{i∈[n]} e_{i} = 1_{Λ}. Throughout this note we assume that G has a Sylow p-subgroup S such that E is closed under the action of S. That is, g(e_{i}) ∈ E for all i ∈ [n] and g ∈ S. In practice this condition is usually satisfied. A trivial case is that |G| is invertible in k, and hence S is the trivial subgroup.

We introduce some notations here. Let Λ^{S} be the space constituted of all elements in Λ fixed by S. Clearly, Λ^{S} is a subalgebra of Λ, and S acts on it trivially. For every M ∈ ΛS-mod, elements v ∈ M satisfying g(v) = v for every g ∈ S form a Λ^{S}-module, which is denoted by M^{S}. Let F^{S} be the functor from ΛS-mod to Λ^{S}-mod, sending M to M^{S}. This is indeed a functor since for every ΛS-module homomorphism f : M → N, M^{S} is mapped into N^{S} by f.

We collect in the following proposition some results taken from [9]

Proposition 2.3. Let S ⩽ G and E be as above and suppose that E is closed under the action of S. Then:

(1) The set E is also a complete set of primitive orthogonal idempotents of ΛS, where we identify e_{i} with e_{i}1_{S}.

(2) Λ^{S} = {∑_{g∈S} g(μ) | μ ∈ Λ}.

(3) The global dimension gl. dim ΛS < ∞ if and only if gl. dim Λ < ∞ and the action of S on E is free.

Moreover, if the action of S on E is free, then

(4) ΛS is the matrix algebra over Λ^{S}, and hence is Morita equivalent to Λ^{S}.

(5) The functor F^{S} is exact.

(6) The regular representation _{ΛS}ΛS ≅ Λ^{|S|}, where Λ is the trivial ΛS-module.

(7) A ΛS-module M is projective (resp., injective) if and only if the Λ-module ΛM is projective (resp., injective).

Proof. See Lemmas 3.2, 3.3, 3.7 and Propositions 3.4, 3.6 in [9]. □

Recall for a finite dimensional algebra A, its finitistic dimension, denoted by fin. dim A, is the supremum of projective dimensions of finitely generated indecomposable A-modules M with pd_{A} M < ∞. If A has finite global dimension, then fin. dim A = gl. dim A. For more details, see [15].

An immediate consequence of Proposition 2.1 and Corollary 2.2 is:

Lemma 2.4. Let H ⩽ G be a subgroup of G. Then fin. dim ΛH ⩽ fin. dim ΛG. If |G : H| is invertible in k, the equality holds.

Proof. Take an arbitrary indecomposable V ∈ ΛH-mod with pd_{ΛH} V < ∞ and consider Ṽ = V ↑_{H}^{G}. We claim that pd_{ΛG} Ṽ < ∞. Indeed, by applying the induction functor to a minimal projective resolution P^{•} → V termwise, we get a projective resolution ΛG ⊗_{ΛH} P^{•} → Ṽ. The finite length of the first resolution implies the finite length of the second one. Therefore, pd_{ΛG} Ṽ < ∞. Consequently, pd_{ΛG} Ṽ ⩽ fin. dim ΛG.

By Corollary 2.2, pd_{ΛG} Ṽ ⩾ pd_{ΛH} Ṽ ↓_{H}^{G}. Since V is isomorphic to a summand of Ṽ ↓_{H}^{G} by Proposition 2.1, we get fin. dim A ⩾ pd_{ΛG} Ṽ ⩾ pd_{ΛH} Ṽ ↓_{H}^{G}⩾ pd_{ΛH} V. In conclusion, fin. dim ΛG ⩾ fin. dim ΛH.

