Modules Of Finite Length Research Articles

Krull-Schmidt fails for Artinian modules

We prove that the Krull-Schmidt theorem fails for artinian modules. This answers a question asked by Krull in 1932. In fact we show that if S is a module-finite algebra over a semilocal noetherian commutative ring, then every nonunique decomposition of every noetherian S-module leads to an analogous nonunique decomposition of an artinian module over a related non-noetherian ring. The key to this is that any such S is the endomorphism ring of some artinian module. Krull asked, in [K '32, p. 38], whether the Krull-Schmidt theorem holds for artinian modules; that is, whether M1 ED ... ED Mm N ED ... ED Nn with each Mi and Ni an indecomposable artinian right module over some ring R, implies that m = n and, after a suitable renumbering of the summands, each Mi -Ni . Warfield showed [W '69, Proposition 5] that the answer is yes when the ring R is either right noetherian or commutative. He did this by showing that, over any ring, any artinian indecomposable module that is a union of modules of finite length has a local endomorphism ring. [Recall that direct sums of indecomposable modules with local endomorphism rings have unique directsum decompositions, even when the direct sum contains infinitely many terms.] Subsequently, Camps and Dicks showed that the endomorphism ring of any artinian module is semilocal (i.e. semisimple artinian modulo its Jacobson radical) [CD '93]. They concluded that artinian modules cancel from direct sums; that is, if M@ A _ M@ B with M artinian, then A _ B. In this note we answer Krull's question by showing that the Krull-Schmidt theorem fails for general artinian modules. The idea of the proof is that decompositions of modules correspond to decompositions of their endomorphism ring, in a natural way. This reduces the question to that of what kinds of Received by the editors October 25, 1993 and, in revised form, May 2, 1994. 1991 Mathematics Subject Classification. Primary 16P20, 16D70.

Wild torsion modules over Weyl algebras and general torsion modules over HNPs

Let A 1 be the first Weyl algebra over a field F of characteristic zero. We show that the category of A 1 -modules of finite length is wild, despite the fact that every A 1 -module of finite length is cyclic. In fact, the category of A 1 -modules (not necessarily finitely generated) of socle-height 2 is wild in a very strong sense. Among the applications, we show: (i) Essentially any F -algebra can occur as the endomorphism algebra of an A 1 -module of socle-height 2; (ii) A 1 has very large indecomposable modules of finite length; (iii) There is an HNP (hereditary Noetherian prime ring) that has indecomposable modules of finite length requiring arbitrary many generators. We also complete the basic theory of finitely generated modules over general HNPs, by showing that every such module is a direct sum of right ideals and homomorphic images of right ideals, and by proving a simultaneous decomposition theorem for an arbitrary projective module and submodule.

On the structure of local cohomology modules for monomial curves in

Our setting for this paper is projective 3-spaceover an algebraically closed fieldK. By a curveC⊂is meant a 1-dimensional, equidimensional projective algebraic set, which is locally Cohen-Macaulay. Letbe the Hartshorne-Rao module of finite length (cf. [R]). HereZis the set of integers andℐcthe ideal sheaf ofC. In [GMV] it is shown that, whereis the homogeneous ideal ofC,is the first local cohomology module of theR-moduleMwith respect to. Thus there exists a smallest nonnegative integerk∊Nsuch that, (see also the discussion on the 1-st local cohomology module in [GW]). Also in [GMV] it is shown thatk= 0 if and only ifCis arithmetically Cohen-Macaulay andCis arithmetically Buchsbaum if and only ifk≤ 1. We therefore have the following natural definition.

Directing projective modules

Let A be an Artin algebra. The A-modules which we consider are always left modules of finite length. If X, Y, Z are A-modules, the composition of maps f : X ~ Y and g : Y ~ Z is denoted by f 9 : X ~ Z. The category of (finite length) A-modules is denoted by A-mod. If X, Y are indecomposable A-modules, we denote by rad (X, Y) the set of non-invertible maps from X to Y. A path in A-mod is a sequence (X o . . . . . Xs) of (isomorphism classes of) indecomposable A-modules X i, 0 1, and X o = Xs, then the path (Xo, . . . , X~) is called a cycle. A indecomposable A-module is called directing if X does not occur in a cycle. Our first aim will be to extend the definition of a directing module to decomposable modules. We show that an indecomposable projective A-module P is directing if and only if the radical of P is directing. In case the top of P is injective it follows that P is directing if and only if the radical of P is directing as a module over the factor algebra of A by the trace ideal of P.

