The theory of motives was created by Grothendieck in the 1960s as
he searched for a universal cohomology theory for algebraic
varieties. The theory of pure motives is well established as far as
the construction is concerned. Pure motives are expected to have a
number of additional properties predicted by Grothendieck's
standard conjectures, but these conjectures remain wide open. The
theory for mixed motives is still incomplete. This book deals
primarily with the theory of pure motives. The exposition begins
with the fundamentals: Grothendieck's construction of the category
of pure motives and examples. Next, the standard conjectures and
the famous theorem of Jannsen on the category of the numerical
motives are discussed. Following this, the important theory of
finite dimensionality is covered. The concept of Chow-Künneth
decomposition is introduced, with discussion of the known results
and the related conjectures, in particular the conjectures of
Bloch-Beilinson type. We finish with a chapter on relative motives
and a chapter giving a short introduction to Voevodsky's theory of
mixed motives.
General
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