Finitely generated module

Summary

In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group. Definition The left R-module M is finitely generated if there exist a1, a2, ..., an in M such that for any x in M, there exist r1, r2, ..., rn in R with x = r1a1 + r2a2 + ... + rnan. The set {a1, a2, ..., an} is referred to as a generating set of M in this case. A finite generating set need not be a basis, since it ne

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Generic direct summands of tensor products for simple algebraic groups and quantum groups at roots of unity

Jonathan Gruber

Let G be either a simple linear algebraic group over an algebraically closed field of characteristic l>0 or a quantum group at an l-th root of unity. The category Rep(G) of finite-dimensional G-modules is non-semisimple. In this thesis, we develop new techniques for studying Krull-Schmidt decompositions of tensor products of G-modules.More specifically, we use minimal complexes of tilting modules to define a tensor ideal of singular G-modules, and we show that, up to singular direct summands, taking tensor products of G-modules respects the decomposition of Rep(G) into linkage classes. In analogy with the classical translation principle, this allows us to reduce questions about tensor products of G-modules in arbitrary l-regular linkage classes to questions about tensor products of G-modules in the principal block of G. We then identify a particular non-singular indecomposable direct summand of the tensor product of two simple G-modules in the principal block (with highest weights in two given l-alcoves), which we call the generic direct summand because it appears generically in Krull-Schmidt decompositions of tensor products of simple G-modules (with highest weights in the given l-alcoves). We initiate the study of generic direct summands, and we use them to prove a necessary condition for the complete reducibility of tensor products of simple G-modules, when G is a simple algebraic group of type A_n.

EPFL2022

Varieties of modules for Z/2Z x Z/2Z

Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety C, that is the set of pairs of (n x n)-matrices (A, B) such that A(2) = B-2 = [A, B] = 0, is equidimensional. C can be identified with the 'variety of n-dimensional modules' for Z/2Z x Z/2Z, or equivalently, for k[X, Y]/(X-2, Y-2). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic > 2. We also prove that if e(2) = 0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of C as a direct sum of indecomposable components of varieties of Z/2Z x Z/2Z-modules. (c) 2007 Elsevier Inc. All rights reserved.

Elsevier2007

Why H Z-algebra Spectra are Differential Graded Algebras?

Varvara Karpova

In homological algebra, to understand commutative rings R, one studies R-modules, chain complexes of R-modules and their monoids, the differential graded R-algebras. The category of R-modules has a rich structure, but too rigid to efficiently work with homological invariants and homotopy invariant properties. It appears more appropriate to operate in the derived category D(R), which is the homotopy category of differential graded R-modules. Algebra of symmetric spectra offers a generalization of homological algebra. In this frame, spectra are objects that take the place of abelian groups; in particular, the analogue of the initial ring Z is the sphere spectrum S. Tensoring over S endows the category of spectra with a symmetric monoidal smash product, analogous to the tensor product of abelian groups. Thus, spectra are S-modules, and ring spectra, which extend the notion of rings, are the S-algebras. To any discrete ring R, one can associate the Eilenberg-Mac Lane ring spectrum HR, which is commutative if R is.

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