We introduce a class of modules, called modules with cores, larger than any previously known fundamental class of indecomposable modules. The core of a.
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 82, Number 4, July 1976
INDECOMPOSABLE MODULES : MODULES WITH CORES BY ROBERT GORDON1 AND EDWARD L. GREEN 2 Communicated by Barbara L. Osofsky, March 8, 1976
We introduce a class of modules, called modules with cores, larger than any previously known fundamental class of indecomposable modules. The core of a module M is the intersection, C(M), of its nonsuperfluous submodules; that is, the intersection of submodules N such that N 4- N'= M for some proper submodule N*. Dually the cocore, C°(M), is the sum of the nonessential submodules of M. Any module with a core (meaning C(M) ¥= 0) or a cocore (C°(M) -=hM) is indecomposable. Our main aim is to describe a classification of radical squared zero Artin algebras such that every indecomposable module of finite length has a core or a cocore. (Finite dimensional algebras over a field are the most important examples of Artin algebras.) The classification shows that there are modules having a core and no cocore. This establishes our claim concerning the size of the class of modules with cores even in the restricted setting of radical squared zero Artin algebras of finite representation type—for modules with waists (see [1]) have a core and a cocore and include, as well as serial modules, modules with simple socles or, dually, simple tops. The proof of the following existence theorem is constructive and uses a result announced in [5] concerning indecomposability of amalgamations of basic modules of finite length. (A nonzero module is basic if it is its own core. For modules of finite length this just means that the module has a simple top.) THEOREM. If a left Artin ring has a basic module containing three nonisomorphic simple submodules then it has an indecomposable module of finite length with neither a core nor a cocore, D
It is an open question whether the theorem is valid without the assumption that the specified simple modules are nonisomorphic. Indeed the assumption is unnecessary when the underlying ring is an Artin algebra with square 0 radical. This fact, which can be verified using techniques of Auslander and Reiten [2], plays a role in the proof of our main result. We denote the length of a module AMS (MOS) subject classifications (1970). Primary 16A64; Secondary 16A46, 16 A4 8. Research partially supported by NSF Grant MPS 75-06327. Research partially supported by NSF Grant MPS 71-03064. Copyright © 1976, American Mathematical Society
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M by l(M), the projective cover by P(M) and the injective envelope by E(M). THEOREM. The following are equivalent properties of a radical squared zero Artin algebra R. (i) Every indecomposable module of finite length has a core or a cocore. (ii) Either every indecomposable module has a core, or else every indecomposable module has a cocore. (iii) (a) Every indecomposable projective module and every indecomposable injective module has length at most 3, and (b) either /(F(rad 0 ) < 5 for each indecomposable projective Q such that l(Q) = 3 or IQXF/xad F)) < 5 for each indecomposable injective F such that 1(F) = 3. (iv) The separated diagram of R is a finite disjoint union of diagrams of the following types:
At:
1 , 1
A2:
1 2«-3
A*:
1 - + 2 3
A4:
1^—2 —> 3 24 3 +~ 4 —> 5 £2:
10 2
, 1 —>2
£*: 1 — 2 , 1 O
2
C 3 : 1 3 , 1 C*: 1 O
3 2 ^ - 3
2 3.
Moreover, ifR has any of the properties (i)-(iv) then (c) R has finite representation type', (d) if R is indecomposable and hereditary then there are at most 5 nonisomorphic indecomposable projective R-modules. D Concerning (d) we should mention that in the nonhereditary case it is not possible to prescribe a bound on the number of nonisomorphic indecomposable projective modules. We should also mention how to find the separated diagram [3] ofR. For this let (Rit T= [
Vr
O \
,
Rlt/9
r = radtf,
and let Px, . . . , Pn be a full set of nonisomorphic indecomposable projective T-modules. Writing down the numbers 1, . . . , n, and drawing the same number of arrows from i to ƒ as there are copies of P/rad P- in rad P., we obtain the
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left diagram, D(TT), of T. Similarly, using the T-duals of the P., we obtain the right diagram, D(TT)\ and then the separated diagram of R is the ordered pair (D(TT), D{TT)). The subscripted letters A, Bf and C indicate the Dynkin diagrams (see [4] and [6] ) the designated separated diagrams come from. As a corollary to the Theorem one obtains that every indecomposable Z?-module has a core if and only if the separated diagram of R is composed of diagrams of types A j , A2, Av A\, A^ A5, B2, Z?|, and C 3 . The corresponding homological criterion is gotten from part (iii) of the Theorem by deleting the second half of the statement in (b). Note that there is a good supply of modules with cores and cocores which do not have waists. The proofs of these results are very long and will appear in a monograph on modules with cores. REFERENCES 1. M. Auslander, E. L. Green, and I. Reiten, Modules with waists. II, J. Math. 19 (1975), 4 6 7 - 4 7 8 . 2. M. Auslander and I. Reiten, Representation theory of Artin algebras, III: Almost split sequences, Communications in Algebra 3 (1975), 239—294. 3. V. Dlab and C. M. Ringel, On algebras of finite representation type, J. Algebra 33 (1975), 3 0 6 - 3 9 4 . MR 50 #9974. 4. P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 7 1 - 1 0 3 ; Correction, ibid. 6 (1972), 309. MR 48 #11212. 5. R. Gordon and E. L. Green, Indecomposable modules: Amalgamations, Bull. Amer. Math. Soc. (to appear). 6. W. Muller, Unzerlegbare Moduln Uber artinschen Ringen, Math Z. 137 (1974), 197-226. MR 50 #4665. DEPARTMENT OF MATHEMATICS, TEMPLE UNIVERSITY, PHILADELPHIA, PENNSYLVANIA 19121 DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS, URBANA, ILLINOIS 61801
