In this post we classify one-dimensional connected group varieties of dimension .
In a previous post I discussed the classification of one-dimensional connected algebraic groups over a field . There an ‘algebraic group’ over meant a smooth affine group of finite type over .
Since then people have asked me directly and/or via the searches that led them to this blog asked what happens in the non-affine case. I’ll answer that question in this post by classifying one-dimensional connected geometrically reduced group schemes over (with some mild assumptions on ).
The answer, as we will see, essentially breaks in to two cases: the affine case (as discussed in the previous post) and the elliptic curve case. The two are very separate and distinct.
The general theory
Let us now clarify exactly the assumptions we want and what basic properties we are able to get from them. Namely, we are interested in classifying one-dimensional groups over but without more adjectives this will be impossible. So, let discuss what adjectives we want, and what they give us.
Throughout a group variety over will mean a finite type group scheme over .
The first assumption that we want to assume about is that it’s connected. The reason for this is silly. If you don’t insist on working with connected groups you have the entire theory of finite groups to contend with. Indeed, for any finite group one can obtain a one-dimensional group (which is even smooth with reductive identity component) as where is the constant group. For this reason we will want to assume that is connected.
Let us note that in general if you don’t want to assume that is connected then you have a short exact sequence of group varieties
where is the identity component of and , the component group of , is a finite etale group scheme. In particular, if is algebraically closed then is essentially the same just an abstract finite group. Thus, one sees that the general non-connected situation comes down to classifying connected groups (which we will essentially do), finite groups (in the form of , and extensions of the latter by the former.
Let us also make an observation about connected groups, or more generally connected varieties with -points, that will be useful later (this was also in the last point):
Observation 1: Let be a field and a finite type -scheme such that . Then, is connected if and only if is geometrically connected.
Proof: Evidently if is connected then so is (since we have a continuous surjection ). Suppose now that is connected. Note that for an extension one has that is disconnected if and only if contains non-trivial idempotents. In particular, since every element of lies in (note that we’re applying `flat base change’ here) for some finite extension we see that disconnected implies that is disconnected for some finite extension .
That said, note that is finite and flat (since is finite and flat) and thus a clopen map. In particular, since is connected we see that every connected component of surjects on to . In particular, fixing we see that every connected component of contains a preimage of . But, since is a -point it has only one preimage. Since connected components are disjoint this implies that has only one connected component. The conclusion follows.
As a corollary we obtain the following:
Corollary 2: Let be a field and a group variety over . Then, is connected if and only if is geometrically connected.
The second assumption we will make is that our group is geometrically reduced. Recall that if is perfect this is the same thing as being reduced. The reason that such assumptions are necessary for a reasonable classification is the prescence of infinitesimal group schemes in characteristic .
Namely, recall that a group scheme over is infinitesimal if is the trivial algebraic group.
Example 3: Define where is the (relative) Frobenius map on which on -points is the additive map . One can see that is, as a scheme, just and so evidently is trivial. So, is infinitesimal.
Infinitesimal groups can be quite complicated. Their difficulty is somewhat comparable to the theory of finite groups. So, they are something we would like to ignore. For this reason we shall want to assume our groups are reduced. Moreover, to ignore issues with base change it’s also useful to assume that our groups are actually geometrically reduced (there exists reduced groups which are not geometrically reduced–see Examples 1.57 and 1.58 of [Mil]).
Let us note that if we are only interested in characteristic then this is no real assumption:
Theorem 4(Cartier): Let be a characteristic field. Then, every group variety over is (geometrically) reduced.
Proof: See [Mil, Theorem 3.23].
The assumption that is one-dimensional is, of course, necessary for a concise classification. Else we have to contend with all of abelian varieties, linear algebraic groups, … Hopefully this is convincing enough.
Let be a group variety over . We see what implications we get if we assume that is a one-dimensional connected geometrically reduced group variety.
In fact, geometrically reducedness for group varieties implies smoothness:
Lemma 5: Let be a field and a group variety over . Then, is geometrically reduced if and only if it’s smooth over .
Proof: If is smooth over then it’s geometrically reduced by standard theory. Conversely, suppose that is geometrically reduced. We need to show that is regular. This means that we need to check that the local rings at all points of are regular. But, by standard theory it suffices to check at only the closed points. By generic smoothness for geometrically reduced varieties we know that there is some point such that is regular. But, for any other closed point the left translation by map is an automorphism of varieties which carries to . Thus, is also regular.
