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Projective Dimensions and Almost Split Sequences

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Projective Dimensions and Almost Split Sequences View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 271 (2004) 652–672 www.elsevier.com/locate/jalgebra Projective dimensions and almost split sequences Dag Madsen 1 Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway Received 11 September 2002 Communicated by Kent R. Fuller Abstract Let Λ be an Artin algebra and let 0 → A → B → C → 0 be an almost split sequence. In this paper we discuss under which conditions the inequality pd B max{pd A,pd C} is strict. 2004 Elsevier Inc. All rights reserved. Keywords: Almost split sequences; Homological dimensions Let R be a ring. If we have an exact sequence ε :0→ A → B → C → 0ofR-modules, then pd B max{pdA,pd C}. In some cases, for instance if the sequence is split exact, equality holds, but in general the inequality may be strict. In this paper we will discuss a problem first considered by Auslander (see [5]): Let Λ be an Artin algebra (for example a finite dimensional algebra over a field) and let ε :0→ A → B → C → 0 be an almost split sequence. To which extent does the equality pdB = max{pdA,pd C} hold? Given an exact sequence 0 → A → B → C → 0, we investigate in Section 1 what can be said in complete generality about when pd B<max{pdA,pd C}.Inthissectionwedo not make any assumptions on the rings or the exact sequences involved. In all sections except Section 1 the rings we consider are Artin algebras. Section 2 gives the necessary background on the theory of almost split sequences. Almost split sequences (also called Auslander–Reiten sequences) were introduced in [1] and have proven to be a valuable tool in the study of finite dimensional algebras. Section 3 contains some preliminary results that we need for our main results. In Section 4 we get to the main results of this paper. We give a characterization of when the inequality is strict. The E-mail address: [email protected]. 1 Supported in part by ERB FMRX-CT97-0100. 0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2003.09.015 D. Madsen / Journal of Algebra 271 (2004) 652–672 653 characterization is given in terms of five conditions that the module A has to satisfy. The description is simplified if max{pdA,pd C} < ∞. In Section 5 we give some examples to show that the five conditions are logically independent. In Section 6 we give some results on the number of almost split sequences with inequality, and see that this has a connection with the finitistic dimension conjecture. We show that the finitistic dimension conjecture implies that there are only finitely many almost split sequences with inequality for a given algebra. We do not have any converse of this result though, so it is quite possible that the number of inequalities is always finite for some other reason. In Section 7 we examine what can be said about projective dimensions of related modules, if we have pd B<max{pd A,pdC}. In a subsequent paper we restrict to the class of Nakayama algebras, and see what can be said about inequalities for these algebras. In that paper we also discuss a related problem concerning projective dimensions of composition factors of a given module. 1. Inequalities for general short-exact sequences In this section we look at what can be said in general about projective dimensions in exact sequences over arbitrary rings. Let R be a ring. If ε :0→ A → B → C → 0 is an exact sequence of R-modules, then pd B max{pdA,pd C}. If ε splits, then obviously pd B = max{pd A,pdC}. The following example shows that the inequality can be strict: Let M be a nonprojective module and consider an exact sequence 0 → ΩM → P(M)→ M → 0 with P(M) projective. Here pd P(M) = 0andmax{pd ΩM,pdM}=pd M 1, so pdP(M)<max{pd ΩM,pdM}. We call a sequence ε :0→ A → B → C → 0asequence with inequality if pd B< max{pdA,pd C}. We say that the inequality is finite if max{pdA,pd C} < ∞. We call the inequality infinite if max{pd A,pdC}=∞. The following lemma shows the relation between pdA and pdC if pd B<max{pdA,pd C}. Lemma 1.1. Let ε :0→ A → B → C → 0 be an exact sequence. Then ε is a sequence with inequality if and only if pd B<pd C. Moreover, (a) ε is a sequence with finite inequality if and only if pd B<pdC = pdA + 1 < ∞. 