Slices
A slice S in mod B consists of a set of indecomposable B-modules with:
- S is a faithful set of modules.
- 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.)
- 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.)