12.1. Structure theory for Lambda-modules
There is a rich structure theory for modules over Iwasawa algebras, closely mirroring the theory of modules over a PID. We state the basic results without proof.
Let M, M' be two \Lam-modules. We say M is pseudo-isomorphic to M',
written M \sim M', if there is a homomorphism M \to M' with finite kernel
and cokernel, equivalently an exact sequence
0 \to A \to M \to M' \to B \to 0
with A and B finite (i.e. of finite cardinality). The relation \sim is
not symmetric in general, but it is an equivalence relation on the class of
finitely generated torsion \Lam-modules.
A polynomial P(T) \in \cO_L[T] is distinguished if it is monic of the form
P(T) = T^n + a_{n-1}T^{n-1} + \cdots + a_1 T + a_0 with every a_i \in \mathfrak{p}
for 0 \le i \le n-1.
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Let M be a finitely generated \Lam-module. Then M is pseudo-isomorphic
to a direct sum
M \sim \Lam^r \oplus \Big( \bigoplus_{i=1}^s \Lam/(p^{n_i}) \Big) \oplus \Big( \bigoplus_{j=1}^t \Lam/(f_j(T)^{m_j}) \Big),
for some integers r,s,t \ge 0, exponents n_i, m_j \ge 1, and irreducible
distinguished polynomials f_j(T) \in \cO_L[T]. This rests on
Definition 12.1.1 and Definition 12.1.2.
This is the structure theorem of Washington (1997). The Iwasawa
algebra \Lam \cong \cO_L[[T]] is a complete regular local ring of Krull
dimension 2 which, while not a PID, is a unique factorisation domain whose
height-one primes are (p) and the ideals (f) generated by irreducible
distinguished polynomials (by the Weierstrass preparation theorem every nonzero
element is, up to a unit, p^\mu times a distinguished polynomial). Localising
M at each height-one prime reduces to the principal-ideal situation, and
patching the resulting elementary divisors over all height-one primes yields the
displayed pseudo-isomorphism; the discrepancy supported at the maximal ideal is
exactly the finite kernel and cokernel allowed by \sim.
It is essential that this holds only for the projective limit \Lam, and not
for the finite-level group algebras \cO_L[\Zp/p^n\Zp]. This is a recurring
theme of Iwasawa theory: one studies the whole tower of objects at once rather
than each finite layer individually.
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Let M be a finitely generated torsion \Lam-module, so that r = 0 in
the structure theorem Theorem 12.1.3. The characteristic
ideal of M is
\Ch_{\Lam}(M) = (p^{n}) \prod_{j=1}^t (f_j^{m_j}) \subseteq \Lam, \qquad n = \sum_{i=1}^s n_i.
Equivalently, \Ch_\Lam(M) is the principal ideal generated by a generator of
the product of the height-one elementary divisors.
We will apply this theory in a slightly more general setting. Suppose
\GG = H \times \Gamma', where H is a finite commutative group of order
prime to p and \Gamma' \cong \Zp; the key example is \GG = \Zpx,
H = \mu_{p-1}, \Gamma' = 1 + p\Zp. Then
\Lam(\GG) \cong \cO_L[H] \otimes_{\cO_L} \Lam.
For each character \omega of H (possibly after enlarging L to contain its
values) define the idempotent projector
e_\omega = \frac{1}{|H|} \sum_{a \in H} \omega^{-1}(a)\,[a] \in \cO_L[H],
which makes sense because |H| is invertible in \cO_L.
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Let M be a finitely generated torsion \Lam(\GG)-module. The group H acts
on M^{(\omega)} := e_\omega M through the character \omega, and there is a
decomposition of \Lam(\GG)-modules
M = \bigoplus_{\omega \in H^\wedge} M^{(\omega)}.
Moreover each M^{(\omega)} is a finitely generated torsion \Lam-module.
Since |H| is a unit in \cO_L, the idempotents e_\omega are orthogonal and
sum to 1 in \cO_L[H], so they decompose any \cO_L[H]-module into its
\omega-isotypic pieces; this is just the semisimplicity of the group algebra of
a prime-to-p group over a ring in which its order is invertible. The
decomposition is \Lam(\GG)-linear because the e_\omega are central. Each
summand is cut out by an idempotent from a finitely generated torsion module, so
is itself finitely generated and torsion over \Lam.
With \GG = H \times \Gamma' as above and M a finitely generated torsion
\Lam(\GG)-module, the characteristic ideal of M is
\Ch_{\Lam(\GG)}(M) := \bigoplus_{\omega \in H^\wedge} \Ch_{\Lam}\big(M^{(\omega)}\big) \subseteq \Lam(\GG),
using the isotypic decomposition Lemma 12.1.5 and the
characteristic ideal Definition 12.1.4 of each \Lam-component.
The characteristic ideal is multiplicative in short exact sequences: if
0 \to M' \to M \to M'' \to 0 is an exact sequence of finitely generated torsion
\Lam(\GG)-modules, then
\Ch_{\Lam(\GG)}(M) = \Ch_{\Lam(\GG)}(M')\cdot \Ch_{\Lam(\GG)}(M''). This uses
Definition 12.1.6.
Reduce to the case of \Lam-modules via the isotypic decomposition
Lemma 12.1.5, which is exact. For \Lam-modules one
localises at each height-one prime \mathfrak{q}: the localisation \Lam_\mathfrak{q}
is a discrete valuation ring, and over a DVR length is additive in short exact
sequences. Hence the \mathfrak{q}-adic exponent of the characteristic ideal —
the length of M_\mathfrak{q} — is the sum of those of M' and M'', which is
exactly multiplicativity of the ideals.