拉回环上导出范畴的伴随函子

Adjoint Functors on Derived Category of the Pullback Rings

设R_1,R_2,R'是3个有单位元的结合环,环R是环同态j_1:R_1→R'和j_2:R_2→R'的拉回环.首先引入了左R_1-模复形范畴与左R_2-模复形范畴的积范畴的一个子范畴C(T),利用拉回函子方法构造了一个加法函子P:C(T)→C(R-Mod),以及S:C(R-Mod)→C(T),证明了(S,P)是一对伴随对函子.其次,在此基础上,研究了相应的左导出范畴,也得到相应左导出范畴之间的伴随对函子.最后通过一个例子说明在同伦范畴上没有相应的伴随对函子.

Let R1 ,R2,R＇ be associative rings with identity, R be a pullback ring along with two homomorphisms of rings j1： R1→R＇ and j2： R2→R＇. It introduces a subcategory C （J） of product category of the left R1-modules complex category and left RE-modules complex category, and then construct additive functors P： C（J）→C（R - Mod） ,S： C（R - Mod）→C（J）, and show that （S, P） is a pair of adjoint functors in terms of the method of pullbacks. We also get the result on the corresponding left-derived categories. Finally it gives an example for illustrating that conclusion on the homotopy category is not hold.