Now suppose that |G : H| is invertible in k. Take an arbitrary indecomposable M ∈ ΛG-mod with pd_{ΛG} M < ∞. By applying the restriction functor ↓_{H}^{G} to a minimal projective resolution Q^{•} → M termwise, we get a projective resolution of M ↓_{H}^{G}, which is of finite length. Therefore, pd_{ΛH} M ↓_{H}^{G}⩽ fin. dim ΛH. By Corollary 2.2, we have pd_{ΛG} M = pd_{ΛH} M ↓_{H}^{G}⩽ fin. dim ΛH. In conclusion, fin. dim ΛG ⩽ fin. dim ΛH. Combining the inequality in the previous paragraph, we have fin. dim ΛG = fin. dim ΛH. □

Lemma 2.5. If a Sylow p-subgroup S ⩽ G acts freely on E, then fin. dim ΛS ⩽ fin. dim Λ.

Proof. Take an arbitrary indecomposable M ∈ ΛS-mod with pd_{ΛS} M = n < ∞ and a minimal projective resolution:

… → 0 → P^{n} → … → P^{1} → P^{0} → M → 0.

Regarded as Λ-modules, we get a projective resolution:

… → 0 →_{Λ} P^{n} → … →_{Λ} P^{1} →_{Λ} P^{0} →_{Λ} M → 0.

Clearly, for every 0 ⩽ s ⩽ n, the syzygy Ω^{s}(M) viewed as a Λ-module is a direct sum of Ω^{s}(_{Λ}M) and a projective Λ-module. We claim that for each 0 ⩽ s ⩽ n, the syzygy Ω^{s}(_{Λ}M) ≠ 0. Otherwise, Ω^{s}(M) viewed as a Λ-module is projective. By (7) of Proposition 2.3, Ω^{s}(M) is a projective ΛS-module, which must be 0. But this implies pd_{ΛS} M < n, contradicting our choice of M. Therefore, for each 0 ⩽ s ⩽ n, Ω^{s}(_{Λ}M) ≠ 0. Consequently, _{Λ}M has a summand with projective dimension n, so n ⩽ fin. dim Λ. In conclusion, fin. dim ΛS ⩽ fin. dim Λ. □

Now we can prove the first statement of Theorem 1.1.

Proposition 2.6. If G has a Sylow p-subgroup S acting freely on E, then ΛG and Λ have the same finitistic dimension.

Proof. Since |G : S| is invertible in k, by Lemma 3.1, fin. dim ΛG = fin. dim ΛS. Also by this lemma, fin. dim ΛS ⩾ fin. dim Λ since S contains the trivial group. The previous lemma tells us fin. dim ΛS ⩽ fin. dim Λ. Putting all these pieces of information together, we get fin. dim ΛG = fin. dim Λ as claimed. □

3. strong global dimensions and piecewise hereditary algebras

A finite dimensional k-algebra A is called piecewise hereditary if the derived category D^{b}(A) is triangulated equivalent to the derived category D^{b}(H) of a hereditary abelian category H [5, 6, 7, 8]. If A is piecewise hereditary, then gl. dim A < ∞ since the property having finite global dimension is invariant under derived equivalence [5]. Therefore, D^{b}(A) is triangulated equivalent to the homotopy category K^{b}(_{A}P), where _{A}P is the full subcategory of A-mod constituted of all finitely generated projective A-modules.

Now let A be a finite dimensional algebra with gl. dim A < ∞. Since D^{b}(A) ≅ K^{b}(_{A}P), we identify these two triangulated categories. For every indecomposable object 0 ≠ P^{•} ∈ K^{b}(_{A}P), consider its preimages P̃^{•} ∈ C^{b}(_{A}P), the category of so-called perfect complexes, i.e., each term of P̃^{•} is a finitely generated projective A-module, and all but finitely many terms of P̃^{•} are 0. We can choose P̃^{•} such that it is minimal. That is, P̃^{•} has no direct summands isomorphic to

\xymatrix{… \ar[r] & 0 \ar[r] & P^{s} \ar[r]^{id} & P^{s+1} \ar[r] & 0 \ar[r] & …} .

Clearly, this choice is unique for each P^{•} ∈ K^{b}(_{A}P) up to isomorphism, so in the rest of this note we identify P^{•} and P̃^{•}. Hopefully this identification will not cause troubles to the reader.

