Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

SGA 3, expose 12, remark 1.6 says that one can easily construct a group scheme over a discrete valuation ring with generic fiber Gm and special fiber Ga.

What is such an example?

share|improve this question

3 Answers 3

up vote 15 down vote accepted

R[x,y]/(y^2-x^3-\pi x^2) gives the coordinate ring of a bad cubic curve, where \pi is a uniformizer in R. Remove the origin (which is the one singular point), and projectivize the curve by adding a point at infinity, so your group has an identity.

The generic fiber (treating \pi as a unit) is then the smooth part of a nodal cubic, yielding Gm, and the special fiber (setting \pi to zero) is the smooth part of a cuspdal cubic, yielding Ga.

share|improve this answer
    
In retrospect, I should have projectivized before removing the point, or maybe just started with a Proj. –  S. Carnahan Mar 10 '10 at 4:39

One more point of view : if X is a 2x2 matrix, its centralizer in PGL(2) is -- for generic X -- isomorphic to Gm; however, it is isomorphic to Ga if X is nonzero and nilpotent. This gives a typical naturally occurring family (you can, of course, restrict it to get a family over a dvr).

share|improve this answer

I like Scott's elliptic curve construction, but here is another construction of essentially the same thing with more explicit formulas. Let your discrete valuation ring be R = k[[b]] and your group scheme be Spec of the ring S = R[t,(1-bt)-1]. Then Spec(S) is a group scheme using the multiplication rule

μ(t1,t2) = t1 + t2 - bt1t2.

and inverse

ν(t) = -t(1-bt)-1.

(Meaning, this is either a Hopf algebra structure or I view it as representing this group-valued functor on rings, depending on your theology.)

When b is invertible, then replacing t with the new coordinate s = 1-bt makes this isomorphic to the multiplicative group scheme (and, in fact, that's how the above formulas for multiplication and inversion are easiest to obtain).

When b is zero, this reduces to the additive group.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.