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AG (Algebraic geometry)
Pure
Notion introduced by Kimura, see also O’Sullivan. A weaker condition is that of Schur finiteness; see Ayoub: The motivic vanishing cycle and the conservation conjecture (in Nagel and Peters, eds).
See Pure motives
Kimura: Chow motives can be finite-dimensional, in some sense.
Guletskii article(s).
Mazza and Del Padrone on Schur finiteness. Looks very good.
Articles by Jannsen and by Kimura in the Nagel-Peters volume II (Algebraic cycles and motives)
Chapter 12 of André.
Bourbaki seminar by André
Andre: Motifs de dimension finie, S´eminaire Bourbaki.Vol. 2003/2004. Ast´erisque No. 299 (2005), Exp.No. 929, viii, 115–145. Very good, and long review in MR
nLab page on Finite-dimensional motives