Take an indecomposable P^{•} ∈ K^{b}(_{A}P) and identify it with P̃^{•}. Therefore, there exist r < s ∈ ℤ such that P^{r} ≠ 0 ≠ P^{s}, and P^{n} = 0 for n > s or n < r. We define its length l(P^{•}) to be s − r. The strong global dimension of A, denoted by sgl. dim A, is defined as sup{l(P^{•}) | P^{•} ∈ K^{b}(_{A}P) is indecomposable}. By taking minimal projective resolutions of simple modules, it is easy to see sgl. dim A ⩾ gl. dim A. Moreover, if A is hereditary, then sgl. dim A = gl. dim A ⩽ 1 (see [5]). It is not clear for what algebras of finite global dimension, sgl. dim A = gl. dim A.

Happel and Zacharia proved the following result, characterizing piecewise hereditary algebras.

Theorem 3.1. (Theorem 3.2 in [7]) A finite dimensional k-algebra is piecewise hereditary if and only if sgl. dim A < ∞.

In [4] Dionne, Lanzilotta, and Smith show that if Λ is a piecewise hereditary algebra, and |G| is invertible in k, then the skew group algebra ΛG is also piecewise hereditary. This result motivates us to characterize general piecewise hereditary skew group algebras. Using the above characterization, we take a different approach by showing that sgl. dim ΛG = sgl. dim Λ under suitable conditions.

We first prove a result similar to Lemma 2.4.

Lemma 3.2. Let H be a subgroup of G and suppose that gl. dim ΛG < ∞. Then sgl. dim ΛH ⩽ sgl. dim ΛG. The equality holds if |G : H| is invertible in k.

Proof. Note that gl. dim ΛH ⩽ gl. dim ΛG < ∞ by (2) of Corollary 2.2 in [9]. Take an indecomposable object P^{•} ∈ K^{b}(_{ΛH}P). Up to isomorphism, its minimal preimage in C^{b}(_{ΛH}P) can be written as:

\xymatrix{… → 0 → P^{r} \ar[r]^{d^{r}} & … \ar[r] & P^{s−1} \ar[r]^{d^{s−1}} & P^{s} → 0 → …}

Applying the exact functor ΛG ⊗_{ΛH} − termwise to the above complex, we get an object P̃^{•} = P^{•} ↑_{H}^{G}∈ C^{b}(_{ΛG}P) as follows:

\xymatrix{… → 0 → ΛG ⊗ P^{r} \ar[r]^{1⊗d^{r}} & … \ar[r] & ΛG ⊗ P^{s−1} \ar[r]^{1⊗d^{s−1}} & ΛG ⊗ P^{s} → 0 → …}

Applying the restriction functor termwise, the second complex gives an object P̃^{•} ↓_{H}^{G}∈ C^{b}(_{ΛH}P). We claim that P^{•} is isomorphic to a direct summand of P̃^{•} ↓_{H}^{G}.

Indeed, there is a family of maps (ι_{i})_{i∈ℤ} defined as follows: for i > s or i < r, ι_{i} = 0; for r ⩽ i ⩽ s, ι_{i} sends v ∈ P^{i} to 1 ⊗ v ∈ ΛG ⊗_{ΛH} P^{i}. The reader can check that (ι_{i})_{i∈ℤ} defined in this way indeed gives rise to a chain map ι : P^{•} → P̃^{•} ↓_{H}^{G}.

We define another family of maps (δ_{i})_{i∈ℤ} as follows: for i > s or i < r, δ_{i} = 0; for r ⩽ i ⩽ s, δ_{i} sends h ⊗ v ∈ ΛH ⊗_{ΛH} P^{i} to hv ∈ P^{i} and sends all vectors in g ⊗ v ∈ ⨁_{1≠g∈G/H} g ⊗ P^{i} to 0. Again, the reader can check that (δ_{i})_{i∈ℤ} gives rise to a chain map δ : P̃^{•} ↓_{H}^{G}→ P^{•}. Moreover, we have δ ∘ ι is the identity map. Therefore, P^{•} is isomorphic to a direct summand of P̃^{•} ↓_{H}^{G}.