In fact, separatedness is true for any group:
Lemma 6: Let be a field and a group variety over . Then, is separated.
Proof: It suffices to show that the diagonal is closed. But, the map given on -points by is clearly a morphism and where is the identity element. Since is a closed point we see that is separated as desired.
From our assumptions that is smooth and geometrically connected we immediately get the following from general theory:
Lemma 7: Let be a field and let be geometrically connected smooth finite type -scheme. Then, is geometrically integral.
Proof: Evidently we may assume that is algebraically closed. Suppose that had more than one irreducible component–say that and are distinct irreducible components of . Note then that if then is not a domain since and give rise to two distinct minimal primes of . But, since is regular we know that is a domain, and this is a contradictoin. Thus, the irreducible components of are disjoint. But, since is finite type we know it has finitely many irreducible components. This implies that the irreducible components are clopen. Since we assumed that had more than one irreducible component this contradicts that is connected.
But, in fact, we can also derive this irreducibility from the theory of group schemes:
Lemma 8: Let be a field and let be a connected group variety over . Then, is geometrically irreducible.
Proof: From Observation 1 it suffices to assume that is algebraically closed. Suppose that were not irreducible. Then, as in the proof of Lemma 7 there must be two irreducible components and of that intersect and, in particular, contain a common -point . Note though that is not contained in the union of the other irreducible components (by irreducibility!) and thus there is some -point of which is contained within a unique irreducible component. Let be the left translation by automorphism. This takes to . But, since lies on the intersection of two irreducible components of and lies in a unique irreducible component of this is a contradiction.
Affineness or projectiveness
One thing that we get for free from our assumptions, in fact just the fact that we’re working with a geometrically connected separated curve, is that our group is either affine or projective. Namely, we have the following:
Proposition 9: Let be a field and let be a one-dimensional geometrically connected separated variety over . Then, either is affine or projective.
The idea of the proof is simple. Namely we will show that is at least contained in a projective curve and then use the fact that since we’re in one dimensional all proper open subvarieties of a projective curve are affine.
We begin by noting the following observation:
Observation 10: In the proof of Proposition 9 it suffices to assume that is smooth over .
Remark 11: We are using the definition of projectiveness as in here. This is contrast to the definition that Hartshorne uses (which is called -projective in loc. cit). This makes no real difference when discussing projectiveness of a variety, but does have difference in general. For instance, in Hartshorne’s definition of a projective morphism it needn’t be true that finite maps are projective (e.g. see the remark here).
We will cite the following lemma for simplicity:
Lemma 12: Let be a field and let be a connected one-dimensional variety over . Then, is projective if and only if it’s proper.
Proof: If one assumes that is smooth and integral (which is the case we’re in for our groups) then this is somewhat easy. Namely, let be an affine open in and let be the common function field between this affine open and . Note that if we embed in to some we can take the closure, which we’ll denote , to obtain a connected projective variety. It may be true that is singular. But, after taking its normalization (see below for references for normalization) we get a smooth projective integral curve which is birational to (since is birational to and contains an open that is also an open in ). One can then use the valuative criterion for propertness to take the birational map and extend it unique to a map which is necessarily an isomorphism.
For the general proper case see this (see Remark 12 for why this phrase ‘-projective’ doesn’t matter here).
Proof (Observation 10): By our assumptions we may clearly assume that is algebraically closed. Then, we know from standard theory that the smoothness of is equivalent to the regularity of . But, since is one-dimensional this is equivalent to the normality of . But, note that we have the normalization map . This is a surjection, this much is clear. What lies significantly deeper is that, in fact, is finite. For example, one can see [Vak, Theorem 9.7.3]. At a deeper level what is happening is that varieties are Nagata since locally their functions are finite type over a field which implies Nagata by basic theory. It is then also well-known that Nagata schemes have finite normalization maps.
Regardless, note that is regular, thus smooth and so to finish it suffices to prove that is affine or projective if and only if is.
If is affine then , being a finite over , is also affine. If is projective, then is projective since finite maps are projective and composition of projective maps are projective.
If is affine then is affine. Indeed, this follows from non-trivial general theory about finite morphisms and affineness (it is really this tag that is most relevant). If is projective then is necessarily proper by standard theory and thus projective by Lemma 12.
