Modules attached to extension bundles

Bulletin of the American Mathematical Society, 1962

has established the rudiments of a dictionary for translating the language of projective modules into that of vector bundles. With this point of departure we have attempted to adapt some of the results and methods of homotopy theory to certain purely arithmetic and even noncommutative settings. Detailed proofs of our results will appear elsewhere. The authors are very grateful to Albrecht Dold who has educated them on the relevant parts of bundle theory. We first recall briefly the topological setting which our theorems parody. [X, Y] denotes the homotopy classes of maps of X to F. If ƒ: A->B induces homotopy isomorphisms in dimensions Sr and if X is a finite CW complex of dimension ^w, then one shows easily (see Appendix] surjective for r^m and injective for r>m. As special cases we have I. [X, Bow]-*[X, Bo(r+i)] is surjective torrent and injective for r>m\ and II. [X, 0(r)]->[X, 0 (r + 1)] is surjective for r^m + 1 and injective for r>m + l. Here Bo( r ) is the classifying space of the orthogonal group 0(r). The classification theorem [ll, §19] says that [X, B 0 ( r )] represents the functor, [(equivalence classes of) real r-plane bundles over X], and then the map in I is obtained by adding a trivial line bundle. Moreover, the argument in [ll, §18] shows that [X, O(r)]/wo(0(r)) represents the functor [real r-plane bundles over SX], where SX is the (reduced) suspension of X. Atiyah and Hirzebruch [l ] define a functor K*(X) = K\X) ®K l (X) as follows: K°(X) is the Grothendieck ring of vector bundles over X. Choosing a base point gives K° an augmentation, and X X (X) is then the augmentation ideal of K°(SX). Thus I and II above define a kind of stable range for K° and K l respectively. Now let R be a commutative ring, X its spectrum of maximal ideals, and À an i^-algebra finitely generated as an jR-module. We require only that X be what might be called a "Zariski complex," i.e. that each closed set be a finite union of irreducible closed sets (see [4, §3]). IfPisa A-module and #GX then P x is the localization of P at x; we say /-rank P^r if P x contains a A^-free direct summand of rank r for all x. Note that P is required to be neither projective nor

2010

A module M is said to be generalized N-projective (or N-dual ojective) if, for any epimorphism g : N → X and any homomorphism f : M → X, there exist decompositions M = M1 ⊕ M2, N = N1 ⊕ N2, a homomorphism h1 : M1 → N1 and an epimorphism h2 : N2 → M2 such that g ◦ h1 = f|M1 and f ◦ h2 = g|N2 . This relative projectivity is very useful for the study on direct sums of lifting modules (cf. [5], [7]). In the definition, it should be noted that we may often consider the case when f to be an epimorphism. By this reason, in this paper we define relative (strongly) generalized epi-projective modules and show several results on this generalized epi-projectivity. We apply our results to the known problem when finite direct sums M1⊕· · ·⊕Mn of lifting modules Mi (i = 1, · · · , n) is lifting.</p

arXiv: K-Theory and Homology, 2017

We provide a framework connecting several well known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras. Finally, we provide a tool to evaluate the possible degrees of a module appearing in a graded projective resolution once the generating degrees for the first term of some particular projective resolution are known.

arXiv (Cornell University), 2013

This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted projective line of weight type (2, 2, n), and investigates the abelianness of the "missing part" from the category of coherent sheaves to the category of finitely generated right modules on the associated tilted algebra for each tilting bundle.

Gazi University Journal of Science

In this note, we investigate extensions of Baer and principally projective modules. Let R be an arbitrary ring with identity and M a right R-module. For an abelian module M, we show that M is Baer (resp. principally projective) if and only if the polynomial extension of M is Baer (resp. principally projective) if and only if the power series extension of M is Baer (resp. principally projective) if and only if the Laurent polynomial extension of M is Baer (resp. principally projective) if and only if the Laurent power series extension of M is Baer (resp. principally projective).

