If the characteristic of the coefficient field is the same as that of the residual characteristic of the local field, we need tools from -adic Hodge theory.
Cohen structure theorem: Suppose is equal characteristic CDVR (complete discrete valuation ring). Then the canonical ring homomorphism splits ring theoretically, i.e. there exists subring such that is an isomorphism, and any choice of induces . When is mixed characteristic , in this case, there is a coefficient subring with the following property: 1. is a CDVR with residue field isomorphic to via . 2. is absolutely unramified, i.e. unramified over , or . This is determined up to isomorphism by , but not uniquely/functorially. Such a coefficient ring is also called Cohen -ring.
If is not perfect, then can be very noncanonical. For example, if we take , and . Then is a CDVR with unique maximal ideal and . But is an automorphism of that reduced to identity mod .
If is perfect, there is a functor of Witt vectors from the category of perfect fields of characteristic to that of Cohen -rings, splitting the functor of taking reduction mod , and it is unique up to unique isomorphism. The functor uniquely extends to a functor from the category of perfect (Frobnius map is an isomorphism) -algebras to that of -adically complete () flat (torsion-free) -algebras with perfect.
Example: , .
-adic expansions: If , for some , then (expand ). There is a unique map preserving multiplication such that and has -th root for all .
The idea is to construct a Cauchy sequence that reduces to . Take such that (which exists since is perfect). Pick any lift , then . By completeness of , converges.
Since is flat, we can divide by and get a -adic expansion .
The question is that if we can write down using and . First . Then . Raising both sides to the -th power, then we get . So on and so forth.
For multiplication , we similarly expand and find out . In general, given , we can find such that is a polynomial over independent of .
The idea is to we can use the representative as coefficient, but they do not have good properties (not closed under addition or multiplication), while the Teichmuller representatives are closed under multiplication.
We now turn the process around and use the universal polynomials to construct from .
Important maps: Teichmuller mapping to which is multiplicative. Then for , we can write it as . Since , we see that is not a zero divisor. Frobenius map given by and Verschiebung given by and by definition we have .
We define (keeping only the first -coordinates) and we have which implies is -adic complete and its reduction mod is . Since the universal polynomials is independent in , the Witt vector construction is functorial in . In particular, is canonically a -algebra. Since is not zero divisor in , it is a flat -algebra.
If is a -adic complete flat -algebra with perfect , then any -homomorphism uniquely lifts to given by . Note that even when is not defined for all elements (e.g. is not perfect), it is defined for for . Moreover, if is flat over or for some (implying or ), then the same universal polynomial for and can still make sense in . Thus is still a ring homomorphism as long as is a -adic complete -algebra and flat over or for some . However, note that is not isomorphic to since it is not flat (see criterion below).
For perfect -algebra , can be thought of as the unique deformation of to a -adic complete flat -aglebra. The argument is based on deformation theory, using (essentially boils down to the derivative of is zero).
Remark: There is an explicit creiterion that is perfect iff flat over (since the flatness is equivalent to being exact, where the first map given by and the second one given by ).
Hodge-Tate and de Rham’s representation: Let and and similarly. We have . It is certainly not discretely valued. Note that we have (allows the use of almost mathematics).
The ring is an example of perfectoid field (a complete topological field with topology induced by a non-discrete valuation such that the arithmetic Frobenius is surjective). Here is a quick summary of some results on perfectoid fields, for details see Scholze’s original paper.
A perfectoid field of characteristic is the same as a complete perfect nonarchimedean field. The non-discrete valuation condition guarantees that the value group is -divisible. It is related to the notion of deeply ramified fields. Next we describe the process of tilting for perfectoid fields, which is a functor that takes as input a perfectoid field and produce a perfectoid field in characteristic .
Choose any such that . Define where is the Frobenius morphism (note that is a highly nonreduced -algebra and by taking inverse limit we have made into a perfect ring of characteristic ). Equip it with the inverse limit topology where each is given the discrete topology. We first claim there is a map . This map is similar to the construction of Teichmuller representative (this just uses that ). So similar to , is multiplicative and continuous. Using we can further define by which is inverse to the projection map. This shows that the two inverse limits are isomorphic as topological multiplicative monoid.
Secondly, There is an element such that (pick any with and choose any sequence ). If we define , then extends to (note that it is harmless to replace by ), which is easily seen to be a homeomorphism. In particular is a field. The topology on is induced .
Fact: Finite extension of perfectoid field is perfectoid and the tilting functor defines an equivalence of category between finite extensions of and finite extensions (Theorem 3.7), the proof of which uses almost mathematics, the key being the following string of equivalences of categories:
In particular, . For example, if we take , then . The reduction is isomorphic (multiplicatively) to mod . Note that the absolute Galois group of is just since taking -th root of gives purely inseparable extensions.
We are interested in representations of the form , but since acts on (since ), it is not a -linear representation, but rather a semi-linear representation. There is a standard recipe to build semi-linear representations, namely if is a ordinary linear representation and is an -algebra such that acts on (e.g. and ) then is a semi-linear representation. In particular, if is a character, then is a semilinear representation defined by .
