The Direct Summand Conjecture in Dimension Three
Copyright policy )The Direct Summand Conjecture in Dimension Three
The direct summand conjecture asserts that if R is a regular local ring and S is a module-finite R-algebra containing R, then R is a direct summand of S as an R-module. It was previously known to be true if R contains a field or if dim R is at most two. In this article, the result is demonstrated for mixed characteristic rings of dimension three. The proof is accomplished by showing that an extension of plus closure has the colon-capturing property in dimension three.
14 pages, no figues
- The University of Texas at Austin United States
monomial conjecture, 13D22, mixed characteristic, Structure, classification theorems for modules and ideals in commutative rings, big Cohen-Macaulay modules conjecture, direct summand conjecture, excellent rings, Integral closure of commutative rings and ideals, Mathematics - Commutative Algebra, Commutative Algebra (math.AC), tight closure, colon-capturing property, FOS: Mathematics, homological conjectures, Homological conjectures (intersection theorems) in commutative ring theory, extended plus closures
monomial conjecture, 13D22, mixed characteristic, Structure, classification theorems for modules and ideals in commutative rings, big Cohen-Macaulay modules conjecture, direct summand conjecture, excellent rings, Integral closure of commutative rings and ideals, Mathematics - Commutative Algebra, Commutative Algebra (math.AC), tight closure, colon-capturing property, FOS: Mathematics, homological conjectures, Homological conjectures (intersection theorems) in commutative ring theory, extended plus closures
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