A ring describes possible points. A module over the ring describes what is carried at those points. A nonempty space of possibilities does not imply a nonzero carrier at every point.
| Symbol | Object and role | Keep separate |
|---|---|---|
| \({\mathcal O},\ k\) | Complete coefficient ring and its residue field. The formal teaching model uses \({\mathcal O}={\mathbb Z}_5\), \(k={\mathbb F}_5\). | Coefficients versus variables. |
| \(R_i,\ I_i\) | A supplied local ring and the ideal imposing a declared local condition. | A quotient condition is not an idempotent. |
| \(R^{\mathrm{loc}}\) | Completed tensor product of the local rings over \({\mathcal O}\). | Completion uses the declared adic topology. |
| \(J\) | Closed ideal generated by the images of all \(I_i\) in \(R^{\mathrm{loc}}\). | Closure, not just a finite list of tests. |
| \(R^{\mathrm{loc},\tau}\) | Completed tensor of the local quotients; identified with \(R^{\mathrm{loc}}/J\). | A constructed comparison, not assumed notation. |
| \(R^\square\) | Supplied global ring receiving the local structural maps. | Representability is arithmetic input. |
| \(R^\tau\) | Completed global base change imposing the simultaneous local conditions. | Does not create a missing residual point. |
| \(R_\infty,\ M_\infty\) | Patched ring and its module; a perfect complex is used in the derived version. | Freeness over an auxiliary ring is weaker than full support. |
| \(R,\ M\) | Finite-level ring and carrier after the specified quotient and control isomorphism. | The module may be zero on a component. |
| \(T,\ H\) | Hecke algebra and its ordinary or derived carrier. | Ring comparison does not imply classicality. |
| \(e_{{\mathfrak m}}H\) | Packet selected by a canonical Artin idempotent. | A generalized packet need not be an eigenspace. |
| \(Y_m\) | Stable image at level \(m\) of an inverse system. | Nonzero levels alone do not give a nonzero limit. |
Why arrows reverse when points appear
A ring map \(f:R\to T\) sends a prime of \(T\) to its inverse image in \(R\):
Similarly, a point \(y:T\to E'\) gives the point \(y\circ f:R\to E'\). The equality \(y\circ f=x\) is an actual requirement in the final theorem, not a consequence of drawing a connecting arrow.
Two kinds of point
Local quotient factorization is stated for continuous local coefficient-algebra maps to deformation test rings. The later field-valued point \(x:R\to E'\) uses its kernel and scalar fiber. A characteristic-zero field point is generally not a local ring map: the coefficient uniformizer becomes a unit. These two uses of “point” must retain their own categories.