This claim has the following consequence: for every indecomposable object X^{•} ∈ K^{b}(_{ΛH}P) (identified with a minimal perfect complex in C^{b}(_{ΛH}P)), there is an indecomposable object X̃^{•} ∈ K^{b}(_{ΛG}P) such that X^{•} is isomorphic to a direct summand of X̃^{•} ↓_{H}^{G}. Clearly, we have l(X̃^{•}) ⩾ l(X^{•}). Therefore, by definition, sgl. dim ΛG ⩾ sgl. dim ΛH.

Now suppose that |G : H| is invertible in k. Take an indecomposable object P^{•} ∈ K^{b}(_{ΛG}P). Up to isomorphism, its minimal preimage in C^{b}(_{ΛG}P) can be written as:

\xymatrix{… → 0 → P^{r} \ar[r]^{d^{r}} & … \ar[r] & P^{s−1} \ar[r]^{d^{s−1}} & P^{s} → 0 → …}

Applying the restriction functor and the induction functor termwise to the above complex, we get another object in P̃^{•} = P^{•} ↓_{H}^{G}↑_{H}^{G}∈ C^{b}(_{ΛG}P) as follows:

\xymatrix{… → 0 → ΛG ⊗ P^{r} \ar[r]^{1⊗d^{r}} & … \ar[r] & ΛG ⊗ P^{s−1} \ar[r]^{1⊗d^{s−1}} & ΛG ⊗ P^{s} → 0 → …}

We claim that P^{•} is isomorphic to a direct summand of P̃^{•}.

Define a family of maps (θ_{i})_{i∈ℤ} as follows: for i > s or i < r, θ_{i} = 0; for r ⩽ i ⩽ s, θ_{i} sends v ∈ P^{i} to \frac{1}{|G:H|} ∑_{g∈G/H} g ⊗ g^{−1}v ∈ ΛG ⊗_{ΛH} P^{i}. We check that (θ_{i})_{i∈ℤ} defined in this way gives rise to a chain map θ : P^{•} → P̃^{•}. Indeed, for r ⩽ i ⩽ s − 1 and v ∈ P^{i}, we have

(θ_{i+1} ∘ d^{i})(v) = θ_{i+1}(d^{i}(v)) = \frac{1}{|G : H|} ∑_{g∈G/H} g ⊗ g^{−1}d^{i}(v)

= \frac{1}{|G : H|} ∑_{g∈G/H} g ⊗ d^{i}(g^{−1}v) = (1 ⊗ d^{i})(θ_{i+1}(v)).

Define another family of maps (ρ_{i})_{i∈ℤ} as follows: for i > s or i < r, ρ_{i} = 0; for r ⩽ i ⩽ s, ρ_{i} sends g ⊗ v ∈ ΛG ⊗_{ΛH} P^{i} to gv ∈ P^{i}. Again, we check that (ρ_{i})_{i∈ℤ} gives rise to a chain map ρ : P̃^{•} → P^{•} as shown by:

(ρ_{i+1} ∘ (1 ⊗ d^{i}))(g ⊗ v) = ρ_{i+1}(g ⊗ d^{i}(v))) = gd^{i}(v) = d^{i}(gv) = d^{i}(ρ_{i}(g ⊗ v)).

Moreover, ρ ∘ θ is the identity map. Therefore, as claimed, P^{•} is isomorphic to a summand of P̃^{•}.

