One interesting thing to note is that unlike the case of groups, here we actually specify the inverse operation as well as the identity element, opposed to just claiming their existence.
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This is important because we want the inverse operation, for example, to actually be a morphism in the category. Now, the group objects in a category form a category themselves. Or, in other words, morphisms should just be morphisms of the underlying -objects that commute with respective multiplications.
In fact, group objects are more often than not presented to us in functorial form—it is the above definition which is less natural. To make rigorous our functorial definition of group objects, we need to recall the ubiquitous Yoneda lemma. For an object , we obtain a morphism by taking to. One can quickly check that this is, indeed, a natural transformation. In this way, we obtain a functor , where is the functor category i. Then, the map taking to is a bijection. Why this has impact on our functor comes from taking. Then, this says that the map sending to is a bijection to.
Well, is the map taking to. This then tells us that our functor is an embedding i.
Jonah , Schreiber : Transitive affine transformations on groups.
Thus, to give a morphism is the same thing as to give a natural transformation. Note first that if and are objects of a category , then we can canonically identify the functors and. Indeed, the natural isomorphism , on an object takes as where and are the projection maps associated to. Well, to give a morphism is the same as giving a morphism , or with the identification we discussed in the last paragraph,. And finally, giving a morphism is the same thing as giving a morphism. But, a group object was more than an object equipped with some maps.
These maps needed to satisfy certain equational properties. For example, we required, for example, that the associativity axioms. But, and are both elements of.
But, to do this, we need only check that for any object of the two set maps are equal. But, the equality of the set maps and is clearly equivalent to the associativity of the set maps. In fact, using exactly the same observations, one sees that the maps , and satisfying the axioms of a group object is equivalent to the statement that , which is a binary function , defines a group structure on with inverse map given by , and identity element note that since is a terminal object is always just a point, so is constant, so just picks out the identity element.
Moreover, note that if is a morphism in , then by definition we obtain a set map defined by. But, since is the multiplication map for the group structure on , and is similarly the multiplication structure on , we find that is a group map. Thus, we see that the group object actually defines a functor whose underlying set functor i.
Tracing the argument backwards shows that the converse is true, and thus we obtain the following theorem:.
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Theorem 2: To give a group object in is the same as giving a factorization of the functor through the forgetful functor. Note that various factorizations of through the forgetful functor correspond to putting different group structures on the same object of.
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Moreover, we can actually take Theorem 2 one step further. Namely, what does it mean to give a morphism from a group object to a group object?
It means to give a morphism in such that. But, this is precisely the statement that for each object of the induced set map is actually a map of groups. Thus, we have actually defined a natural transformation where these are the associated functors to group. Moreover, once again by Yoneda, all such morphisms of group objects arise in this way:. Theorem 3: If and are group objects in , with associated factorization and , then to give a morphism of group objects is the same thing as giving a natural transformation.
So, for example, instead of the group , we will now write.
- Jonah , Schreiber : Transitive affine transformations on groups..
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Another way to phrase the above is that a group object in is that it is a representable functor or, more technically, representable when composed with the forgetful functor to , and to to give a morphism between group objects is simply to give a natural transformation between these functors. Also, from now on, as is customary, we shall drop any reference to the operations of a group object and refer to it only as , unless confusion would arise.
Now that we have set up all of this notation, it is very simple to define group schemes. Note then that the morphisms of a group object take the form of -morphisms , , and i. One nice property about group schemes, is that it suffices to actually give their values on affines. To make this rigorous, let us define, for a scheme , the category to be the full subcategory of affine schemes in.
The claim is then the following:.
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Then, any natural isomorphism can be lifted to an isomorphism. Moreover, if and are given a factorization through , and is a natural isomorphism of functors to , then so is. To do this, let be an open cover of by affine open subschemes, and for each pair , let be an open cover of by affine open subschemes. We then have the following diagram. So, in essence, it takes a tuple and sends it along the first arrow to the tuple which, in the entry has and along the second arrow sends it to the tuple which in the entry has. The maps for are defined similarly. The vertical arrows are the arrows and.
One can check, by refining both open covers, that the map is independent of open cover. And, of course, since each And is an isomorphism, so is the map. Lastly, to check the naturality of follows by definition, and the fact that the are natural on the affine schemes. We must now show the surjectivity of this map.
For an aribtrary -scheme define the group structure on to be , where is the set of all affine open subschemes of , made into a directed set by definiing to be the opposite of inclusion maps.
We must now show that this group structure is functorial in , in the sense that for every arrow , the map is a group map, where each are given the above described group structure. But, for each affine open of , we obtain a map , where are the affine open subschemes of which is a group map.
One then recovers the induced map. To conclude we need to check that this functor agrees with the original group structure when restricted to. But, this is just because is a sheaf. It really does tell us that group functors are really determined by their action on affine schemes. To keep things simple, we will assume, in what follows, that , for some ring. That said, all parts that make sense for a general base e. On morphisms sends an -morphism to the associated map of abelian groups. It is clear to see that this is functorial. But, to conclude that is actually an affine group scheme over we need to show that , composed with the forgetful functor , is representable.
To do this, merely note that is the same thing as. Thus, is an affine group scheme over , with representing object.
For completeness, we should mention what the explicit operations on are. To this end, we need to define maps , , and note that [or more technically ] is the terminal object]. Well, to give such an is the same thing as giving an -algebra map , is the same as giving a -algebra map , and is the same thing as giving a -algebra map. So, let be defined by , as , and. One can easily check that the desired relations hold, and that this does, in fact, define the same affine group scheme as.
The next fundamental example is the next obvious choice after the additive group. Define the functor on objects by with the usual product , and for an -morphism associate the group map coming from the associated ring map. It is clear to see that really is a functor. To see that is an affine group scheme, we need to show that its composition with the forgetful functor is representable.
But, one can quickly check that this is the case, in particular, that is isomorphic to where is more precisely written as. Once again, we need to define maps , , and.
As before, this is equivalent to defining -algebra map , , and. Well, define to be such that , as such that , and as such that. Lluis - Acerca de un resultado de Segre. Anales del Inst. Danielewski - On the cancellation problem and automorphism group of affine algebraic varieties.
Preprint Appendix by K. Max-Planck Institut Bonn, preprint Dimca - Singularities and topology of hypersurfaces. Dovermann , M.