We call a -semilinear representation is trivial if it is isomorphic to for some . Note that -semi-linear representation of is trivial if and only if it admits a basis of vectors which are fixed by . In particular, it is quite possible that a nontrivial semi-linear representation becomes trivial after scalar extension. Given , we denote by the subset of consisting of fixed points under , Clearly is a module over . Moreover scalar extension provides a canonical morphism in : This is useful for recognizing trivial semi-linar representations since if is trivial, will be an isomorphism by virtue of . The converse holds when and are free of finite rank over and respectively (the intuition is that we can detect a trivial -semi-linear representation arise from a trivial -linear representation).
If is finite acting on a field , then is a finite Galois extension with Galois group . Hilbert 90 can be reformulated by saying that is always surjective, and if is finite-dimensional, then is bijective, is trivial semi-linear representation, see this survey paper, Theorem 1.3.3 for details. Note that this fails if is an infinite extension, e.g. , since there is no such that for every , see Example 1.3.5 for detail.
Now we come to an important definition, a finite-dimensional representation is -admissible if is trivial. A numerical criterion for recognizing -admissible representations is that , provided satisfies some requirements (Proposition 1.4.4).
Fact: Let be a -linear finite dimensional representation of . Then is -admissible if and only if the inertia subgroup of acts on through a finite quotient (Theorem 1.4.6). Thus, -admissibility detects those representations which are potentially unramified. In particular, the cyclotomic character are not -admissible. Then is Hodge-Tate iff it is -admissible.
A larger class of -representations: A -linear finite-dimensional representation of is Hodge-Tate if . This fits into the framework of -admissibility as follows: Let and .
Theorem 1.4.6 is the starting point for studying Hodge–Tate representations. For example, it implies that the integers ’s that appeared above are uniquely determined up to permutation (Proposition 2.2.8). They are called the Hodge–Tate weights of the representation . Finally, Hodge-like decomposition theorems show that many representations coming from geometry are Hodge–Tate.
Unfortunately, Hodge–Tate representations have several defaults. First, they are actually too numerous and, for this reason, it is difficult to describe them precisely and design tools to work with them efficiently. The second defect of Hodge–Tate representations is of geometric nature. Indeed, tensoring the etale cohomology with (or equivalently, with ) captures the graded module of the de Rham cohomology. However, it does not capture the entire complexity of de Rham cohomology, the point being that the de Rham filtration does not split canonically in the -adic setting.
In order to work around this issues, Fontaine defined other period rings ‘finer’ than . The most classical period rings introduced by Fontaine are ; the corresponding admissible representations are called crystalline, semi-stable and de Rham respectively. Moreover, is a filtered field whose graded ring can be canonically identified with . This property, together with the aforementionned inclusions, imply the following implications (since if , or is an algebra over and is -admissible, then it is admissible): crystalline implies semi-stable implies de Rham implies Hodge–Tate.
Rapidly, let us say here that representations coming from the geometry, i.e. of the form where is a smooth projective algebraic variety over , are all de Rham. By definition, this means that the space has the correct dimension. It turns out that this space has a very pleasant cohomological interpretation: it is canonically isomorphic to the de Rham cohomology of , namely . We thus get an isomorphism:
The introduction of resolves elegantly the geometric issue we have pointed out earlier. However, the class of -admissible representations is still rather large and not easy to describe. The ring is a subring of which is equipped with more structures and provides very powerful tools for describing crystalline representations. On the geometric side, crystalline representations correspond to the etale cohomology of varieties with good reduction and the space is related to the crystalline cohomology of (the special fibre of a proper smooth model of , equipped with its Frobenius endomorphism.
Let and . It has a strong geometrical interpretation observed first by Colmez and then by Fargues–Fontaine and Scholze that appears at a mixed characteristic analogue of the ring of bounded analytic functions on the open unit disc. There is an important map lifting . Concretely, it is given by The kernel is a principal ideal generated by . It turns out the element also generates , where is a compatible system of -th power roots of unity.
We now define and . The key feature of is that it’s a CDVR with residue field (see Tony Feng’s thesis, Prop. 8.18). We equip with the topology from (so that it induce the usual topology on . There is a special element in that is a period for the cyclotomic character, i.e. acts by multiplication by , namely . Note that the -line generated by is independent of the choice of , which can be thought of as analogous to in complex analysis, and the element as analogous to a choice of .
Fact: is a uniformizer for and thus the associated graded algebra of is isomorphic (Galois-equivariantly) to .
One idea to create is that we want to functorially build a complete discrete valuation ring with residue field of characteristic 0. Naturally the Witt vector construction comes to mind, but we need to be more artful here since we are in the equicharacteristic zero situation. Note that any complete discrete valuation ring with residue field of characteristic is abstractly isomorphic to by commutative algebra, such a structure will not exist for in a -equivariant manner. Rather than trying to directly make a canonical complete discrete valuation ring with residue field , we observe that which is closely related to -power torsion rings. Hence, it is more promising to try to adapt Witt-style constructions for than for . We will make a certain height- valuation ring of equicharacteristic whose fraction field is algebraically closed (hence perfect) such that there is a natural -action on and a natural surjective -equivariant map . (Note that , so is a domain of characteristic 0.) We would then get a surjective -equivariant map . Since is like a 1-dimensional ring, is like a 2-dimensional ring and so is like a 1-dimensional ring. The ring structure of is generally pretty bad if is not a perfect field of characteristic , but as long as the maximal ideal is principal and nonzero we can replace with its -adic completion to obtain a canonical complete discrete valuation ring having residue field .
Reference: https://math.stanford.edu/~conrad/papers/notes.pdf