This claim tells us that for every indecomposable object X̃^{•} ∈ K^{b}(_{ΛG}P) (identified with a minimal perfect complex in C^{b}(_{ΛG}P)), there is an indecomposable object X^{•} ∈ K^{b}(_{ΛH}P) such that X̃^{•} is isomorphic to a direct summand of X^{•} ↑_{H}^{G}. Therefore, l(X^{•}) ⩾ l(X̃^{•}), and sgl. dim ΛH ⩾ sgl. dim ΛG. The two inequalities force sgl. dim ΛG = sgl. dim ΛH. □

Remark 3.3. In this lemma we actually proved a stronger conclusion. That is, the induction and restriction functor actually induce an ‘induction’ functor and a ‘restriction’ functor between the homotopy categories of perfect complexes. Moreover, every indecomposable object X^{•} ∈ K^{b}(_{ΛH}P) can be obtained by applying the ‘restriction’ functor to an indecomposable object in K^{b}(_{ΛG}P) and taking a direct summand. When |G : H| is invertible, every indecomposable object X̃^{•} ∈ K^{b}(_{ΛG}P) can be obtained by applying the ‘induction’ functor to an indecomposable object in K^{b}(_{ΛH}P) and taking a direct summand.

In the next lemma we show that if G has a sylow p-subgroup S acting freely on E, then sgl. dim ΛS = sgl. dim Λ. Since we already know that ΛS is Morita equivalent to Λ^{S} (Proposition 2.3), instead we show sgl. dim Λ = sgl. dim Λ^{S}. Note that Λ^{S} is a subalgebra of Λ, so we can define the corresponding restriction functor from Λ-mod to Λ^{S}-mod and the induction functor Λ⊗_{Λ^{S}} − from Λ^{S}-mod to Λ-mod.

Since S acts on E, we take one representative idempotent from each S-orbit and let ϵ be the sum of these idempotents. Note that 1_{Λ} = ∑_{g∈S} g(ϵ) since S acts on E freely.

For M ∈ Λ^{S}-mod, we define two linear maps:

ψ : M → (Λ ⊗_{Λ^{S}} M) ↓_{Λ^{S}}^{Λ}, v ↦ ∑_{g∈S} g(ϵ) ⊗ v, v ∈ M;

φ : (Λ ⊗_{Λ^{S}} M) ↓_{Λ^{S}}^{Λ}→ M, λ ⊗ v ↦ (∑_{g∈S} g(λϵ))v, v ∈ M, λ ∈ Λ.

Since ∑_{g∈S} g(λϵ) ∈ Λ^{S}, its action on v is well defined. We also observe that when λ ∈ Λ^{S},

φ(λ ⊗ v) = ( ∑_{g∈S} g(λϵ))v = ( ∑_{g∈S} λg(ϵ))v = λv.

Lemma 3.8 in [9] shows that both maps are actually Λ^{S}-module homomorphisms. Moreover, φ ∘ ψ is the identity map.

Lemma 3.4. Suppose that gl. dim Λ < ∞. If a Sylow p-subgroup S ⩽ G acts freely on E, then sgl. dim Λ = sgl. dim ΛS.

Proof. Note that gl. dim ΛS = gl. dim Λ < ∞ by Theorem 3.9 in [9]. By the previous lemma, sgl. dim Λ ⩽ sgl. dim ΛS. Therefore, it suffices to show sgl. dim Λ ⩾ sgl. dim ΛS. Since ΛS is Morita equivalent to Λ^{S}, we show sgl. dim Λ ⩾ sgl. dim Λ^{S}. The proof is similar to that of the previous lemma, using the corresponding induction and restriction functors for the pair (Λ^{S}, Λ) and the two maps ψ, φ defined above.

Take an indecomposable object P^{•} ∈ K^{b}(_{Λ^{S}}P) and write its minimal preimage in C^{b}(_{Λ^{S}}P) as follows:

\xymatrix{… → 0 → P^{r} \ar[r]^{d^{r}} & … \ar[r] & P^{s−1} \ar[r]^{d^{s−1}} & P^{s} → 0 → …}

Applying Λ ⊗_{Λ^{S}} − and the restriction functor termwise, we get another object in P̃^{•} = P^{•} ↑_{Λ^{S}}^{Λ}↓_{Λ^{S}}^{Λ}∈ C^{b}(_{Λ^{S}}P) :

\xymatrix{… → 0 → Λ ⊗ P^{r} \ar[r]^{1⊗d^{r}} & … \ar[r] & Λ ⊗ P^{s−1} \ar[r]^{1⊗d^{s−1}} & Λ ⊗ P^{s} → 0 → …}

We claim that P^{•} is isomorphic to a direct summand of P̃^{•}.