654 D. Madsen / Journal of Algebra 271 (2004) 652–672 (b) ε is a sequence with infinite inequality if and only if pdB<pdC = pd A =∞. Proof. We use the fact that for an R-module M,pdM i if and only if the functor i − = = ∞ ExtR(M, ) 0. Suppose ε is a sequence with inequality. Let pd B m< .Weget a long-exact sequence m − → m − → m+1 − → → m+1 − ExtR (B, ) ExtR (A, ) ExtR (C, ) 0 ExtR (A, ) → m+2 − → ExtR (C, ) 0 m − → m+1 − → We have an epimorphism of functors ExtR (A, ) ExtR (C, ) 0 and natural equivalences p − p+1 − ExtR(A, ) ExtR (C, ) for all p>m. We see that pd A =∞if and only if pdC =∞. Suppose pd A and pdC are finite. We must have pd A>mor pdC>m.IfpdC>m,thenpdA = pd C − 1. If pdA>m,then pdC = pd A + 1. So if both dimensions are finite, then pd A + 1 = pd C. Therefore if we have a finite inequality, then pdB<pdC = pd A + 1 < ∞. The converse is obvious. If we have an infinite inequality, then pd B<pdC = pdA =∞. The converse is obvious. From these two cases we see that ε is a sequence with inequality if and only if pdB<pd C. ✷ We now prove a simple but useful lemma about inequalities for general short-exact sequences. Lemma 1.2. Let 0 → A → B → C → 0 be an exact sequence. Suppose pdA = n<∞. (a) pd(B) pd(A) if and only if 1 n n+1 = ExtΛ Ω B,Ω C 0. (b) pd(B) < max{pd(A), pd(C)} if and only if + = 1 n n+1 = pd(A) 1 pd(C) and ExtΛ Ω B,Ω C 0. Proof. (a) We have exact sequences + 0 → Ωn 1C → P ΩnC → ΩnC → 0and0→ ΩnA → ΩnB P → ΩnC → 0, where P is some projective module. Taking pullback we get the following commutative diagram: D. Madsen / Journal of Algebra 271 (2004) 652–672 655 0 0 ΩnA ΩnA 0 Ωn+1C E ΩnB P 0 0 Ωn+1C P(ΩnC) ΩnC 0 00 The left vertical sequence splits, and therefore E is projective. We see that if 1 n n+1 = ExtΛ Ω B,Ω C 0, then the upper horizontal sequence will split, and ΩnB must be projective. Therefore 1 n n+1 = pd(B) n in this case. If pd(B) n, then ExtΛ(Ω B,Ω C) 0. (b) We know that pd(B) < max{pd(A), pd(C)} if and only if pd(B) pd(A) and pd(A) + 1 = pd(C). Combining this with (a) we see that pd(B) < max{pd(A), pd(C)} + = 1 n n+1 = ✷ if and only if pd(A) 1 pd(C) and ExtΛ(Ω B,Ω C) 0. If 0 → A → B → C → 0 is an exact sequence with finite inequality and pd A = n<∞, then 0 → ΩnA → ΩnB P → ΩnC → 0 is a projective resolution of ΩnC.Itis not necessarily minimal, but we must have ΩnA Ωn+1C P for some projective module P . If we have one exact sequence with inequality, we can use pushout and pullback diagrams to find more such sequences as in the following example. Example 1.3. In the diagram below if the bottom row has inequality, then the top row or the right column must have inequality: 0 0 0 X Y C 0 0 A B C 0 Z Z 00 656 D. Madsen / Journal of Algebra 271 (2004) 652–672 Proof. Suppose that the bottom row has inequality. Suppose that the top row and the right column have equality. Then pd C pdY pdB<pd C, a contradiction. So the top row or the right column must have inequality. ✷ 2. Background on almost split sequences Let Λ be an Artin algebra, that is a finitely generated algebra over some commutative Artin ring R. This will be the setup for the rest of the paper. (In the first version of the paper the setup was finite dimensional algebras over a field k, but as the referee pointed out my results hold for the more general case of Artin algebras.) We denote the category of finitely generated Λ-modules by modΛ. In this section we recall the definition of an almost split sequence and give some well known results about such sequences. An exact sequence f g ε :0→ A → B → C → 0 of finitely generated Λ-modules is called an almost split sequence if the following conditions hold: (1) ε is not split. (2) All morphisms X → C in modΛ that are not split epi factor through g. (3) All morphisms A → Y in modΛ that are not split mono factor through f . We can illustrate the definition with the following diagram: X not split epi f g 0 A B C 0.
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