Abstract
It is well known that a partial tilting module may not be completed to a tilting module. However, it is still unknown whether a partial tilting module can be completed to a silting complex. The affirmative answer to this question will give an affirmative answer to the well-known rank question for tilting modules. In this paper, we prove that a partial tilting simple module can always be completed to a silting complex. More generally, we give the sufficient conditions for a partial tilting module to be completed to a silting complex.
Highlights
In this paper, we always let R be an artin algebra
In [4], Rickard and Schofield proved that the Complement Question for partial tilting modules has a positive answer for algebra of finite representation type
We extend the above two questions on tilting modules to silting complexes
Summary
We always let R be an artin algebra. As usual, modR denotes the category of all finitely generated left R-modules, D b (modR) denotes the correspondent bounded derived category, and K b (P R ). In [4], Rickard and Schofield proved that the Complement Question for partial tilting modules has a positive answer for algebra of finite representation type. They gave a counter-example to the question. Note that the counter-example by Rickard and Schofield is a partial tilting simple module of projective dimension 2 over a finite dimensional algebra of global dimension 4, so the Complement. It was shown that tilting complexes satisfy the rank-condition, i.e., the number of distinct indecomposable direct summands of a tilting complex is just the rank of K0 ( R) [5]. Rickard [5] gave a simple counter-example to the Complement Question for partial tilting complexes. We will suggest a new idea to consider the Rank Question for tilting modules via silting complexes and show that the above mentioned counter-examples will not be counter-examples to our new questions
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