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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 21st 2010

started stub for Chow group

hoping I got this right...

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeJan 21st 2010
• (edited Jan 21st 2010)

Well, as I warned you in a parallel discussion there are MANY different versions of Chow groups, depending on particular adequate equivalence relation used. You here forced rational equivalence as THE equivalence in your exposition, what is OK for some purposes, but the general story is also important. It is like putting the condition "Zariski cover" in the definition of what a sheaf is. Another thing is that one can label them by dimension or by codimension of the cycles, and the distinction is of course crucial in their interpretation. Chow groups are rather difficult subject, especially in hi codimension. The business of filtrations is here important and more subtle than with algebraic K-theory. Of course, between the two there is again a version of the Chern character, as I mentioned earlier.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 21st 2010

Ah, thanks. Okay, I need to add at least a warning. Better yet, I try to better understand this! :-)

Which precise definition of Chow group should be the one that Denis-Charles Cisisnki refers to in the passage that I included into motivic cohomology?

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeJan 21st 2010

It is not obvious from that "Nisnevich" infinity stack description, I don't know which one.

• CommentRowNumber5.
• CommentTimeMar 21st 2013

I updated the page with definitions and references.

• CommentRowNumber6.
• CommentTimeMar 21st 2013

Zoran, in the literature I haven't seen the term "Chow group" being used for other equivalence relations than rational equivalence. Even in André's book on motives, where other relations are also discussed, he specifically says that Z_k(X)/~ is called the Chow group in the case ~ = ~_rat.

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeMar 21st 2013
• (edited Mar 21st 2013)

Already the very first google hit disproves you. Look at 2.7 of

http://jfresan.files.wordpress.com/2010/11/lectures-murre.pdf

and you will see variants $C H^*_{hom}$, $C H^* = C H^*_{rat}$, $C H^*_{num}$ etc. used (in this notation) and compared. Of course, the DEFAULT case is the Chow groups for rational equivalence which are almost always used without saying “rational”. If we do not put any subscript to $C H$ we mean rational, of course. But in the development of the theory, the other Chow groups are sometimes compared, with great importance and often with such names and notation.

• CommentRowNumber8.
• CommentTimeMar 21st 2013
• (edited Mar 21st 2013)

Indeed he uses the notation CH_hom, CH_num, etc., but he never calls these "Chow groups". This term is reserved for the rational case, see page 9. I believe this is because historically Chow introduced and studied rational equivalence. Anyway, I will add a note about this on the page.

• CommentRowNumber9.
• CommentTimeMay 30th 2013
• (edited May 30th 2013)

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeMay 30th 2013

Thanks!

It would be good to add brief explanation of some of the symbols that appear

• at algebraic cycle: “$long$” ?

• at rational equivalence I added a pointer to Weil divisor where $div(-)$ appears. An explaination of $j_{\alpha, \ast}$ is missing (I added: direct image of …)

At graph morphism I added a pointer to graph of a function.

• CommentRowNumber11.
• CommentTimeMay 30th 2013

Sorry, good catches. "long" denoted "longeur" and I changed it to "length".

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeMay 30th 2013

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