A Tiled Order of Finite Global Dimension with No Neat Primitive Idempotent

Abstract

Let R be a discrete valuation ring with residue class field F. For an arbitrary prime p, we construct a tiled R-order Λ such that global dimension gld Λ = 5 if characteristic char F ≠ p and gld Λ = ∞ if char F = p. If char F ≠ p, Λ is a tiled R-order of finite global dimension with no neat primitive idempotent, so that a question posed in Fujita (2002) is solved.

Key Words:

• Global dimension
• Neat primitive idempotent
• Representation of poset
• Tiled order

2000 Mathematics Subject Classification:

• 16E10
• 16G20
• 16G30
• 16S50

ACKNOWLEDGMENTS

The authors would like to thank the referee for helpful comments to improve the article. The first-named author was partially supported by JSPS Grant-in-Aid for Scientific Research ((C) 18540011).

Notes

Communicated by J. Kuzmanovich.