Homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth.
Facts
Partially Attested
Progress Toward ResolutionThe direct summand conjecture was proven by Yves Andre using a theory of perfectoid spaces. 1 Some of the interrelated conjectures have been proven (such as the direct summand conjecture, by Andre using perfectoid spaces), while others in the group remain open; the group as a whole is not fully resolved. StatementThe homological conjectures are a number of interrelated conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Homological Conjectures in Commutative Algebra (Wikipedia)
Sources
1. Homological Conjectures in Commutative Algebra (Wikipedia)
Lead section
They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth.
Direct summand conjecture section
The conjecture was proven by Yves Andre using a theory of perfectoid spaces.
In Branch: Commutative Algebra, Lead sentence
ectures have been a focus of research activity in commutative algebra since the early 1960s.
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