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Entirely new invariant in commutative (and non-commutative) algebra

World Scientific

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Credit: World Scientific, 2015

World Scientific's newly published book A Non-Hausdorff Completion: The Abelian Category of C-complete Left Modules over a Topological Ring, introduces an entirely new invariant in commutative (and non-commutative) algebra and in homological algebra, and opens up a whole new area for study, with also probable applications to algebraic geometry and algebraic topology - including p-adic cohomology in algebraic geometry.

It leaves many directions that both young and experienced researchers can pursue to create new results and applications, in an entirely new field. There are no competing books on the subject. The new invariant is the C-completion of a module over a commutative topological ring - or, more generally, of an abstract left module over a not-necessarily-commutative topological ring.

The C-completion and associated spectral sequences can be very useful in computing the p-adic cohomology of varieties, or schemes, in characteristic p>0. Also, the category of C-complete left modules over a topological ring is a very interesting new abelian category: It always has enough projectives, and direct products and inverse limits are as expected: But, although infinite direct sums and direct limits exist, they are unusual. For example, the category of C-complete modules over a discrete valuation ring O that is not a field, is an exact full abelian subcategory of the category of abstract O-modules, but the Eilenberg-Moore Axiom (S1) but not (S2). Examples of abelian categories obeying (S1) but not (S2) are very difficult to find - and C-complete O-modules are a very natural such example.

The new invariants and tools in this book are expected to be used in the study of p-adic cohomolgy in algebraic geometry; and also in the study of p-adic Banach spaces -- by replacing the cumbersome "complete tensor product" of p-adic Banach spaces, with the more sophisticated "C-complete tensor product", discussed in this book.

It is also not unlikely that the further study of these new invariants may well develop into a new branch of abstract mathematics - connect with commutative algebra, homological algebra, and algebraic topology.

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The book retails US$88 / £58 at leading bookstores.. For further information regarding the book, please visit http://www.worldscientific.com/worldscibooks/10.1142/9553. If you require a copy of the book for media review or would like to interview the author(s), please contact Jason CJ, cjlim@wspc.com.sg, +65-64665775, ext 247

ABOUT THE AUTHOR

Saul Lubkin's major fields of interest are Algebraic Geometry, Homological Algebra, Algebraic Topology and Commutative Algebra.

He did his undergraduate work at Columbia College, especially enjoying the homological algebra course taught by Samuel Eilenberg. He graduated Summa Cum Laude with major in Mathematics. While an undergraduate at Columbia, he researched and wrote a paper, "Imbedding of Abelian Categories", which was published in the Transactions of the American Mathematical Society, and is widely used when working in abelian categories today. He also wrote the paper, "Theory of Covering Spaces", in the field of algebraic topology, also published in the Transactions, which he used as his Ph.D. thesis at Harvard University. His Ph.D. supervisor at Harvard was John Tate. While at Harvard, he profited from the algebraic geometry course presented by Alexandre Grothendieck, who was visiting Harvard at the time, but declined Grothendieck's offer to join him in Paris.

After completing his graduate work at Harvard, Lubkin spent a year as an NSF Postdoctoral Fellow, visiting Oxford University, UK, under the guidance of Michael Atiyah, and then Stanford University. He has spent several terms at the Institute for Advanced Study, and was a Professor at Berkeley, then a Sloane Foundation Fellow at Harvard, a Professor at Penn State, finally settling down at the University of Rochester in upstate NY, where he is currently located.

Lubkin has presented colloquia and invited talks at Harvard, Berkeley, the Institute for Advanced Study, Princeton, NJ, University of Oxford, England, University of Padua in Italy, and as a Senior Visiting Fellow of the Science Research Council of the UK, gave talks at several British universities, including Liverpool, Cambridge, Nottingham, Sussex, Warwick, Exeter, Kings College London and Edinburg Scotland. He also delivered talks at the BMC in Manchester, the University of Toronto, Canada, the Instituto Polytechnica National in Mexico City, the University of the Witwatersrand in Johannesburg, the University of Cape Town, and the CSIR in Pretoria, South Africa, the University of Rochester, and many other places. He has supervised several Ph.D. theses at the University of Rochester.

About World Scientific Publishing

World Scientific Publishing is a leading independent publisher of books and journals for the scholarly, research and professional communities. The company publishes about 600 books annually and about 130 journals in various fields. World Scientific collaborates with prestigious organisations like the Nobel Foundation, US National Academies Press, as well as its subsidiary, the Imperial College Press, amongst others, to bring high quality academic and professional content to researchers and academics worldwide. To find out more about World Scientific, please visit http://www.worldscientific.com.

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