Discussion

The key idea is to separate arithmetic on an underlying approximation with the computation of the precision of the result. To achieve this goal, we define classes for precision objects in different styles: e.g. flat, jagged, convex... See below for more details.

Motivation: We want to work with complete discrete valuation rings. Such rings are equal to the projective limit of the quotients by ideals, and these quotients are algebraic simpler (in practice, often finite). For rings, we can compute with elements by working in these quotients. But for modules, there are more quotients, and we need to choose a distinguished set of quotient to compute with.

Definition: Let M be a finite type module over a complete discrete valuation ring R with uniformizer p. A precision type for M is a set T of distinguished submodules of M. T is a partially ordered set under reverse inclusion, and we require the following:

  1. For any P in T, computation in M/P is feasible. Usually this comes down to having finite representations of elements and being able to say whether two representations correspond to the same element of M/P. Note that this condition implies that P is nonzero.
  2. If P and Q are elements of T then they have an infimum in T, namely an element R that contains both P and Q and is contained in all other elements of T with this property (such a poset is generally referred to as a semilattice, but we will avoid this terminology because of its conflict with the definition of lattice in vector spaces over discrete valuation fields).

We say that a precision type is complete if in addition

  1. The intersection of all P in T is 0.

By a precision we mean an element of some precision type.

Examples:

Assume that we can compute in the residue field k=R/pR of R.

  1. The set of all submodules P of M such that M/P has finite length over R.

  2. The set of all submodules of the form p^k*M. We will call such a precision type flat.

  3. Suppose M is free, with a distinguished basis {e_i}_{i in I}. For each tuple (n_i)_{i in I} define P_{n_i} to be the submodule generated by p^{n_i}e_i. As the n_i range over non-negative integers, the resulting set of submodules is an example of a precision type. We will call such a precision type jagged.

  4. The empty set of submodules is an example of a precision type which is not complete. The previous examples are all complete.

Examples when M is the space of matrices of a given size over a complete discrete valuation ring:

  1. All of the above, since we can just forget the matrix structure on M.

  2. A row precision, where we specify the same precision for each row, which is a precision for the row module of M.

  3. A column precision, where we specify the same precision for each column, which is a precision for the column module of M. Note that this has a natural interpretation in terms of morphisms, giving a common precision for the image of any vector.

We'd like to discuss precisions for modules such as R[x] which are not of finite type.

Definition: Let M be a module over R. Let U be a set of pairs (N, P) of submodules with N of finite type and P contained in N. We order U by reverse inclusion (so (N, P) <= (N', P') if N contains N' and P contains P'). For any submodule N of M, let T(N) be the set of P contained in N with (N, P) in U. We say that N appears in U if T(N) is nonempty. We say that U is a regional precision type if:

  1. For any submodule N appearing in U, T(N) is a precision type for N.
  2. Infimums exist in the poset U.

In addition, we say that

  1. U is exhaustive if the union of all N appearing in U is M.

  2. U is complete if for any element x in M and any integer k > 0 there is an (N, P) in U such that x is in N and (p^k*M intersect N) is contained in P. (implies exhaustive)

Examples:

  1. The flat regional precision is the set of all (N, p^k*M intersect N). It is complete. We can also restrict N to an exhaustive set of submodules of M.

  2. Suppose that M is free with a chosen basis {e_i}. For all tuples (n_i) of elements of NN_infinity (the union of the non-negative integers with infinity) that are almost always infinite, define N_{n_i} to be the submodule of M generated by the e_i with n_i finite and define P_{n_i} to be the submodule of M generated by the p^{n_i}*e_i. Then the set of all pairs (N_{n_i}, P_{n_i}) is the jagged regional precision.

  3. Suppose that M = R[x] with the canonical basis (indexed by the natural numbers). Given a finite list L of pairs (a, v) with a and v non-negative integers, set N_L to be the submodule of polynomials with zero coefficients outside the interval defined by the smallest and largest values of a appearing in L. Set P_L to be the submodule of polynomials where all (i, v(c_i)) lie in the convex hull of L. We will call this the

  4. Let M be a finite dimensional vector space over a discrete valuation field. We let U be set of (N, P) with N a lattice in M and P a precision (in some precision type) for N. Note that you can extend this for infinite dimensional vector spaces.

We've come a long way, but we have another problem to solve. We want to talk about the precision of a submodule (so that we can give the precision for the kernel of a matrix for example). We can think about a submodule as a point on an appropriate Grassmanian, so we're led to the task of defining precisions for not-necessarily affine schemes. As a first step, we'd like to make the following definition.

Definition: Let M be a module over R. Let X be a set of triples (A, N, P), where A is an affine subspace of M, N is the submodule of M lying under A and P is a submodule of N. We assume that all N have finite type. We define U(X) to be the set of all pairs (N, P) with (A, N, P) in X for some A. We say that X is an affine regional precision type if:

  1. U(X) is a regional precision type.
  2. Some more conditions, but we're not sure what. We don't even know if this makes sense. 

class PrecisionObject

Tasks

  1. Define precision classes for vectors (e.g. flat, jagged, concave, submodule) and for matrices (e.g. flat, jagged, planar, column (submodule of codomain), row (submodule of domain))

  2. Define a vector class that separates data from precision. The approximation could be a vector over another QuoDVR or over ZZ for example. Override vector operations to compute an approximation separately from the precision of the answer (mostly arithmetic).

  3. Define a matrix class that separates data from precision. The appoximation could be a matrix over another (finite) QuoDVR or over ZZ for example. Override necessary matrix methods (quite a few).