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This book provides a rigorous and self-contained review of
desingularization theory. Focusing on arbitrary dimensional
schemes, it discusses the important concepts in full generality,
complete with proofs, and includes an introduction to the basis of
Hironaka's Theory. The core of the book is a complete proof of
desingularization of surfaces; despite being well-known, this
result was no more than folklore for many years, with no existing
references. Throughout the book there are numerous computations on
standard bases, blowing ups and characteristic polyhedra, which
will be a source of inspiration for experts exploring bigger
dimensions. Beginners will also benefit from a section which
presents some easily overlooked pathologies.
The relations that could or should exist between algebraic cycles,
algebraic K-theory, and the cohomology of - possibly singular -
varieties, are the topic of investigation of this book. The author
proceeds in an axiomatic way, combining the concepts of twisted
PoincarA(c) duality theories, weights, and tensor categories. One
thus arrives at generalizations to arbitrary varieties of the Hodge
and Tate conjectures to explicit conjectures on l-adic Chern
characters for global fields and to certain counterexamples for
more general fields. It is to be hoped that these relations ions
will in due course be explained by a suitable tensor category of
mixed motives. An approximation to this is constructed in the
setting of absolute Hodge cycles, by extending this theory to
arbitrary varieties. The book can serve both as a guide for the
researcher, and as an introduction to these ideas for the
non-expert, provided (s)he knows or is willing to learn about
K-theory and the standard cohomology theories of algebraic
varieties.
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