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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeMay 4th 2016

Started the stub for semilinear map. More to come.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeMay 4th 2016

Of course the notion makes sense and is important in the generality of modules over rings. I haven’t encountered the term “semilinear map” before though, I think usually I’ve seen it called an “equivariant” map or something like that.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeMay 4th 2016

I am reading a lot through synthetic projective and affine geometry books these days and they are about uniform in calling this notion in said generality semilinear. As far as the equivariance, in traditional mathematics it is customary to call it that way in the case of group actions, rather than for general ring actions. Of course, in some cases like actions of group algebras one still uses the term equivariant.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeMay 4th 2016

Besides, it is not entirely clear to me if the word equivariance of the map with respect to ring actions would include addivity of the map or not (semilinearity includes it). Of course, if we work internally to abelian groups and look at it as at an equivariant internal monoid action then it includes the additivity automatically. But the geometries I have in mind are the world where nonadditive maps often appear as well, so it is not beneficial to phrase everything internally.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeMay 4th 2016

Well, what would you call it for general rings? I don’t like “semilinear” because it suggests that it is “like a linear map, but weaker” whereas actually it is fully linear, it’s just that its linearity is mediated by the extra datum of a ring homomorphism. I’ve definitely seen “equivariant” used here.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeMay 5th 2016
• (edited May 5th 2016)

Well, you are completely right that the point of view that it is like a “linear map but weaker” is not good as some relation between the ground rings is needed anyway. But the main cases of interest in geometry are endofunctions on a fixed vector space where the identity is the preferred to general automorphism of a ground skewfield. So many theorems say that under some useful conditions the semilinear endofunction is in fact linear. So it is like a stronger property thanks to the preferred role of the identity isomorphism.

Similar situation is in algebra where one introduces skew-derivations by modified Leibniz fules, using endomorphism of the ground ring.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeMay 5th 2016

what would you call it for general rings?