So, we are now free to assume that is smooth. It is now not hard to justify our intuitive sketch of proof after the statement of Proposition 9:
Proof(Proposition 9): We may assume that is algebraically closed.
Let be an affine open subscheme of . As in the first paragraph of the proof of Lemma 12 we can find an open embedding where is smooth projective geometrically connected curve over . By the valuative criterion for properness we can uniquely extend to a map . We claim that is an open embedding.
Indeed, let us note that is certainly quasi-finite since it’s quasi-finite on (it’s an open embedding) and is finite (since is one-dimensional). Thus, by Zariski’s main theorem (evidently our maps and schemes are quasi-compact and separated) we can find a factorization
where is a dense open embedding and is finite. It’s clear we may assume that is integral since both and are integral. Moreover, note since the normalization map is an isomorphism on the smooth locus and finite we can, up to replacing with , assume that is smooth over . Note then that is a birational map of smooth proper curves, so an isomorphism. Indeed, since is surjective we know that it’s flat (e.g. see [Qin, Proposition 3.9]). Note then that is vector bundle on . So, to show that is an isomorphism (which is clearly all we need to do) it suffices to show it’s an isomorphism generically, but this is true since is birational. The claim that is an open embedding then immediately follows.
Now, since is an open subscheme of , a smooth projective integral curve, it suffices to show that any such variety is affine or projective. This follows from the lemma following this proof.
Lemma 13:Let be a field and a smooth geometrically integral projective curve over . Then, any open subscheme is affine.
Remark 14: This was also contained in the previous post.
Proof: We may assume that is algebraically closed. If is empty, we’re done. So, assume that is not empty. Then, for some -points . Consider then the line bundle , where for .
Now, choosing the sufficiently large, we know that will have negative degree, and so we may assume that . So, by choosing the sufficiently large, we can made sufficiently large. In particular, we can find such that has poles precisely at the . Thus, we obtain a non-constant rational map off of the . Extending this to a morphism , we obtain a finite map with , and thus is affine.
Remark 15: There is almost certainly an easier way to do the above. For example, if one uses this your life becomes much easier. In the above I don’t really use this proposition (save I use part of it in the proof of the non-smooth case of Lemma 12). The reason I opted to not use it is that while I think this approach is more elementary, it’s somehow trickier than what I did above. A lot of the techniques used above are incredibly common and worth internalizing (albeit maybe overkill).
The broad classification
We are now ready to start classifying one-dimensional connected geometrically reduced group varieties over a field . We will break our discussion in to its two separated, and distinct cases: the affine case and the projective case.
The projective case
In short, the projective one-dimensional geometrically reduced group varieties are just elliptic curves. This is what we endeavor to prove. But, before we state this precisely let us remind ourselves what an elliptic curve is.
Definition 16: Let be a field. Then, an elliptic curve over is a pair where is variety of the form is a smooth curve where has homogenous degree and .
Of course, we know that elliptic curves over have a group law. More explicitly, let be distinct. There exists a unique line (i.e. a closed subscheme of the form where is a degree one homogenous polynomial) such that passes through and . If we set to be the unique line in passing through and which is tangent to at (i.e. is the line where ).
Note then that by Bezout’s theorem that for any (possibly with ) we have that (by which we rigorously mean the fiber product ) is a finite -scheme with -dimensional global sections. If then it’s easy to see that is the disjoint union of three -points which are and a third point . If then is the disjoint union of (where for a closed subscheme of we denote by the ideal of induced by ) and an -point . In either case we see that we obtain a third point from .
Note then the exact same ideas yield associated to a line and a third point of intersection of the line and . We denote this third -point of by .
We then have the following:
Theorem 17: Let be a field and an elliptic curve over .
- The operation given by described above endows with the structure of a group.
- There exists a unique group variety structure on such that the group structure on agrees with that from 1.
Proof: The proof of 1. is [Sil, Proposition 2.2]. The unicity of the group scheme structure in 2. is clear. Indeed, let be a multiplication map. Since is a dense subset of , and all schemes in consideration are separated, the map is determined by its value on . The existence of such (i.e. the algebraicity of the group operation from 1. and the fact that it’s defined over ) follows from [Sil, Group Law Algorithm 2.3].
