Inertia Groups Research Articles

On purity of inertia

This paper considers the phenomenon of inertia groups jumping at regular points of the branch locus of a cover, especially in mixed characteristic. After giving conditions under which this cannot occur, these ideas are applied to the problem of obtaining groups as Galois groups of unramified covers of the affine line in finite characteristic.

THE GALOIS GROUP OF A MULTIDIMENSIONAL LOCAL FIELD OF POSITIVE CHARACTERISTIC

Let be an arbitrary field, Henselian relative to a discrete valuation of finite rank with residue field . If , where () is a discrete valuation of rank 1, then, setting , we denote by the residue field of the valuation of the field , where . A description of the absolute Galois group of the field , the inertia group and the ramification group of the valuation are obtained in terms of the absolute Galois group of the field of residues, its action on the roots of unity in the separable closure of the field , and the cardinalities of the fields and .Bibliography: 12 titles.

A note on ramification of the Galois representation on the fundamental group of an algebraic curve

Abstract Let X be a smooth proper connected algebraic curve defined over an algebraic number field K. Let πl( X )l be the pro-l completion of the geometric fundamental group of X = X ⊗K K . Let p be a prime of K, which is coprime to l. Assuming that X has bad reduction at p and the Jacobian variety of X has good reduction at p , we describe the action of the inertia group I p on the quotient groups of π1( X )l by the higher commutator subgroups.

Clifford theory of simple modules

“Clifford theory” primarily is concerned with the representations of normal subgroups and quotient groups of (finite) groups, the basic contribution being due to A. H. Clifford [3]. Modern developments, founded on the concepts of group graded rings and crossed products, have shown close relations with (commutative and non-commutative) Galois theory. This is most conspicuous in (stable) Clifford theory of simple modules. Suppose N w H -++ G is an extension of finite groups and V is a simple F,N-module where F,, is any field. Since passage from the inertia group to the whole group usually is easy to handle, we may and do assume that V is G-invariant (or H-invariant, stable). Then the annihilator I, = Ann,,,(V) of V generates a (homogeneous) ideal Z of F,-,H and A = F,H/Z is a fully G-graded algebra with simple l-component R = F. N/Z,. General Clifford theory is concerned with such G-graded algebras A and their modules (cf. Dade [7, lo]). Assume the l-component R = A, is simple Artinian as above. Then the centre K = Z(R) has in a natural way the structure of a G-field. The main object of study is the endomorphism ring E = End,( V”) of the A-module induced from the simple R-module V. This is a fully G-graded algebra as well. Generalizing Wedderburn’s theorem we show that A z M,(E) as G-graded algebras, where d is the dimension of V over the division algebra D = End, (V) (Theorem 2.1). In particular, K= Z(D) has the same G-structure with respect to E, and A and E are Morita equivalent in some strong sense. Now assume that dim, R = d2m2 is finite and that the (Schur) index m =m( V) is relatively prime to the order of the automorphism group G/C,(K) of K induced by G. Then the graded structure of A, E can be described by some distinguished cohomology class w,(V) E H*(G, K’). More precisely, w = oA( V) determines a crossed product Sz of K with G

Vanishing Cycles and Geometry of Curves Over a Discrete Valuation Ring

0. Introduction. Geometry of curves over a discrete valuation ring is reflected in the action of the monodromy on the f-adic cohomology groups. The stable reduction theorem ([D-M]) gives an equivalent geometrical condition for the action of the inertia group to be unipotent in the situation, where global means that the curve in consideration is proper. In this paper we will generalize it in two directions. One is to the local situation with the isolated singularity. The other is to consider the geometrical condition for the action of the wild ramification group to be trivial. We will also give a purely cohomological proof (without using Picard schemes) of the stable reduction theorem, and a new proof of the formula of Deligne ([La], Theorem (5.1.1)) based only on the stable reduction theorem. Here we only treat a strictly local discrete valuation ring with algebraically closed residue field; however the results apply to any discrete valuation ring with perfect residue field since they are etale local. We introduce some notations and terminologies.

Representation groups of characters

The subject we are concerned with has its roots in the work of Schur, who introduced the notion of a “representation group” (Darstellungs- gruppe). These groups have the property to lift (“linearize”) all projective representations of some (finite) group over an algebraically closed field [ 16). Fong [8 J noticed that Schur’s method applies in context with Clif- ford’s work, exhibiting some single invariant (stable) character or module. The familiar proof for the famous Fong-Swan theorem gives the model for this kind of representation groups (see Serre [17] or Feit [7]). Throughout we fix a finite group G with normal subgroup H and quotient group S= G/H. We also fix an absolutely irreducible (Frobenius) character x of H and a field F containing the values of x. We assume that x is G-invariant. As passage from the inertia group to the whole group usually is easy to handle, this is not too restrictive. It is a necessary con- dition for x to be extendible to G.

