Examples of Projective Resolution of Lengths 3n/4n That Do Not Satisfy Homological Properties of Nakayama Algebra

2018 ◽  
Vol 28 (2) ◽  
pp. 1-9
Author(s):  
Kwasi Gyam ◽  
Abraham Aidoo ◽  
Yarhands Arthur
Keyword(s):  
Projective Resolution ◽  
Nakayama Algebra

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.

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.

The stable group of a selfinjective nakayama algebra

1991 ◽  
Vol 19 (2) ◽  
pp. 509-517 ◽  
Author(s):  
Roberto Martínez Villa
Keyword(s):  
Stable Group ◽  
Nakayama Algebra

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.

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.

scholarly journals A NOTE ON RESOLUTION QUIVERS

2013 ◽  
Vol 13 (03) ◽  
pp. 1350120 ◽  
Author(s):  
DAWEI SHEN
Keyword(s):  
Connected Component ◽  
Nakayama Algebra

Recently, Ringel introduced the resolution quiver for a connected Nakayama algebra. It is known that each connected component of the resolution quiver has a unique cycle. We prove that all cycles in the resolution quiver are of the same size. We introduce the notion of weight for a cycle in the resolution quiver. It turns out that all cycles have the same weight.

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