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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Correction to 'isogeny'
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obscurity of morphism by PopeA
Correction id: 7490
Filed on: 2006-02-01 19:42:02
Status: Accepted on 2006-02-17 11:09:47
Type: Meta/Minor
Correction text:
To understand the term morphism in the given context requires one to know which category is implied. Chasing the definition back through the links (finite morphism etc) makes the defintiion more and more obscure as the level of abstraction is constantly increased.
And yet the objects we are discussing are at a very concrete level. An isogeny is a (set-)function which preserves some structure. The problem is I cannot understand readily what the structure is that is preserved. I understand that it is a group homomorphism. But this is cited as a _consequence_ of the definition, so the definition must be different from this. (Of course group homomorphism implies preservation of the point at infinity, so this also suggests the definition is other than merely a group homomorphism.)
So my request is for clarification: can we please have a concrete definition of isogeny? Recognising that it is a morphism in (various) categories can come later.
I cannot suggest what is needed as I do not know the answer!
Thanks
Alun |
Comment from object owner mathcam:
I agree that the term morphism here is ambiguous. It refers to a morphism in the category of varieties (now specified in the entry). The answer to your question about what structure it preserves is exactly that...the structure of an elliptic curve as a (projective) variety!
Cam |
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