Cornell University - arXiv, 2020

The string group acts on the category of coherent sheaves over a weighted projective line by degree-shift actions. We study the equivariant equivalence relations induced by degree-shift actions between weighted projective lines. We prove that such an equivariant equivalence is characterized by an admissible homomorphism between the associated string groups. We classify all these equivariant equivalences for the weighted projective lines of domestic and tubular types.

Indian Journal of Pure and Applied Mathematics

Projective modules are a link between geometry and algebra as established by the theorem of Serre-Swan. In this paper, we define the super analog of projective modules and explore this link in the case of some particular super geometric objects. We consider the tangent bundle over the supersphere and show that the module of vector field over a supersphere is a super projective module over the ring of supersmooth functions. Also, we discuss a class of super projective modules that can be constructed from a projection map on modules defined over the ring of supersmooth functions over superspheres.

Journal of Commutative Algebra, 2009

Suppose R and S are local rings and (R, m) → (S, n) is a flat local homomorphism. Given a finitely generated S-module N , we say N is extended (from R) provided there is an R-module M such that S ⊗ R M is isomorphic to N as an S-module. If such a module M exists, it is unique up to isomorphism (cf. [EGA, (2.5.8)], and it is necessarily finitely generated. The m-adic completion R → R and the Henselization R → R h are particularly important examples. One reason is that the Krull-Remak-Schmidt uniqueness theorem holds for directsum decompositions of finitely generated modules over a Henselian local ring. Indeed, failure of uniqueness for general local rings stems directly from the fact that some modules over the Henselization (or completion) are not extended. Understanding which R h-modules are extended is the key to unraveling the direct-sum behavior of R-modules. Throughout, we assume that (R, m) and (S, n) are Noetherian local rings and that R → S is a flat local homomorphism. Many of our results generalize easily to a mildly noncommutative setting. Moreover, it is not always necessary to assume that our rings are local. Thus, we assume that A is a commutative ring, that B is a faithfully flat commutative A-algebra, and that Λ is a module-finite A-algebra. Given a finitely generated left B ⊗ A Λ-module N , we say that N is extended (from Λ) provided there is a finitely generated left Λ-module M such that B ⊗ A M is isomorphic to N as a B ⊗ A Λ-module. In Sections 1 and 2 of the paper, we examine how the extended modules sit inside the family of all finitely generated modules. In Sections 3-4 we consider rings of dimension 2 and 1, respectively, find criteria for a module to be extended , and show how the extendedness problem for one-dimensional rings reduces to the Artinian case. In Section 5, we find situations where every finitely generated B-module is a direct summand of an extended module, and in Section 6 we make a few observations about the Artinian case. 1. Two out of three: direct sums Our goal in this section is to prove the following theorem, which generalizes Proposition 3.1 of [FSW]: The research of W. Hassler was supported by the Fonds zur Förderung der Wissenschaftlichen Forschung, project number P20120-N18. Wiegand's research was partially supported by NSA Grant H98230-05-1-0243.

International Mathematics Research Notices, 2018

Let $\textrm{coh}\ \mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\textrm{coh}\ \mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\textrm{coh}\ \mathbb{X})$ attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star-shaped quiver ${Q}$ associated with $\mathbb{X}$. By further dealing with the Ringel–Hall algebra of $\mathbb{X}$, we show that these functors provide a realization for Tits’ automorphisms of the Kac–Moody algebra ${\mathfrak g}_{Q}$ associated with ${Q}$, as well as for Lusztig’s symmetries of the quantum enveloping algebra of ${\mathfrak g}_{Q}$.

Symmetry

A new category equivalence is proved in this paper, involving the two distinct categories of modules, the covariant and the contravariant, respectively, released by Higgins and Mackenzie. The equivalence of the two categories is given when restricting to almost finitely generated projective modules and their allowed morphisms, defined in the paper. The equivalence is expressed by using generators. In particular, we obtain the well-known equivalence of the two categories of projective finitely generated modules; thus, our main result extends this classical one.