Mark Kleiner

Almost split sequences for relatively projective modules

The paper deals with almost split sequences. Introduced in [2] for the category mod A of finitely generated modules over an artin algebra A, almost split sequences were later found in the category of lattices over an order [l, 43, as well as in certain subcategories of mod A [6, l&3]. It is generally recognized that if almost split sequences exist, the subcategory has nice properties. We are concerned with the subcategory of relatively projective modules. Let R be a field or a Dedekind domain with the field of quotients k, and let A and A be finite-dimensional R-algebras or R-orders, respectively, with A mapped into A via an R-algebra map i: A -+ A. Here we understand orders and lattices in the sense of [ 1, p. 85, Example (b)]. Namely, A is an R-order if it is a noetherian R-algebra projective as an R-module, and x = k OR A is a self-injective ring. A-mod denotes the category of finitely generated left A-modules if R is a field, or the category of left A-lattices if R is a Dedekind domain, where a left A-module h4 is a lattice if it is a finitely generated projective R-module such that k Ox M is a projective

Computation of Almost Split Sequences with Applications to Relatively Projective and Prinjective Modules

Let Λ be an Artin algebra, let modΛ be the category of finitely generated Λ-modules, and let A⊂modΛ be a contravariantly finite and extension closed subcategory. For an indecomposable and not Ext-projective module C∈A, we compute the almost split sequence 0→A→B→C→0 in A from the almost split sequence 0→D TrC→E→C→0 in modΛ. Since the computation is particularly simple if the minimal right A-approximation of D TrC is indecomposable for all indecomposable and not Ext-projective C∈A, we manufacture subcategories A with the desired property using orthogonal subcategories. The method of orthogonal subcategories is applied to compute almost split sequences for relatively projective and prinjective modules.

Abelian categories, almost split sequences, and comodules

The following are equivalent for a skeletally small abelian Horn-finite category over a field with enough injectives and each simple object being an epimorphic image of a projective object of finite length. (a) Each indecomposable injective has a simple subobject. (b) The category is equivalent to the category of socle-finitely copresented right comodules over a right semiperfect and right cocoherent coalgebra such that each simple right comodule is socle-finitely copresented. (c) The category has left almost split sequences.

Local theory of almost split sequences for comodules

SuntoDimostriamo che le successioni che quasi spezzano nella categoria dei comoduli su una coalgebra Γ con termine di destra di dimensione finita sono limiti diretti di successioni che quasi spezzano su sottoalgebre di dimensione finita. In un lavoro precedente abbiamo dimostrato che tali successioni che quasi spezzano esistono se il termine di destra ha un duale lineare quasi-finitamente copresentato. Viceversa, prendendo il limite delle successioni che quasi spezzano su categorie di comoduli di dimensione finita, dimostriamo che, per coalgebre di dimensione numerabile, esistono alcune successioni esatte che soddisfano una condizione più debole di essere quasi spezzanti, che noi chiamiamo “finitamente quasi spezzanti”. Sotto ipotesi aggiuntive, si dimostra che queste successioni quasi spezzano nelle opportune categorie.AbstractWe show that almost split sequences in the category of comodules over a coalgebraΓ with finite-dimensional right-hand term are direct limits of almost split sequences over finite dimensional subcoalgebras. In previous work we showed that such almost split sequences exist if the right hand term has a quasifinitely copresented linear dual. Conversely, taking limits of almost split sequences over finte-dimensional comodule categories, we then show that, for countable-dimensional coalgebras, certain exact sequences exist which satisfy a condition weaker than being almost split, which we call “finitely almost split”. Under additional assumptions, these sequences are shown to be almost split in the appropriate category.

The Graded Preprojective Algebra of a Quiver

The paper introduces a new grading on the preprojective algebra of an arbitrary locally finite quiver. Viewing the algebra as a left module over the path algebra, the author uses the grading to give an explicit geometric construction of a canonical collection of exact sequences of its submodules. If a vertex of the quiver is a source, the above submodules behave nicely with respect to the corresponding reflection functor. It follows that when the quiver is finite and without oriented cycles, the canonical exact sequences are the almost split sequences with preprojective terms, and the indecomposable direct summands of the submodules are the non-isomorphic indecomposable preprojective modules. The proof extends that given by Gelfand and Ponomarev in the case when the finite quiver is a tree.

Pairs of partially ordered sets of tame representation type

Abstract A criterion for tameness of representation type of a pair of partially ordered sets is obtained. An application to the study of torsionless modules over the tensor product of finite-dimensional associative l -hereditary algebras of dominant dimension ⩾ 1 is considered.

Schur's Lemma for partially ordered sets of finite type

It follows, for example, from the explicit description of the indecomposable representations of posets of finite type (51. Being interesting per se, Schur’s Lemma easily implies a bijection between the dimensions of indecomposable representations of a poset of finite type and the positive integral roots of its Tits form. That was shown in [ 1 ] according to the idea of [2] where the same implication was established for representations of quivers of finite type. This bijection allows one to prove directly certain qualitative results that otherwise follow only from the classification of representations. There are two published proofs of Schur’s Lemma for posets. One of them has a gap [7], and the other [I] is rather cumbersome. The purpose of this paper is to give a simple proof of the Lemma. Together with the well-known fact that one can canonically associate an indecomposable representation of a poset of finite type to every indecomposable representation of a quiver of finite type [2], the proof below constitutes perhaps the shortest known proof of Schur’s Lemma for quivers of finite type as well.