scholarly journals On minimal projective resolution

2007 ◽  
Vol 2 ◽  
pp. 3079-3084
Author(s):  
M. A. Al Shumrani
Keyword(s):  
Projective Resolution ◽  
Minimal Projective Resolution

scholarly journals Complexity for modules over the classical Lie superalgebra

2012 ◽  
Vol 148 (5) ◽  
pp. 1561-1592 ◽  
Author(s):  
Brian D. Boe ◽  
Jonathan R. Kujawa ◽  
Daniel K. Nakano
Keyword(s):  
Lie Algebra ◽  
Lie Superalgebra ◽  
Projective Resolution ◽  
Lie Superalgebras ◽  
Simple Modules ◽  
Minimal Projective Resolution ◽  
Finite Dimensional ◽  
Reductive Lie Algebra ◽  
Completely Reducible ◽  
Kac Modules

AbstractLet ${\Xmathfrak g}={\Xmathfrak g}_{\zerox }\oplus {\Xmathfrak g}_{\onex }$ be a classical Lie superalgebra and let ℱ be the category of finite-dimensional ${\Xmathfrak g}$-supermodules which are completely reducible over the reductive Lie algebra ${\Xmathfrak g}_{\zerox }$. In [B. D. Boe, J. R. Kujawa and D. K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), 696–724], we demonstrated that for any module M in ℱ the rate of growth of the minimal projective resolution (i.e. the complexity of M) is bounded by the dimension of ${\Xmathfrak g}_{\onex }$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\Xmathfrak {gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.

τ-Rigid Modules for Algebras with Radical Square Zero

2021 ◽  
Vol 28 (01) ◽  
pp. 91-104
Author(s):  
Xiaojin Zhang
Keyword(s):  
Projective Resolution ◽  
Representation Type ◽  
Tilting Modules ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Minimal Projective Resolution ◽  
Nakayama Algebras ◽  
Radical Square Zero Algebra

For a radical square zero algebra [Formula: see text] and an indecomposable right [Formula: see text]-module [Formula: see text], when [Formula: see text] is Gorenstein of finite representation type or [Formula: see text] is [Formula: see text]-rigid, [Formula: see text] is [Formula: see text]-rigid if and only if the first two projective terms of a minimal projective resolution of [Formula: see text] have no non-zero direct summands in common. In particular, we determine all [Formula: see text]-tilting modules for Nakayama algebras with radical square zero.

Gröbner-Shirshov Basis and Minimal Projective Resolution of Uq+(B2)

2018 ◽  
Vol 25 (04) ◽  
pp. 713-720
Author(s):  
Lingling Mao ◽  
Jingqian Wang
Keyword(s):  
Global Dimension ◽  
Projective Resolution ◽  
Type B ◽  
Enveloping Algebra ◽  
Minimal Projective Resolution ◽  
Trivial Module ◽  
Quantized Enveloping Algebra

In this paper, by using the Anick resolution and Gröbner-Shirshov basis for quantized enveloping algebra of type B2, we compute the minimal projective resolution of the trivial module of [Formula: see text], and as an application we compute the global dimension of [Formula: see text].

scholarly journals Regular projective resolutions OF SEMIMODULES

2017 ◽  
Vol 126 (1B) ◽  
Author(s):  
Nguyễn Xuân Tuyến
Keyword(s):  
Comparison Theorem ◽  
Projective Resolution ◽  
Projective Resolutions

In this paper, we present an approach version of semimodule homologies by regular projective resolutions such as define a concept of a regular projective resolution, prove the comparison theorem for semimodules by these resolutions and based them provide cohomology monoids of semimodules.

Neat-phantom and clean-cophantom morphisms

2020 ◽  
pp. 2150172
Author(s):  
Lixin Mao
Keyword(s):  
Exact Sequence ◽  
Projective Resolution ◽  
Injective Envelope ◽  
Finitely Generated ◽  
The Relationship ◽  
Exact Sequences

Let [Formula: see text] be the class of all left [Formula: see text]-modules [Formula: see text] which has a projective resolution by finitely generated projectives. An exact sequence [Formula: see text] of right [Formula: see text]-modules is called neat if the sequence [Formula: see text] is exact for any [Formula: see text]. An exact sequence [Formula: see text] of left [Formula: see text]-modules is called clean if the sequence [Formula: see text] is exact for any [Formula: see text]. We prove that every [Formula: see text]-module has a clean-projective precover and a neat-injective envelope. A morphism [Formula: see text] of right [Formula: see text]-modules is called a neat-phantom morphism if [Formula: see text] for any [Formula: see text]. A morphism [Formula: see text] of left [Formula: see text]-modules is said to be a clean-cophantom morphism if [Formula: see text] for any [Formula: see text]. We establish the relationship between neat-phantom (respectively, clean-cophantom) morphisms and neat (respectively, clean) exact sequences. Also, we prove that every [Formula: see text]-module has a neat-phantom cover with kernel neat-injective and a clean-cophantom preenvelope with cokernel clean-projective.

The B∞-Structure on the Derived Endomorphism Algebra of the Unit in a Monoidal Category

2021 ◽  
Author(s):  
Wendy Lowen ◽  
Michel Van den Bergh
Keyword(s):  
Monoidal Category ◽  
Projective Resolution ◽  
Homotopy Category ◽  
Endomorphism Algebra ◽  
Coalgebra Structure ◽  
The Right ◽  
Hochschild Complex

Abstract Consider a monoidal category that is at the same time abelian with enough projectives and such that projectives are flat on the right. We show that there is a $B_{\infty }$-algebra that is $A_{\infty }$-quasi-isomorphic to the derived endomorphism algebra of the tensor unit. This $B_{\infty }$-algebra is obtained as the co-Hochschild complex of a projective resolution of the tensor unit, endowed with a lifted $A_{\infty }$-coalgebra structure. We show that in the classical situation of the category of bimodules over an algebra, this newly defined $B_{\infty }$-algebra is isomorphic to the Hochschild complex of the algebra in the homotopy category of $B_{\infty }$-algebras.

scholarly journals Regular projective resolutions OF SEMIMODULES

2017 ◽  
Vol 126 (1B) ◽  
pp. 43
Author(s):  
Ho Xuan Thang ◽  
Nguyen Xuan Tuyen
Keyword(s):  
Comparison Theorem ◽  
Projective Resolution ◽  
Projective Resolutions

In this paper, we present an approach version of semimodule homologies by regular projective resolutions such as define a concept of a regular projective resolution, prove the comparison theorem for semimodules by these resolutions and based them provide cohomology monoids of semimodules.

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