Character stabilizer limits relative to a normal nilpotent subgroup

In his recent paper [ 11, Dade established the existence of some (rather unexpected) invariants of certain irreducible characters of finite groups. The purpose of the present paper is to state and to provide a relatively easy proof for a somewhat weakened and less general form of Dade’s result. In fact, Dade, in his introduction to [ 11, explicitly encouraged me to publish this simplified approach. To explain and motivate our theorem, we need to discuss “stabilizer limits.” Let x E Irr(G). (We work only with characters over the complex numbers. Dade, on the other hand, considers other characteristic zero fields too.) If Ma G and 8 is an irreducible constituent of xM, let T= Z&O), the inertia group. Then there is a unique character VE Irr(TI 0) such that qG =x. We call q the Clzfford correspondent of x with respect to 0. (Note that by taking M = 1, we see that x is one of its own Clifford correspondents.) We can repeat this process and consider Clifford correspondents (in T) for the Clifford correspondent q E Irr( T) of x. A character $ arising via any number of such iterations, we call a compound ClifSord correspondent (CCC) of x and we denote the set of these objects by CCC(x). Note that if Ic/ ECCC(X), then Ic/“=x. Consider “minimal” element of CCC(x). These are the compound Clifford correspondents $ of x which themselves have no proper Clifford correspondents. (In other words, they are quasiprimitive.) Following T. Berger and Dade, we call these characters the stabilizer limits of x. What do the different stabilizer limits for a given x E Irr(G) have in common? In general, the answer is “not much”.

On the Inertia Groups of h-Cobordant Manifolds

On the Inertia Groups of h-Cobordant Manifolds

On the inertia groups of ℎ-cobordant manifolds

It is shown that if M M and N N are h h -cobordant manifolds of dimension at least eight, then their special inertia groups are equal. (G. Brumfiel has shown that special inertia groups are not homotopy type invariants.) Examples are constructed for which the inertia group is an h h -cobordism invariant.

On the inertia groups of fibre bundles

A subgroup I ~ ( M × S i ) \tilde I(M \times {S^i}) of the inertia group I ( M × S i ) I(M \times {S^i}) is defined and shown to lie in I ( C ) I(C) for every fibre bundle M n → C → N i {M^n} \to C \to {N^i} . For certain M M , examples of nontrivial elements in I ~ ( M × S i ) \tilde I(M \times {S^i}) are constructed using the τ \tau -pairing of Milnor-Munkres-Novikov. For compact mapping tori M g {M_g} it is shown that I ( M g ) = I ( M × S 1 ) I({M_g}) = I(M \times {S^1}) if π 1 M {\pi _1}M is finite and Wh ( π 1 M ) = 0 {\text {Wh}}({\pi _1}M) = 0 .

Fields with two incomparable henselian valuation rings

Let K be a field and C, C' be two incomparable valuation rings of the separable closure of K, Theorem 1.2 states that the intersection of the decomposition groups of C, C', with respect to K, is precisely the inertia group of the composition ring C·C'. We apply this theorem in the study of two special cases of valued fields (L,B). In the first case, B is henselian and there is a subfield K of L such that L|K is a normal extension and B ∩ K is not henselian. The second case is that in which B has exactly two prolongations in the separable closure of L. We call these rings semihenselian rings, and they are characterized through Theorems 2.6 and 2.12.

On inertia groups and bordism.

On inertia groups and bordism.

On the Inertia Groups of Certain Manifolds

On the Inertia Groups of Certain Manifolds

Addendum: Π-principal hereditary orders (Nagoya Math. J. 32 (1968), 41–65)

Let R denote a complete discrete rank one valuation ring of unequal characteristic, and let p denote the characteristic of the residue class field R̅ of R. Consider the integral closure S of R in a finite Galois extension K of the quotient field k of R. Recall (see Prop. 1.1 of [3]) that the inertia group G0 of K over k is a semi-direct product G0 = J × Gp, where J is a cyclic group of order relatively prime to p and Gp is a normal p-subgroup of G.

On the Inertia Group of a Product of Spheres

In this paper it is proved that the smooth connected sum of a product of ordinary spheres with an exotic combinatorial sphere is never diffeomorphic to the original product. This result is extended and compared to certain related examples.

On the inertia group of a product of spheres

In this paper it is proved that the smooth connected sum of a product of ordinary spheres with an exotic combinatorial sphere is never diffeomorphic to the original product. This result is extended and compared to certain related examples.

Inertia Groups of Manifolds and Diffeomorphisms of Spheres

Inertia Groups of Manifolds and Diffeomorphisms of Spheres

On the inertia groups of homology tori

On the inertia groups of homology tori

Inertia groups of low dimensional complex projective spaces and some free differentiableactions on spheres, I

Inertia groups of low dimensional complex projective spaces and some free differentiableactions on spheres, I

On the Inertia Group of π-Manifolds

On the Inertia Group of π-Manifolds