Endomorphism

Abstract: Let X(Se) be the set of all bisequences over a symbol set 6 a, where 1 < card S# < 0% and let cr be the shift transformation. If the product topology induced by the discrete topology of 6: is assigned to X(6a), X(6 a) is homeomorphic to the Cantor discontinuum and ~ is a homeomorphism of X(6Q onto X(6Q. The discrete flow (X(SQ, a) is the symbolic flow over 5: or the shift dynamical system over S a. The shift dynamical system (X(S:), a) has been analyzed rather thoroughly, both in its topological and in its measure-theoretic aspects [1, 7, 9, 10, 13, 14]. Recent work of Smale [27] shows that the shift dynamical system is ubiquitous. Any closed subset of X(SQ which is invariant under a defines a subdynamical system. There is an endless variety of these and they have served as useful models to indicate possible structures of dynamical systems, particularly minimal sets [7, 8, 9, 10, 11, 12, 15, 20, 22, 23, 24, 25]. These systems are characterized by the fact that the phase space is totally disconnected and the transformation is expansive (Section 2). However, there are dynamical systems, notably geodesic flows on compact manifolds of negative curvature, for which the phase space is a manifold, yet the orbits can be characterized by symbolic bisequences [2, 3, 4, 5, 20, 21, 28]. Properties of such dynamical systems and their subdynamical systems can be determined from knowledge of the properties of symbolic flows. A natural question in connection with any dynamical system is that of the existence and properties of continuous transformations which commute with the group action. In the case of the system (X(S:), ~), an obvious example of such a transformation is obtained by simply permuting the symbols. A generalization of this is to define a mapping of blocks (words) of symbols of a specified length into single symbols and to extend this mapping in a natural manner to infinite sequences. It has been shown by Curtis, Hedlund and Lyndon that these mappings, composed with powers of the shift, constitute the entire class of continuous transformations which commute with the shift (Section 3). This fundamental

Abstract: Let R be a regular local ring, and let A = R/(x), where x is any nonunit of R. We prove that every minimal free resolution of a finitely generated A -module becomes periodic of period 1 or 2 after at most dim A steps, and we examine generalizations and extensions of this for complete intersections. Our theorems follow from the properties of certain universally defined endomorphisms of complexes over such rings. Let A be a commutative ring, and let x G A be a nonzero divisor. How does homological algebra over A/(x) = B differ from that over AI In this paper we will study a certain natural endomorphism / of complexes of free A / (x)-modu\es which seems to reflect some of the difference. For example, the (homotopic) triviality of t is an obstruction (closely related to the usual one in Ext2,) to the lifting of a complex of free 5-modules to a complex of free /I-modules. More generally, if x,, . . . , xn is an A -sequence, we study « natural endomorphisms /,,..., tn of complexes of free A/(xx, . . . , x")-modules, and try to use them to explain the way in which free resolutions over A/(xx, . . . , x") differ from free resolutions over A (the construction and elementary properties of these endomorphisms is given in §1). In this paper, we will study the case in which A is a regular local ring and B = A/(xx, . . . , x") is not regular. (It would also be very interesting to understand the case in which both A and A/(x) = B were regular-with, say, A of mixed characteristic and B ramified or of characteristic p.) In this case, the homological algebra over A is dominated, roughly speaking, by the fact that minimal /I-free resolutions are finite; we seek to understand the eventual behavior of minimal 5-free resolutions in terms of the tt. For example, if « = 1, so that B = A/(x), we prove that / is eventually an isomorphism, so that every minimal 5-free resolution becomes periodic of period 2 after at most 1 + dim B steps (§6). We also show that the 5-modules with periodic resolutions are the maximal Cohen-Macaulay modules without free direct summands. Since the periodic part of a periodic resolution over A/(x) (or more generally, over A/(xx, . . . , xn), if x,, . . . , x" is an A -sequence) is easy to describe explicitly (§5), this yields information on maximal Cohen- Macaulay modules.

Abstract: This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

Abstract: Many basic definitions and results in the theory of near-rings can be found in G. Pilz (4). We follow these for the most part, except that we use left near-rings rather than right near-rings. We follow exactly an earlier paper, Meldrum (2), where there are detailed definitions and many results relating to faithful d.g. near-rings. Let R be a d.g. near-ring, distributively generated by the semigroup S, which need not be the semigroup of all distributive elements. Denote such a d.g. near-ring by (R, S). Then (R, +) = Gp where is a set of defining relations in S. Let (T, U) be a d.g. near-ring. Then a d.g. homomorphism θ from (R, S) to (T, U) is a near-ring homomorphism from R to T which satisfies Sθ ⊆ U. If (G, +) is a group, let T0(G) be the near-ring of all maps from G to itself with pointwise addition and map composition. Let End G be the semigroup of all endomorphisms of G. Then (E(G), End G) is a d.g. near-ring. A d.g. near-ring (R, S) is faithful if there exists a d.g. monomorphism θ:(R, S) → (E(G), End G) for some group G.