Wild ramification and a vanishing cycles formula

Article On formal groups over complete discrete valuation rings with application to a theorem of Lutz. was published on January 1, 1982 in the journal Journal für die reine und angewandte Mathematik (volume 1982, issue 334).

Let (R,m) be a complete discrete valuation ring with the finite residue field R∕m= Fq. Given a monic polynomial P(t)∈R[t] whose reduction modulo m gives an irreducible polynomial P‾(t)∈Fq[t], we initiate an investigation of the distribution of coker(P(A)), where A∈Matn(R) is randomly chosen with respect to the Haar probability measure on the additive group Matn(R) of n×n R-matrices. In particular, we provide a generalization of two results of Friedman and Washington about these random matrices. We use some concrete combinatorial connections between Matn(R) and Matn(Fq) to translate our problems about a Haar-random matrix in Matn(R) into problems about a random matrix in Matn(Fq) with respect to the uniform distribution. Our results over Fq are about the distribution of the P‾-part of a random matrix A‾∈Matn(Fq) with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime p, any finite abelian p-group (i.e., Zp-module) H occurs as the p-part of the class group of a random imaginary quadratic field extension of Q with a probability inversely proportional to |AutZ(H)|. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems.

In this note we study a certain formal group law over a complete discrete valuation ring F [[ u n -1 ]] of characteristic p > 0 which is of height n over the closed point and of height n -1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law H n -1 of height n - 1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group S n -1 of H n -1 . We show that the automorphism group S n of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of S n -1 to the cohomology of S n with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed.

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].

Representation types of group rings over complete discrete valuation rings

We apply the G-algebra theory to the tensor product of algebras. These considerations are applied to extend the results of Alghamdi and Khammash [1], Khammash [4] and Külshammer [5, Proposition 1.2] on the tensor product of group algebras and modules over an algebraically closed field to lattices over a complete discrete valuation ring. This places these results in the standard integral finite group modular representation theory of G-algebras as pioneered by Puig (cf. [8]). We also study some aspects of covering homomorphisms and the Green correspondence in this context (cf. [8, Sections 20 and 25]).

A characterization of blocks with vertices

Let R be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let L and k be its fraction field and residue field respectively. Let Ω R be the set of rank 1 discrete valuations of L corresponding to codimension 1 points of regular proper models of Spec R . We prove that a quadratic form q over L satisfies the local-global principle with respect to Ω R in the following two cases: (1) q has rank 3 or 4; (2) q has rank ≥ 5 and R = A [ [ y ] ] , where A is a complete discrete valuation ring with a not too restrictive condition on the residue field k , which is satisfied when k is C 1 .

Special Quasi-triads and Integral Group Rings of Finite Representation Type, I

4 - Indecomposable Modules

Throughout the paper p is a prime number, k a field of characteristic p, assumed algebraically closed except in Appendix 1, and O a complete discrete valuation ring with residue field k (we allow the possibility O = k). All the O-algebras we consider are associative and unitary except in Remark 11.12 below, but the homomorphisms between O-algebras are not required to be unitary. The unity elements are simply denoted by 1 whenever they have no previous name as idempotents in bigger O-algebras; abusively, 1 denotes also either the trivial element or the trivial subgroup in any group. All the O-modules and O-algebras we consider are assumed to be finitely generated as O-modules (in short O-finite) except when we mention explicitely the contrary; a significant exception is O-algebras ℱ and D that we introduce in Section 9 and employ in all the subsequent sections. If A is an O-algebra, we denote by Aut(A) the group of automorphisms of O, by A* the group of invertible elements of A, by J(A) the Jacobson radical of A, by Z(A) the center of A and by A° the opposite O-algebra; recall that $$A = \sum\limits_{a \in {A^*}} {\mathcal{O}a} $$ (2.1.1)

Morita Equivalence for Blocks of the Schur Algebras

An integral domain R with quotient field Q is defined to be a Q-ring if Ext R 1 ⁡ ( Q , R ) ≅ Q \operatorname {Ext}_R^1(Q,R) \cong Q . It is shown that R is a Q-ring if and only if there exists an R-module A such that Hom R ( A , R ) = 0 {\operatorname {Hom}_R}(A,R) = 0 and Ext R 1 ⁡ ( A , R ) ≅ Q \operatorname {Ext}_R^1(A,R) \cong Q . If A is such an R-module and t ( A ) t(A) is its torsion submodule, then it is proved that A / t ( A ) A/t(A) necessarily has rank one. There are only three kinds of Q-rings, namely, Q 0 - , Q 1 - {Q_0}{\text {-}},{Q_1}{\text {-}} , or Q 2 {Q_2} -rings. These are described by the fact that if R is a Q-ring, then K = Q / R K = Q/R can only have 0, 1, or 2 proper h-divisible submodules. If H is the completion of R in the R-topology, then R is one of the three kinds of Q-rings if and only if H ⊗ R Q H{ \otimes _R}Q is one of the three possible kinds of 2-dimensional commutative Q-algebras. Examples of all three kinds of Q-rings are produced, and the behavior of Q-rings under ring extensions is examined. General conditions are given for a ring not to be a Q-ring. As an application of the theory, necessary and sufficient conditions are found for the integral closure of a non-complete Noetherian domain to be a complete discrete valuation ring.

It is a classical result that if ( R , M ) (R,\mathfrak {M}) is a complete discrete valuation ring with quotient field K K , and if R / M R/\mathfrak {M} is perfect, then any finite dimensional central simple K K -algebra Σ \Sigma can be split by a field L L which is an unramified extension of K K . Here we prove that if ( R , M ) (R,\mathfrak {M}) is any regular local ring, and if Σ \Sigma contains an R R -order Λ \Lambda whose global dimension is finite and such that Λ / Rad ⁡ Λ \Lambda /\operatorname {Rad} \Lambda is central simple over R / M R/\mathfrak {M} , then the existence of an “ R R -unramified” splitting field L L for Σ \Sigma implies that Λ \Lambda is R R -separable. Using this theorem we construct an example which shows that if R R is a regular local ring of dimension greater than one, and if its characteristic is not 2, then there is a central division algebra over K K which has no R R -unramified splitting field.

Weak completions, bornologies and rigid cohomology