Slices

A slice S in mod B consists of a set of indecomposable B-modules with:
  1. S is a faithful set of modules.
  2. S is path-closed
    (this means: If X0 → X1 → ... → Xn are indecomposable modules and non-zero maps,
    with X0 and Xn in S, then all Xi are in S.)
  3. If 0 → M' → M → M" → 0 is exact, and M has an indecomposable direct summand which belongs to S, then either M' or M" (but not both) belong to S.

Let T be a tilting A-module, A hereditary and B = End(T).
Then Hom(T,DA) is a slice in mod B. And any slice occurs in this way.

Thus: B is a tilted algebra if and only if mod B contains a slice.


The slice S is always contained in Y. Write Y' = Y \ S.


The triple (Y',S,X) is left-right symmetric.
(Note that the torsion pair (Y,X) may lack such a symmetry: Y is always faithful, but X may not be so.)