On the structure theory of the Iwasawa algebra of a p-adic Lie group

Venjakob, Otmar

doi:10.1007/s100970100038

Cited by 104 publications

(144 citation statements)

References 33 publications

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“…Thus there is a dimension theory for A-and B-modules, for a thorough treatment of which we refer the reader to [10] or [28]. The previous lemma shows that the canonical (co-)dimension filtrations of M ǫ A-mod R considered as A- similarly for B).…”

Section: Assume Now Thatmentioning

confidence: 99%

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A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory

Venjakob¹,

Vogel²

2003

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. In this paper and a forthcoming joint one with Y. Hachimori [15] we study Iwasawa modules over an infinite Galois extension k ∞ of a number field k whose Galois group G = G(k ∞ /k) is isomorphic to the semidirect product of two copies of the p-adic integers Z p . After first analyzing some general algebraic properties of the corresponding Iwasawa algebra, we apply these results to the Galois group X nr of the p-Hilbert class field over k ∞ . The main tool we use is a version of the Weierstrass preparation theorem, which we prove for certain skew power series with coefficients in a not necessarily commutative local ring. One striking result in our work is the discovery of the abundance of faithful torsion Λ(G)-modules, i.e. non-trivial torsion modules whose global annihilator ideal is zero. Finally we show that the completed group algebra F p [[G]] with coefficients in the finite field F p is a unique factorization domain in the sense of Chatters [8].

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Section: Assume Now Thatmentioning

confidence: 99%

“…2.3]). This should be compared also with Chamarie's result 3.3-3.5 in [7]: While A S is "responsible" for the faithful modules the bounded ring A (X) covers the bounded modules (in the quotient category of A-mod modulo the Serre subcategory of pseudo-null A-modules, see [10] or [28]). …”

Section: Every Height-1 Prime Ideal Of a Is Principalmentioning

confidence: 99%

A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory

Venjakob¹,

Vogel²

2003

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

Self Cite

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“…Assume that the last statement was not satisfied; among all codimensions of H i which violate this inequality, choose the maximal one, c. Among all i < d for which this codimension is achieved, choose the minimal one. Thus, the codimension of H i is c < d − i, but the codimension of H k for k < i is greater than c. The results of [63] imply that if X is of codimension c, then E j (X) = 0 for j < c, E c (X) is of codimension (exactly) c, and E j (X) is of codimension ≥ j for j > c. Now look at the Poincaré duality spectral sequence [17, Section 1.3]:…”

Section: P-adic Automorphic Formsmentioning

confidence: 75%

On torsion in the cohomology of locally symmetric varieties

Scholze

2015

Ann. Math.

134

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Abstract. The main result of this paper is the existence of Galois representations associated with the mod p (or mod p m ) cohomology of the locally symmetric spaces for GL n over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for GL n as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of Shimura varieties.Thus, we prove new results on the p-adic geometry of Shimura varieties (of Hodge type). Namely, the Shimura varieties become perfectoid when passing to the inverse limit over all levels at p, and a new period map towards the flag variety exists on them, called the Hodge-Tate period map. It is roughly analogous to the embedding of the hermitian symmetric domain (which is roughly the inverse limit over all levels of the complex points of the Shimura variety) into its compact dual. The Hodge-Tate period map has several favorable properties, the most important being that it commutes with the Hecke operators away from p (for the trivial action of these Hecke operators on the flag variety), and that automorphic vector bundles come via pullback from the flag variety.

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“…It is well known that Z p G is an Auslander regular ring (cf. [32,Theorems 3.26]). Furthermore, the ring Z p G has no zero divisors (cf.…”

Section: Completely Faithful Modules Over Completed Group Algebrasmentioning

confidence: 99%

On completely faithful Selmer groups of elliptic curves and Hida deformations

Lim

2015

Journal of Algebra

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In this paper, we study completely faithful torsion Zp G -modules with applications to the study of Selmer groups. Namely, if G is a nonabelian group belonging to certain classes of polycyclic pro-p group, we establish the abundance of faithful torsion Zp G -modules, i.e., non-trivial torsion modules whose global annihilator ideal is zero. We then show that such Zp G -modules occur naturally in arithmetic, namely in the form of Selmer groups of elliptic curves and Selmer groups of Hida deformations. It is interesting to note that faithful Selmer groups of Hida deformations do not seem to appear in literature before. We will also show that faithful Selmer groups have various arithmetic properties. Namely, we show that faithfulness is an isogeny invariant, and we will prove "control theorem" results on the faithfulness of Selmer groups over a general strongly admissible p-adic Lie extension.

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On the structure theory of the Iwasawa algebra of a p-adic Lie group

Cited by 104 publications

References 33 publications

A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory

A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory

On torsion in the cohomology of locally symmetric varieties

On completely faithful Selmer groups of elliptic curves and Hida deformations

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