The Sub-arrow Graph And The Overall Dimension

Homological algebra is one of most powerful tool to study the algebraic objects,which occupies a very significant position in the research of algebras.Homological di-mension is an important theme in homological algebra.The global dimension,as an important homological invariant,has been extensively studied,and it links to many conjectures.For an given quiver,clarifying the relation between ideals over the quiver and the global dimension is a natural problem.In the research of the global dimension of algebras which admits Am or Xm as a subquiver,Poettering provided a construction of admissible ideal I,such that the global dimension of algebra kQ/I is the given value.In this thesis,we mainly consider the global dimension of algebras and subalgebras,and by generalizing Poettering’s construction,we reveal the global dimension of algebras admits Am or Xm as a subquiver.The construction of the thesis is as follows:the second chapter introduce the related concepts in representation theory,and in the third chapter,we characterize the global dimension of Am type algebra and Xm type algebra in terms of relations.Finally,we provide a construction to determine the global dimension of algebras by the relations of the subquiver.To be precise,if a quiver Q admits Am or Xm as a subquiver,then the admissible global dimension on Am or Xm is also admissible on Q.