The Duality Between Subcategories Induced By Higher Dimendsional Cotilting Modules

Tilting theory is a central topic in the representation theory of finite dimensional ( Artin algebra).It can be seen the generalization of Morita Equivalence and has extensive interaction with various research directions in representation theory. Cotilting theory can be seen as the duality of tilting theory in finite dimensional algebra.It is useful to give the cotilting theory directly. We can not get the cotilting modules by duality in general case ,and as the main parts of higher dimensional cotilting theory ,the basic theorems of higher dimensional cotilting modules have not been stated concretely in previous papers.In this paper We give the basic theorems of higher dimensional cotilting modules directly ,they are the followings( A be a finite dimensional algebra over a field k):1 Let T ∈A-mod be a r-cotilting module B = End_A(T)^op, then we have (i) T_B is also a r-cotilting module .(ii) A ≌ End_B(T_B),a |→ (t |→at), a ∈ A,t ∈ T.2 Let T ∈A-mod be a r-cotilting module B = End_A(T)^op,0 ≤ e ≤ r be an integer.We denote_AT^e={_AX ∈ A-mod| Ext_Aⁱ(X,T) = 0,(?) i ≥ 0, i≠e } andT_B^e={Y_B ∈mod-B| Ext_Bⁱ(Y,T) = 0,(?) i≥0,i≠e}.then we have Ext_A^e(—,T) |_AT^e: _AT^e → T_B^e is a duality functor and its inverse functor is Ext_B^e(—,T) |_TB^e :T_B^e→ _AT^e.In fact the above theorems are right in the finite generated module categories of arbitrary associate rings.By using the previous theorems ,We give an direct proof application of the corollary 5.10 of [AR] which is the theorem IV of this papers:If T ∈A-mod is a r-cotilting module,then we have X_T is functorially finite in A-mod.