Preorders on canonical families of modules of finite length

AbstractLet R be an artinian ring. A family, ℳ, of isomorphism types of R-modules of finite length is said to be canonical if every R-module of finite length is a direct sum of modules whose isomorphism types are in ℳ. In this paper we show that ℳ is canonical if the following conditions are simultaneously satisfied: (a) ℳ contains the isomorphism type of every simple R-module; (b) ℳ has a preorder with the property that every nonempty subfamily of ℳ with a common bound on the lengths of its members has a smallest type; (c) if M is a nonsplit extension of a module of isomorphism type II1 by a module of isomorphism type II2, with II1, II2 in ℳ, then M contains a submodule whose type II3 is in ℳ and II1 does not precede II3. We use this result to give another proof of Kronecker's theorem on canonical pairs of matrices under equivalence. If R is a tame hereditary finite-dimensional algebra we show that there is a preorder on the family of isomorphism types of indecomposable R-modules of finite length that satisfies Conditions (b) and (c).

A deformation theory approach to betti numbers of finite length modules

Let R=k[x,, . xd] be a polynomial ring in d variables with d at least 2 over a field k and let A4 be a finite length R module. Buchsbaum and Eisenbud [B.E.] conjecture that the ith betti number, pi, should be at least (f). In their work on linkage Huneke and Ulrich [H.U.] prove this if M= R/I, where I is a licci ideal, by observing on pages 317-318 that after deforming and localizing they can assume that Z is a complete intersection. This paper extends this remark to ideals generated by monomials. The technique works because deformation preserves the betti numbers exactly while localization can only make them go down. In this paper we consider a sequence of localizations and deformations until we arrive at a module whose betti numbers we can compute. In [C] one of us showed that if R = kCx I, ..., xd] and Z was an ideal generated by monomials in the xi such that /(R/Z) < co and Z is not an R sequence then the pi satisfy stronger inequality that pi 2 (f) + (II::). In this paper we give a deformation theoretic proof of this by showing [Theorem l] that if Z is a finite colength ideal generated by monomials and is not a complete intersection then after a sequence of deformations and localizations we arrive at a new monomial ideal generated by d + 1 elements consisting of powers of each of the xi and one other monomial,

Betti Numbers for Modules of Finite Length

Let $R$ be a Gorenstein local ring of dimension $d < 5$ and let $M$ be a module of finite length and finite projective dimension. If $M$ is not isomorphic to $R$ modulo a regular sequence, then the Betti numbers of $M$ satisfy ${\beta _i}(M) > (_i^d)$ for $0 < i < d$, and $\sum \nolimits _{i = 0}^d {{\beta _i}(M) \geq {2^d} + {2^{d - 1}}}$.

Betti numbers for modules of finite length

Let R R be a Gorenstein local ring of dimension d &gt; 5 d &gt; 5 and let M M be a module of finite length and finite projective dimension. If M M is not isomorphic to R R modulo a regular sequence, then the Betti numbers of M M satisfy β i ( M ) &gt; ( i d ) {\beta _i}(M) &gt; (_i^d) for 0 &gt; i &gt; d 0 &gt; i &gt; d , and ∑ i = 0 d β i ( M ) ≥ 2 d + 2 d − 1 \sum \nolimits _{i = 0}^d {{\beta _i}(M) \geq {2^d} + {2^{d - 1}}} .

