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John Pardon
John Pardon
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For the equation you are asking about, you don't want $MA$ to be dualizable (which is lucky, because it's not), you want $MA\wedge X_+$ to be dualizable as an $MA$-module. This is true in the situation you are interested in but it is not trivial. This is proven in the following paper:

Marc Hoyois, Shane Kelly, Paul Arne Østvær - The motivic Steenrod algebra in positive characteristic.

Unfortunately a shift appears, which for smooth projective varieties is given by the dimension of the normal bundle, but I am not completely sure of what happens for non-proper varieties.

(I'm not sure if there is a simpler proof in your special case that appeared earlier in the literature. The idea is that $MX$ is always dualizable when $X$ is smooth and projective and then you can use alterations to "complete" any $X$ up to a factor prime to $l$).

For the equation you are asking about you don't want $MA$ to be dualizable (which is lucky, because it's not), you want $MA\wedge X_+$ to be dualizable as an $MA$-module. This is true in the situation you are interested in but it is not trivial. This is proven in the following paper:

Marc Hoyois, Shane Kelly, Paul Arne Østvær - The motivic Steenrod algebra in positive characteristic.

Unfortunately a shift appears, which for smooth projective varieties is given by the dimension of the normal bundle, but I am not completely sure of what happens for non-proper varieties.

(I'm not sure if there is a simpler proof in your special case that appeared earlier in the literature. The idea is that $MX$ is always dualizable when $X$ is smooth and projective and then you can use alterations to "complete" any $X$ up to a factor prime to $l$).

For the equation you are asking about, you don't want $MA$ to be dualizable (which is lucky, because it's not), you want $MA\wedge X_+$ to be dualizable as an $MA$-module. This is true in the situation you are interested in but it is not trivial. This is proven in the following paper:

Marc Hoyois, Shane Kelly, Paul Arne Østvær - The motivic Steenrod algebra in positive characteristic.

Unfortunately a shift appears, which for smooth projective varieties is given by the dimension of the normal bundle, but I am not completely sure of what happens for non-proper varieties.

(I'm not sure if there is a simpler proof in your special case that appeared earlier in the literature. The idea is that $MX$ is always dualizable when $X$ is smooth and projective and then you can use alterations to "complete" any $X$ up to a factor prime to $l$).

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Denis Nardin
Denis Nardin
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For the equation you are asking about you don't want $MA$ to be dualizable (which is lucky, because it's not), you want $MA\wedge X_+$ to be dualizable as an $MA$-module. This is true in the situation you are interested in but it is not trivial. This is proven in the following paper:

Marc Hoyois, Shane Kelly, Paul Arne Østvær - The motivic Steenrod algebra in positive characteristic.

Unfortunately a shift appears, which for smooth projective varieties is given by the dimension of the normal bundle, but I am not completely sure of what happens for non-proper varieties.

(I'm not sure if there is a simpler proof in your special case that appeared earlier in the literature. The idea is that $MX$ is always dualizable when $X$ is smooth and projective and then you can use alterations to "complete" any $X$ up to a factor prime to $l$).