We are now able to state our desired broad classification in the projective case:
Proposition 18: Let be a field. Every elliptic cruve is a one-dimensional geometrically connected proper group variety over . Conversely, if is a one-dimensional geometrically connected projective group variety over then is isomorphic (as a group variety) to an elliptic curve.
Proof: Let us first observe that elliptic curves are certainly one-dimensional, smooth, and projective (by definition). Thus, it remains to show why they are connected. There a multitude ways to prove this (Bezout’s lemma, [Vak, Exercise 11.3.F],…) but we list here a cohomological one. It suffices to show that is a one-dimensional space since this then forces the ring to be and thus to have no non-trivial idempotents. But, note the ideal sheaf of is evidently and, in particular, is a line bundle. We have the short exact sequence of sheaves
where is the tautological closed embedding. We then get a long exact sequence on cohomology group that contains the portion
But, evidently (e.g. see [Vak, Theorem 18.1.3]) and equally evident is and thus we see that
Conversely, suppose that is a one-dimensional connected geometrically reduced group variety. By the Rigidity Lemma (which immediately succeeds this proof) it suffices to show that there exists an isomorphism of pointed varieties where is an elliptic curve and and are the respective identity elements. Of course, it suffices to show that there exists an isomorphism of varieties since we could postcompose with translation by an appropriate element of to guarantee that mapsto .
We begin by noting that necessarily has genus . There are several ways to see this, but we proceed with the one that is simplest. Namely, it’s not hard to show that any group variety is parallelizable (i.e. has cotangent bundle which is free) by using the theory of invariant differentials (e.g. see [BLR, §4.2 Corollary 3]). In particular, we see that and thus . But, (e.g. see [Har, Example 1.3.3]). Thus, as desired.
We now need to show that is isomorphic to an actual variety of the form where is a smooth cubic function. To do this let us begin by noting that since we have by standard theory (e.g. see [Vak, Conclusion 19.2.11]) that is very ample. Let us note also that, by Riemann-Roch, we have that
but since , which has negative degree, the left-hand side of the above simplifies to . So, . Thus, we see determines a closed embedding . So, is isomorphic to a closed subscheme of . Since we know that we have that corresponds to a height one homogenous prime in which is automatically principal. Thus, (or more precisely the image of ) is actually a hypersurface, say .
Thus, it remains to show that . But, let us note that if then the ideal sheaf of is . Using the long exact sequence in cohomology for
(as we’ve already discussed in this proof) we see that (again using [Vak, Theorem 18.3.1]) that
But, note that since is a closed embedding we know from standard theory (e.g. [Har, Exercise 8.2]) that
and the right-hand side is one dimensional since has genus . Thus, in conclusion we see that . But, (e.g. see again [Vak, THeorem 18.3.1]). Thus, as desired.
Lemma 19(The Rigidity Lemma): Let be a field. Let and be geometrically integral schemes of finite type over and a separated -scheme. Let be a morphismof -schemes, and assume further that
- is proper.
- For some algebraically closed extension there exists some such that the restriction to is a constant map to some .
Then is independent of ; i.e. there exists a unique morphism of -schemes such that .
In particular, if are smooth geometrically integral proper -varieties with identity elements then any morphism of pointed -varieties is a morphism of group varieties.
Proof: See [Con, Theorem 1.7.1] for the first statement. The second statement follows from the first by noting that the map given on points by is constant on and thus is trivial. This implies that is a group morphism.
The affine case
This was taken care of in the previous post. In particular, we have the following:
Theorem 20: Let be a field and let be an affine one-dimensional connected geometrically reduced group variety over . Then, or . If then the assumption that is geometrically reduced is unnecessary.
The two cases combined
Combining our two cases we arrive that the following:
Theorem 21: Let be a field and let be a one-dimensional connected geometrically reduced group variety over . Then, one of the following holds:
- is an elliptic curve.
If then the assumption that is geometrically reduced is unnecessary.
A finer result
Theorem 21 is lacking in two orthogonal ways. The proper case gives us a nice classification over but it’s pretty inexplicit (how do we explicitly parameterize elliptic curves up to isomorphism?). The affine case is very explicit, but only works over . Our goal now is to remedy both of these issues in (essentially) full generality. In particular, the main assumption we will often make is that is perfect.
