Failure of splitting from module-finite extension rings. (English) Zbl 0976.13006
The direct summand conjecture of Hochster asserts that if \(R\) is a regular subring of a commutative ring \(S\), then \(R\) is a module direct summand. This paper deals with a generalisation in which the regularity of \(R\) is replaced by the weaker condition that \(S\) has finite projective dimension as \(R\)-module. The authors use methods of algebraic geometry to show that the strengthened conjecture is false for rings of both prime and mixed characteristic. They also find several conditions under which the conjecture holds.
Reviewer: Phillip Schultz (Nedlands)
MSC:
13C05 | Structure, classification theorems for modules and ideals in commutative rings |
13D05 | Homological dimension and commutative rings |
13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |
13B02 | Extension theory of commutative rings |