On the catenarity of virtually nilpotent mod- p Iwasawa algebras
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ON THE CATENARITY OF VIRTUALLY NILPOTENT MOD-p IWASAWA ALGEBRAS
BY
William Woods Department of Mathematics, Ben-Gurion University, Be’er Sheva 84105, Israel e-mail: [email protected]
ABSTRACT
Let p > 2 be a prime, k a finite field of characteristic p, and G a nilpotentby-finite compact p-adic analytic group. Write kG for the completed group ring of G over k. We show that kG is a catenary ring.
Introduction Fix a prime p, a commutative pseudocompact ring k (e.g., Fp or Zp ) and a compact p-adic analytic group G. (Such groups are perhaps most accessibly characterised as those groups G which are isomorphic to a closed subgroup of GLn (Zp ) for some n.) The completed group ring kG (sometimes written k[[G]]) is defined by kG := ← lim − k[G/N ], N
where the inverse limit ranges over all open normal subgroups N of G, and k[G/N ] denotes the ordinary group algebra of the (finite) group G/N over k. This ring satisfies an obvious universal property [24, Lemma 2.2], and modules over it characterise continuous k-representations of G (which has the profinite topology). When k = Fp , Zp or related rings, this is often called the Iwasawa algebra of G. Received August 21, 2017 and in revised form June 29, 2018
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W. WOODS
Isr. J. Math.
Iwasawa algebras (and related objects, such as locally analytic distribution algebras [21]) have recently become a very active research area due to their number-theoretic interest, for instance in the p-adic Langlands programme: see [20], for example. They are also interesting objects of study in their own right, as an interesting class of noetherian rings: see [3] for a 2006 survey of what is known about these rings. Our main result is the following. Theorem A: Take p > 2. Let G be a nilpotent-by-finite compact p-adic analytic group, and let k be a finite field of characteristic p. Then kG is a catenary ring. Recall that a ring R is said to be catenary if any two maximal chains of prime ideals with common endpoints have the same length, i.e., whenever P = P1 P2 · · · Pr = P 0 , P = Q1 Q2 · · · Qs = P 0 are two chains of prime ideals of R which cannot be refined further (i.e., by adding an extra prime ideal Pi I Pi+1 or Qi I Qi+1 ), we have that r = s. This is a “well-behavedness” condition on the classical Krull dimension of kG: it says that, whenever P P 0 are adjacent prime ideals and the height h(P ) of P is finite, then we have h(P 0 ) = h(P ) + 1. This result goes some way towards redressing the long-standing gap between Iwasawa algebras and similar algebraic objects; for instance, similar catenarity results had already been established for classical group rings of virtually polycyclic groups (in a special case in [19], in full generality in [14]), for universal enveloping algebras of finite-dimensional soluble Lie algebras over C [6]; for quantised coordinate rings over C [8], and over more general fields in [26]; for q-commutative power series rings [11]; and so on. In proving this result, we crucially use the prime extension theorem, [23, Theorem A]. Before we can state this,
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