The proper case
We begin by trying to understand how to explicitly parameterize elliptic curves over . We begin with a well-known and simple first case:
Proposition 22: Let be an algebraically closed field. Then, associated to every elliptic curve over is an element . This integer only depends on the isomorphism class of and induces a bijection
The element from Proposition 22 is the so-called -invariant of . Its definition can be given purely in terms of a defining equation for as in [Sil, §3.1]. The proof of Proposition 22 is then the contents of [Sil, Proposition III.1.4.(b)]. If we want to explicitly construct an inverse for the bijection in Proposition one associates to the elliptic curve with the polynomial
at least if . If one can take the elliptic curve with equation
and if one can take the elliptic curve with equation
which covers all cases. We denote these explicit elliptic curves as for any
In fact, from the simple observation that has a model over (that with the same equation) we actually deduce the following strengthening of Proposition 22:
Proposition 23: Let be a field. Then, the map
is a surjection with fibers the sets
Thus it really remains to explicitly understand the sets . These sets are exactly the sets of ‘twists’ of (e.g. see the discussion of twists in the notes of this post). For our sanity, we now assume that is perfect so that is Galois. We then have the following:
Lemma 24: Let be a perfect field. Then, there is a natural bijection
Proof: For a down-to-earth proof see [Sil, §X.5]. The high level proof is as follows. Let be the category fibered in groupoids over the small etale site for . This is a stack (e.g. see [Ols, Theorem 13.1.2]). Thus, by Theorem 5.1 of the notes from this post the claim essentially follows. Namely, the only other reduction is the observation that by standard theory one has that
from where the conclusion follows.
Remark 25: I don’t truthfully know what happens over non-perfect fields. I suspect that as long as the characteristic is not or then the fact that is smooth should imply that all isomorphisms which occur over actually occur over in which case Lemma 24 is still valid. Even if this is true, I don’t know what can happen in the case of characteristic or . Please feel free to enlighten me if you know the answer.
So, all that remains to do to completely classify elliptic curves over , at least in the case when is perfect, is to calculate the groups for all . This turns out to be quite simple in the case when the characteristic of is not or . Indeed, we have the following:
Lemma 26: Let be a perfect field of characteristic not or . Then,
where this is isomorphisms as group schemes over .
Proof: This is precisely [Sil, III.10.2].
Now, the Galois cohomology of is very simple in general. Namely, by Hilbert’s theorem 90 we know that in we have a canonical bijection and . In particular, we see that we have bijections
Thus, we can enhance Proposition 22 to the following:
Theorem 27: Let be a perfect field of characteristic not nor . Then, there is an explicit bijection
where if and where
The explictness in the statement of Theorem 27 means that there is an explicit inverse (since the map itself just sends to where is the element of that comes from our discussion). For an explicit description see [Sil, Corollary 5.4.3]. We mention though, since it’s the most common and most simple case, that if then as soon as one writes where is the homogenization of a polynomial of the form (which is always possible since we’re in characteristic different from or ) then the element that corresponds to is the elliptic curve whose affine equation is –this is the so-called quadratic twist of .
The affine case
The affine case was covered in the previous post, so we just summarize the results here. Namely, we have the following:
Theorem 28: Let be a perfect field of characteristic not . Then, a one-dimensional connected affine geometrically reduced group variety over is either or a torus. Moreover, there is a bijection
such that that identity element of corresponds to and non-identity corresponds to the torus
Putting all together
Summarizing everything above we get the following:
Theorem 29:Let be a field. Then every one-dimensional connected geometrically reduced group variety over is either affine or an elliptic curve. If then the geometrically reduced assumption is automatically satisfied.
Moreover, we have the following parameterization of these two families:
- If is perfect and of characteristic not or then there is an explicit bijection
where if and where
- If is perfect and of characteristic not then an affine is either or a one-dimensional torus. Moreover, there is a bijection
such that that identity element of corresponds to and non-identity corresponds to the torus
Remark 30: One can almost certainly remove the perfectness hypotheses from 1. in Theorem 29. Indeed, let and be elliptic curves over . We need only show that if then . But, note that is an fppf torsor for . But, is for by Lemma 26. Its simple to show that any fppf torsor for is representable and is smooth since is. So, is a representable by a smooth (in fact etale) -scheme over . So, it evidently has a point from where the conclusion follows.
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