Tilting modules of finite projective dimension

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書誌事項

公開日
1986-03
権利情報
  • http://www.springer.com/tdm
DOI
  • 10.1007/bf01163359
公開者
Springer Science and Business Media LLC

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説明

A unital left A-module \({}_ AT\) over a ring A is said to be a tilting module of projective dimension \(\leq r\) (where \(r\geq 0\) is an integer) if it satisfies the following three conditions: (1) \({}_ AT\) has a projective resolution \[ 0\to_ AP_ r\to...\to_ AP_ 0\to_ AT\to 0 \] such that each \(P_ i\) is finitely generated. (2) \(Ext^ i_ A(T,T)=0\), if \(1\leq i\leq r\). (3) There exists an exact sequence \[ 0\to_ AA\to_ AT_ 0\to_ AT_ 1\to...\to_ AT_ r\to 0 \] such that each \(T_ i\) is a direct summand of a finite direct sum of copies of \({}_ AT\). (If \(r=1\) then \({}_ AT\) is exactly a tilting module.) Then the following hold. Firstly, a generalization of a theorem of Brenner-Butler holds. Namely, if \({}_ AT\) is a tilting module of projective dimension \(\leq r\), and \(End(_ AT)=B\), then \(T_ B\) also is a tilting module of projective dimension \(\leq r\), \(End(T_ B)\overset \sim \rightarrow A\), and \({}_ AT_ B\) induces category equivalences between some categories. If A and B are artinian algebras then the Grothendieck group of A and the Grothendieck group of B are isomorphic to each other. There results an inequality l.gl.dim \(A\leq l.gl.\dim B+r\). There are tilting modules associated with some simple modules. Those are generalizations of tilting modules of Auslander- Platzeck-Reiten and Brenner-Butler. Dualizing the above generalization of the theorem of Brenner-Butler, there results a generalized duality theorem.

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