Keep the coefficient ring, its reduction, and the carrier separate. A residue class is a finite-precision address. A vector in a carrier is a different kind of object, even when both are selected by something called a packet.
| Object | What it records | What it does not supply |
|---|---|---|
| \({\mathcal O},\varpi,k\) | Complete valuation ring, uniformizer, and residue field \(k\) of size \(q\). | A represented deformation functor. |
| \(\bar x\) | One residual object in the declared moduli problem. | Other residual roots or their transports. |
| \(X_m(\bar x)\) | Admissible objects modulo \(\varpi^m\) reducing to \(\bar x\). | A lift of every individual member. |
| \((x_m)_m\) | A branch satisfying \(x_{m+1}\bmod\varpi^m=x_m\). | A global object without the stated reconstruction hypotheses. |
| \(w_i\) | A coordinate constrained to begin in \(\varpi^{w_i}{\mathcal O}\). | A classification of all singular deformation rings. |
| \(Z(T)\) | The formal series \(\sum_{m\ge1}\#X_mT^m\). | The cardinality of the inverse limit. |
| \(C^\bullet,d\) | Cochains and a differential with \(d^2=0\). | Its arithmetic identification with a Galois problem. |
| \(G\) | The graph of supplied admissible residual transports. | A proof that a proposed arithmetic edge exists. |
| \(\mathscr C\) | Common algebraic polynomials and declared local data. | A member realizing those data at every prime. |
| \(C\boldsymbol m=\boldsymbol b\) | A finite trace system for carrier multiplicities. | Integrality, positivity, or available capacity automatically. |
| \(e_TV\) | The joint eigenspace inside an existing finite carrier \(V\). | Nonzero occupancy for every desired joint label. |
Three indices that must not be conflated
The depth \(m\) counts retained digits. A coordinate index \(i\) selects a parameter or a packet. A coefficient place \(\lambda\) selects a different local field. Increasing \(m\) at one \(\lambda\) neither changes \(i\) nor constructs the member at another \(\lambda\).
Read this as
Specify the ambient object before applying a projector, a reduction, a rank, or a counting formula. The same-looking symbol can refer to quite different spaces.