Extending Tilting Modules to One-Point Extensions by Projectives

Abstract

Let k be an algebraically closed field, B be a finite dimensional k-algebra and A be the one-point extension of B by the projective B-module P ₀. We compare the posets 𝒯 _A and 𝒯 _B of tilting A-modules and B-modules, respectively. We prove that the restriction and the extension functors define morphisms of posets r: 𝒯 _A  → 𝒯 _B and e: 𝒯 _B  → 𝒯 _A such that re = id. Moreover, e induces a full embedding of the quiver of 𝒯 _B into that of 𝒯 _A , whose image is closed under successors, and mapping distinct connected components of the first into distinct connected components of the second.

Key Words:

• One-point extension
• Posets of tilting modules
• Restriction and extension functors

1991 Mathematics Subject Classification:

• 16G70
• 16G20
• 16E10

ACKNOWLEDGMENTS

The first author gratefully acknowledges partial support from the NSERC of Canada. This work was done while the second named author was visiting the Université de Sherbrooke. He would like to thank the first named author for the invitation and kind hospitality during his stay. The third author is a researcher of CONICET. She gratefully acknowledges partial support from ANPCyT of Argentina and NSERC of Canada. She also thanks the members of the algebra group in Sherbrooke for their hospitality.

Notes

Communicated by D. Zacharia.