Complexes with homology of finite length and Frobenius functors

In this paper R is a Noetherian local ring of prime characteristic p, m its maximal ideal, and k its residue field. Let f: R + R(x H xP) be the Frobenius homomorphism. For an R-module M the R-module M,,, is A4 as an additive group with the scalar multiplication given by rOm=f”(r)m=rP”m for rER, REM. F; denotes the nth Frobenius functor, i.e., F”,(M) = MOR R,,, for each R-module M, where the R-module structure of F”,(M) is given by Y(M @ s) = m @ (rs) for r, s E R, m E M. For basic properties of these functors see [2, 5, and 91. Remember that the functors (n)~ (m) resp. F;“Ro Fz and (n + m) resp. Fi+“’ are isomorphic. The length of an R-module M is denoted by &(M), its Krull dimension by dim M or dim, M, and its projective dimension by pd, M. The ith homology of a complex M. will be written as Hi(M.); the support supp M. of M. is the set of all p E Spec R such that M. OR R, is not exact. A complex M. is called bounded if and only if Mi # 0 for only a finite number of iE Z. For a bounded complex M. with homology modules of finite length the Euler characteristic x(M.) = CieZ (l)‘&(Hi(M.)) is defined. The following theorem based on arguments of Dutta [2] and Monsky [8] is the key to our results:

Canonical Extensions of Harish-Chandra Modules to Representations of G

Let G be the group of R-rational points on a reductive, Zariskiconnected, algebraic group defined over R, let K be a maximal compact subgroup, and let g be the corresponding complexified Lie algebra of G. It is a curious fault of the current representation theory of G that for technical reasons one very rarely works with representations of G itself, but rather with a certain category of simultaneous representations of g and K. The reasons for this are, roughly speaking, that for a given (g,K)-module of finite length there are clearly any number of overlying rather distinct continuous G-representations, whose ‘essence’ is captured by the (g, K)-module alone. At any rate, this paper will propose a remedy for this inconvenience, and define a category of smooth representations of G of finite length which will, I hope, turn out to be as easy to work with as representations of (g, K) and occasionally much more convenient. It is to be considered a report on what has been to a great extent joint work with Nolan Wallach, and is essentially a sequel to [38].

On descent in dimension two and non-split Gorenstein modules

In this paper we present a ring with a Gorenstein module but no dualizing module. The context of this example will be illustrated in the historical overview presented below. In [21] Sharp furnished the definition of a Gorenstein module G of rank t over a local, finite-dimensional ring A: G is finitely generated and for p~Spec A the Bass number, ,LL~(#z, G), equals t if i= htA(fi) and equals zero otherwise. Sharp went further to show that any ring admitting a Gorenstein module must be Cohen-Macaulay, and also must have Gorenstein formal fibres [21]. We observe that a Gorenstein module G of rank one over an n-dimensional local ring (A, m, k) induces a natural equivalence of functors Ext;( -, G)=Hom,( -, E(k)) on modules of finite length. For this reason Fossum, Foxby, Griffith, and Reiten assigned to Gorenstein modules of rank one the name dualizing module [S]. In the same paper, it was obtained that any Gorenstein A-module is a direct sum of copies of the Gorenstein A-module of minimal rank (of which, up to isomorphism, there is only one) [S]. It is clear that if a ring has a dualizing module, then any of its Gorenstein modules are direct sums of the dualizing module. Foxby [S], Reiten [19], and Sharp [21] showed that a local, CohenMacaulay ring A admits a dualizing complex if and only if A has a dualizing module. On the other hand, Grothendieck, who initiated the notion of a dualizing complex, provided the following result:

Quelques propriétés des représentations sphériques pour les espaces symétriques réductifs

Let G H be a reductive symmetric space and suppose V is an admissible ( g , K )-module of finite length possessing a linear functional T ϵ Vsu which is fixed by h and H ∩ K . We prove that V can be mapped equivariantly into C ∞ ( G H ) such that T becomes the pull-back of the Dirac measure at the origin. Essential in the proof is the fact that the formal power series of certain matrix coefficients of V satisfy a system of differential equations with regular singularities.

Torsion-free pre-images of modules of finite length

The purpose of this note is to provide a simplified proof of the following result due to Crawley-Boevey, Kropholler and Linnell.

