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How is the Ogus conjecture explicitly stated, which is a variant of the Hodge and the Tate conjectures for crystalline cohomology ?

How do we build its class cycle map, and how do we formulate its Poincaré duality theorem ? In other words, how do we define the class cycle map that Ogus conjecture predicts its surjectivity, and for which field of coefficients should we establish this surjectivity ?

Thanks in advance for your help.

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    $\begingroup$ I'm not sure, but if you google "the Ogus conjecture" you'll find a paper that refers to Ogus' paper "Hodge cycles and crystalline cohomology". So why don't you look at that? Indeed the introduction says "This paper ... contains many more conjectures than proofs....". Once you figure it out, let us know. $\endgroup$
    – Donu Arapura
    Commented Mar 31, 2021 at 23:44
  • $\begingroup$ @Donu Arapura, Thank you. I took a look at this paper, but it's hard to understand it. I look for a course which summurize kindly this conjecture, adapted to my level of study. I'm still novice in this field. $\endgroup$
    – Angel65
    Commented Apr 1, 2021 at 2:14
  • $\begingroup$ @Donu Arapura, I've read right now that paper you mentioned, and I can't find any mention of the Ogus conjecture which says Every Ogus cycle is algebraic. Can you help me to find the right reference on this conjecture ? Thank you. $\endgroup$
    – Angel65
    Commented Apr 1, 2021 at 18:21
  • $\begingroup$ Unfortunately, I don't have any more information. Since no one else here has suggested anything either, perhaps you should ask Ogus. $\endgroup$
    – Donu Arapura
    Commented Apr 2, 2021 at 14:08

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Look at Yves André's book "Une introduction aux motifs", subchapter 7.4.

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