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Phantom Homology
Melvin Hochster and Craig Huneke
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Memoirs of the American Mathematical Society
1993; 91 pp; softcover
Volume: 103
ISBN-10: 0-8218-2556-9
ISBN-13: 978-0-8218-2556-3
List Price: US$34
Individual Members: US$20.40
Institutional Members: US$27.20
Order Code: MEMO/103/490
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This book uses a powerful new technique, tight closure, to provide insight into many different problems that were previously not recognized as related. The authors develop the notion of weakly Cohen-Macaulay rings or modules and prove some very general acyclicity theorems. These theorems are applied to the new theory of phantom homology, which uses tight closure techniques to show that certain elements in the homology of complexes must vanish when mapped to well-behaved rings. These ideas are used to strengthen various local homological conjectures. Initially, the authors develop the theory in positive characteristic, but it can be extended to characteristic 0 by the method of reduction to characteristic \(p\). The book would be suitable for use in an advanced graduate course in commutative algebra.

Readership

Algebraists and algebraic geometers interested in a deeper understanding of commutative algebra.

Table of Contents

  • Minheight and the weak Cohen-Macaulay property
  • Acyclicity criteria with denominators for complexes of modules
  • Vanishing theorems for maps of homology via phantom acyclicity
  • Regular closure
  • Intersection theorems via phantom acyclicity
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