In Galois Deformation Rings we introduced the concept of a Galois deformation ring, and how it is used to prove “R=T” theorems. In this post we will look at a very simple example to help make things more concrete. Then we will explore more about the structure of Galois deformation rings, in particular we want to relate the tangent space of such a Galois deformation ring to the Selmer group in Galois cohomology (which also shows up in a lot of contexts all over arithmetic geometry and number theory).
Let be a finite extension of , and let be some finite field, with ring of Witt vectors (for example if then ). Let our residual representation be the trivial representation, i.e. the group acts as the identity. A lift will be a Galois representation , where is a complete Noetherian algebra over . Then our Galois deformation ring is given by the completed group ring
where means the pro-p completion of the abelianization of the Galois group . Using local class field theory, we can express this even more explicitly as
Let us now consider a useful fact about the tangent space (see also Tangent Spaces in Algebraic Geometry) of such a deformation ring. Let us first consider the framed deformation ring . It is local, and has a unique maximal ideal . There is only one tangent space, defined to be the dual of , but this can also be expressed as the set of its dual number-valued points, i.e. , which by the definition of the framed deformation functor, is also . Any such deformation must be of the form
where is some matrix with coefficients in . If and are elements of , if we substitute the above form of into the equation we have
from which we can see that
and, multiplying by on the right,
In the language of Galois cohomology, we say that is a -cocycle, if we take the matrices to be a Galois module coming from the “Lie algebra” of . We call this Galois module .
Now consider two different lifts (framed deformations) and which give rise to the same deformation of . Then there exists some matrix such that
Plugging in and we obtain
which will imply that
In the language of Galois cohomology (see also Etale Cohomology of Fields and Galois Cohomology) we say that and differ by a coboundary. This means that the tangent space of the Galois deformation ring is given by the first Galois cohomology with coefficients in :
More generally, when our Galois deformation ring is subject to conditions, it will be given by a subgroup of the first Galois cohomology known as the Selmer group (note that the Selmer group shows up in many places in arithmetic geometry and number theory, for instance, in the proof of the Mordell-Weil theorem where the Galois module used comes from the torsion points of an elliptic curve – in this post we are considering the case where the Galois module is , as stated earlier). The advantage of expressing the tangent space in the language of Galois deformation ring using Galois cohomology is that in Galois cohomology there are certain formulas such as Tate duality and the Euler characteristic formula that we can use to perform computations.
Finally to end this post we remark that under certain conditions (namely that for every open subgroup of the space of continuous homomorphisms from to has finite dimension) this tangent space is going to be a finite-dimensional vector space over . Then the Galois deformation ring has the following form
i.e. it is a quotient of a -power series in variables, where the number is given by the dimension of as a -vector space, while the number of relations is given by the dimension of as a -vector space.
Knowing the structure of Galois deformation rings is going to be important in proving R=T theorems, since such proofs often reduce to commutative algebra involving these rings. More details will be discussed in future posts on this blog.
Group cohomology on Wikipedia
Galois cohomology on Wikipedia
Selmer group on Wikipedia
Tate duality on Wikipedia
Modularity Lifting Theorems by Toby Gee
Modularity Lifting (Course Notes) by Patrick Allen
Motives and L-functions by Frank Calegari
Beyond the Taylor-Wiles Method by Jack Thorne