Some Researches On ?-tilting Theory

This paper mainly discussed the problem about theory of support ?-tilting,such as support ?-tilting pairs,the Hasse quiver of support ?-tilting module of an algebra,the torsion pairs induced by support ?-tilting modules or r-rigid modules,etc.In this paper,I will study such problems as follows:(1)The number of arrows which start and end at the vertex of the support ??tilting pair in the Hasse quiver of support ?-tilting modules for a finite dimensional algebra is fixed.It is similar to the mutation between tilting modules,support?-tilting modules(?-tilting pairs)also have the mutation relationships.The Hasse quiver is defined based on mutation.In chapter one we should prove:when the dimension of an algebra is finite,we make sure that the number of arrows which start and end at the vertex of a support ?-tilting pair is fixed,i.e,the number of nonisomorphic simple modules.(2)We consider the heredity of support ?-tilting modules of trivial extension of finite algebra.For the trivial extensional algebra R=?×R,if T is a support ?-tilting module of ?-mod,then T could induce a support ?-tilting module T(?)M of R-module.If T is a right ?-module,R=?×R,M is a ?-?-bimodule,then(T,P)is a support ?-tilting pair,TOM is generated by T if and only if(T(?)R,P(>)R)is a support ?-tilting pair of right R-module.(3)?-rigid modules on the pullback rings.For the category of pullback rings,the ?-rigid module on pullback ring could induce a ?-rigid module on the initial ring.(4)In the recollement of derived categories,presilting complex in the leftmost or rightmost could induce presilting complex in the derived category which in middle of the recollement.In recollement of module categories,some ?-rigid modules in the rightmost module category could induce two torsion pairs in it,these two torsion pairs then induce a thick subcategory in this module category,this thick subcategory then also induce thick subcategories in leftmost and middle respectively.