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Phantom Homology

Available Formats:
Electronic ISBN: 978-1-4704-0067-5
Product Code: MEMO/103/490.E
91 pp
List Price: $36.00 MAA Member Price:$32.40
AMS Member Price: $21.60 Click above image for expanded view Phantom Homology Available Formats:  Electronic ISBN: 978-1-4704-0067-5 Product Code: MEMO/103/490.E 91 pp  List Price:$36.00 MAA Member Price: $32.40 AMS Member Price:$21.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1031993
MSC: Primary 13;

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.

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

• Chapters
• 1. Introduction
• 2. Minheight and the weak Cohen-Macaulay property
• 3. Acyclicity criteria with denominators for complexes of modules
• 4. Vanishing theorems for maps of homology via phantom acyclicity
• 5. Regular closure
• 6. Intersection theorems via phantom acyclicity
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Volume: 1031993
MSC: Primary 13;

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.