Binomial behavior of Betti numbers for modules of finite length

The problem which we address in this article is that of providing bounds for the Betti numbers of modules M of finite length over a regular local ring (R,m,k) of dimension n. The conjectured lower bound is that the /th Betti number βi{M), or β*{M) if we wish to call attention to the ring R, is at least (). We establish this inequality for finite length modules of monomial type (see definition below).

Modules of finite length and Chow groups of surfaces with rational double points

Modules of finite length and Chow groups of surfaces with rational double points

Modules of finite projective dimension with negative intersection multiplicities

Let x~ be the image of X~ in R. The main object of this paper is to study modules of finite length and finite projective dimension over R. One consequence of our study will be counterexamples to several conjectures concerning the behavior of intersection multiplicities for modules of finite projective dimension. Let / denote length. If S is a Noetherian ring, M, N are finitely generated S-modules, l(M| ) is finite, and d=pdsM<~, we may define, following Serre IS] d Zs(M, N)= z(M, N) = ~ ( 1)'/(TorS(M, N)).

Ring of Endomorphisms of a Finite Length Module

An example of a uniserial module ${M_R}$ of composition length 2, such that $S = \operatorname {End}({M_R})$ acting on the left is not right artinian, is given. An elementary proof of a known result, that the ring of endomorphism of a finite length quasi-injective module ${M_R}$ acting on the left is left artinian, is also given.

Equivalence of metric homomorphisms

Suppose K is an algebra with involution overk and A, B are K-modules on which are defined ɛ-Hermitian K-invariant forms with values ink. Metric homomorphisms of the module A into the module Β are called equivalent in the broad sense if one can be obtained from the other by multiplying by automorphisms of both modules, and equivalent in the narrow sense if one can be obtained from the other by multiplying by an automorphism of B. Necessary and sufficient conditions are given for the broad and narrow equivalence of two metric homomorphisms of one semisimple module of finite length into another. As a consequence, a classification of representations of one quadratic form by means of another is obtained.

On a theorem of kulikov for artinian rings

For a right artinian Ring A maximal submodules of pure projective right A-modules (i.e. of direct sums of modules of finite length) are again pure projective if and only if the category of finitely generated right A-modules is right noetherian, equivalently, if right glob. dim . In this case pure projectivity generally is hereditary in Mod-A (Kulikov-property). The Kulikov-property can be tested by generalized almost split sequences, the test also works in the case of Oedekind domains.

Partial Coxeter functors and right pure semisimple hereditary rings

An important problem in the representation theory of artinian rings is to obtain a charcterization of rings of finite representation type (see [2, 7, 16, 201). We recall that a ring R is of finite representation type if R is both left and right artinian and there is only a finite number of isomorphism classes of indecomposable finitely generated R-modules. The main tools for studying the representation theory of hereditary artinian rings are partial Coxeter functors and Coxeter functors (see [4, 8, 10, 141). They are successfuly applied in the investigation of hereditary rings of finite representation type [8, 9-12, 141. In the present paper we study partial Coxeter functors for hereditary artinian rings and we use them as a tool for studying right pure semisimple hereditary rings (which are close to rings of finite representation type). We recall that for any ring R the right pure global dimension r.P.gl.dim R and the left pure global dimension l.P.gl.dim R are defined [19-21, 261. There is an interesting problem how to characterize rings R with r.P.gl.dim R = 0 which are exactly those right artinian rings R all of whose right R-modules are algebraically compact [ 19, 20, 261 or, equivalently, all of whose right R-modules are direct sums of modules of finite length. A ring R with r.P.gl.dim R = 0 is called right pure semisimple [25-271. It is well known that any ring of finite representation type is both left and right pure semisimple 12, 20, 23, 251 and the converse implication also holds true [2, 15, 201. However, it is still an open question if a right pure semisimple ring R is of finite representation type or, equivalently, if r.P.gl.dim R = 0 implies l.P.gl.dim R = 0 (see [29]). A positive solution of this problem is given by Auslander [3] for artin algebras. He also discusses the problem for arbitrary left artinian rings (see also 1181). We recall that a 195 0021.8693/81/070195~24$02.00/0