As did in the proof of the previous lemma, (ψ_{i})_{i∈ℤ} gives rise to a chain map ψ : P^{•} → P̃^{•}. We check the commutativity: for r ⩽ i ⩽ s − 1 and v ∈ P^{i},

(ψ_{i+1} ∘ d^{i})(v) = ψ_{i+1}(d^{i}(v)) = ∑_{g∈S} g(ϵ) ⊗ d^{i}(v)

= ∑_{g∈S}(1 ⊗ d^{i})(g(ϵ) ⊗ v) = (1 ⊗ d^{i})(ψ_{i}(v)).

Similarly, the family of maps (φ_{i})_{i∈ℤ} gives rise to a chain map φ : P̃^{•} → P^{•} as shown by:

(φ_{i+1} ∘ (1 ⊗ d^{i}))(λ ⊗ v) = φ_{i+1}(λ ⊗ d^{i}(v)) = (∑_{g∈S} g(λϵ))d^{i}(v)

= d^{i}((∑_{g∈S} g(λϵ)v) = d^{i}(φ_{i}(v)).

Moreover, ρ ∘ θ is the identity map, so as claimed, P^{•} is isomorphic to a summand of P̃^{•}.

Therefore, for every indecomposable object X^{•} ∈ K^{b}(_{Λ^{S}}P), there is an indecomposable object X̃^{•} ∈ K^{b}(_{Λ}P) such that X^{•} is isomorphic to a direct summand of X̃^{•} ↓_{Λ^{S}}^{Λ}. Consequently, l(X^{•}) ⩽ l(X̃^{•}), and sgl. dim Λ^{S} ⩽ sgl. dim Λ. □

Now we are ready to prove the main result.

Theorem 3.5. Let Λ be a finite dimensional algebra with gl. dim Λ < ∞, G be a finite group whose elements act on Λ as algebra automorphisms. Let E be a complete set of primitive orthogonal idempotents in Λ, which is closed under the action of a Sylow p-subgroup S of G. Then:

(1) If S acts freely on E, then sgl. dim ΛG = sgl. dim Λ.

(2) ΛG is a piecewise hereditary algebra if and only if so is Λ and S acts freely on E.

Proof. Suppose that S acts freely on E. By Lemma 3.2, sgl. dim ΛG ⩾ sgl. dim Λ, and sgl. dim ΛG = sgl. dim ΛS. Moreover, since ΛS is Morita equivalent to Λ^{S}, we get sgl. dim ΛS = sgl. dim Λ^{S}. By the previous lemma, sgl. dim Λ ⩾ sgl. dim Λ^{S}. Put all these pieces together, we have sgl. dim ΛG = sgl. dim ΛS = sgl. dim Λ = sgl. dim Λ^{S}.

We use the homological characterization of piecewise hereditary algebras described in Theorem 3.1 to prove the second statement. If ΛG is piecewise hereditary, then sgl. dim ΛG < ∞. In particular, gl. dim ΛG < ∞. By (4) of Proposition 2.3, the action of S on E must be free. By the first statement of this theorem, sgl. dim Λ = sgl. dim ΛG < ∞, so Λ must be piecewise hereditary as well. On the other hand, if Λ is piecewise hereditary and S acts freely on E, then sgl. dim ΛG = sgl. dim Λ < ∞, so ΛG is also piecewise hereditary. The proof is completed. □

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Department of Mathematics, University of California, Riverside, CA, 92507.

E-mail address: lipingli@math.ucr.edu