An injectivity result for Hermitian forms over local orders

We show that PD4-complexes with free fundamental group are determined by their intersection pairings and that every hermitian form on a finitely generated free module over the group ring of a free group is realized by some such complex. The purpose of this article is to show that some of the basic properties of PD4-complexes with free fundamental group can be derived homologically, without reference to the topology of 4-manifolds or stabilization by connected sums, as used in (4, 8, 13). We also avoid explicit calculations of obstructions, relying instead on the easily verified fact that the 3-skeletons of the complexes considered have su‰ciently many self homotopy equivalences. In particular we give a new proof of the fact that such complexes are determined by their intersection pairings, and that every hermitian form on a finitely generated free module over the group ring of a free group is realized by some such complex. In the final section we consider briefly the classification (up to s-cobordism) of closed 4-manifolds with free fundamental group.

This chapter deals with a particular Hermitian module, namely that of the ring of integers in a tame normal extension of a global or local field, viewed as a Galois module, together with the Hermitian form coming from the trace. It was this application which originally motivated the making of a general Hermitian theory.KeywordsLocal FieldGalois GroupNumber FieldPrime DivisorRoot NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let R be an associative ring with a unit and N be a left R-module. The set MR(N) = {f: N → N | f(rx) = rf(x), r ∈ R, x ∈ N} is a near-ring with respect to the operations of addition and composition and contains the ring ER(N) of all endomorphisms of the R-module N. The R-module N is endomorphic if MR(N) = ER(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2.

The computation of the knot cobordism groups reduces to the computation of the L-theory of various fields and integral domains. The main properties of the L-groups of a field with involution F are summarized in 37A. The computations of the L-groups of ℤ, ℤ m , \({\widehat {\Bbb Z}_m}\), ℚ and the localization exact sequence are used in 37B to prove a general result: the natural maps L* (A) → L* (ℚ ⊗ℤ A) are isomorphisms modulo 8-torsion for any additively torsion-free ring involution A (e.g. a group ring A = ℤ[π]). In Chap. 42 the version of this result for torsion L-groups will be used to prove that the natural maps $${C_*} = LIs{o^{* + 1}}({\Bbb Z}) \to LIso_{{U_{\Bbb Z}}}^{* + 1}({\Bbb Q})$$ are isomorphisms modulo 8-torsion, for an appropriately defined U ℤ.

Louis Solomon introduced the notion of a zeta function ${\zeta _\Theta }(s)$ of an order $\Theta$ in a finite-dimensional central simple K-algebra A, with K a number field or its completion ${K_P}$ (P a non-Archimedean prime in K). In several papers, C. J. Bushnell and I. Reiner have developed the theory of zeta functions and they gave explicit formulae in some special cases. One important property of these zeta functions is the Euler product, which implies that in order to calculate ${\zeta _\Theta }(s)$, it is sufficient to consider the zeta function of local orders ${\Theta _P}$. However, since these local orders ${\Theta _P}$ are in general not principal ideal domains, their zeta function is a finite sum of so-called ’partial zeta functions’. The most complicated term is the ’genus zeta function’, ${Z_{{\Theta _P}}}(s)$, which is related to the free ${\Theta _P}$-ideals. I. Reiner and C. J. Bushnell calculated the genus zeta function for hereditary orders in quaternion algebras (i.e., $[A:K] = 4$). The authors mention the general case but they remark that the calculations are cumbersome. In this paper we derive an explicit method to calculate the genus zeta function ${Z_{{\Theta _P}}}(s)$ of any local hereditary order ${\Theta _P}$ in a central simple algebra over a local field. We obtain ${Z_{{\Theta _P}}}(s)$ as a finite sum of explicit terms which can be calculated with a computer. We make some remarks on the programming of the formula and give a short list of examples. The genus zeta function of the minimal hereditary orders (corresponding to the partition (1, 1, ... , 1) of n) seems to have a surprising property. In all examples, the nominator of this zeta function is a generating function for the q-Eulerian polynomials. We conclude with some remarks on a conjectured identity.

Non-isomorphic direct decompositions of torsion-free abelian groups are reflected in their endomorphism ring decompositions which admit matrix representations. The set of possible direct decompositions of a special kind matrix rings into direct sums of one-sided indecomposable ideals is described. This leads to the combinatorial constructions of isomorphisms between non-commutative differently decomposable ring structures.

In this note we deal with a question raised by R. S. Pierce in 1963: Determine the elements of the Jacobson radical of the endomorphism ring of a primary abelian group by their action on the group. We concentrate on separable abelian p p -groups and give a counterexample to a conjecture of A. D. Sands. We also show that the radical can be pinned down if the endomorphism ring is a split-extension of its ideal of all small maps.

The Green ring and modular representations of finite groups of Lie type

Some years ago A. L. S. Corner proved that every countable and cotorsion-free ring can be realized as the endomorphism ring of some torsion-free abelian group. This result has many interesting consequences for abelian groups. Using a set-theoretic axiom ∇k., which follows for instance from V = L, we can drop the countability condition in Corner's theorem.

In ring and module theory, the cleanness property is well established. If any element of R can be expressed as the sum of an idempotent and a unit, then R is said to be a clean ring. Moreover, an R-module M is clean if the endomorphism ring of M is clean. We study the cleanness concept of coalgebra and comodules as a dualization of the cleanness in rings and modules. Let C be an R-coalgebra and M be a C-comodule. Since the endomorphism of C-comodule M is a ring, M is called a clean C-comodule if the ring of C-comodule endomorphisms of M is clean. In Brzezi´nski and Wisbauer (2003), the group ring R[G] is an R-coalgebra. Consider M as an R[G]-comodule. In this paper, we have investigated some sucient conditions to make M a clean R[G]-comodule, and have shown that every G-graded module M is a clean R[G]-comodule if M is a clean R-module.

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν‎ and uses the usual convention that ν‎(0) = infinity. The chapter starts with a notation regarding the ring of integers of K and the natural map from it to the residue field, followed by a number of propositions regarding an anisotropic quadratic space. These include an anisotropic quadratic space with residual quadratic spaces, an unramified quadratic space of finite dimension, unramified finite-dimensional anisotropic quadratic forms over K, unramified anisotropic quadratic forms and a bilinear form, and a round quadratic space over K. The chapter concludes with a theorem that there exists an anisotropic quadratic form over K, unique up to isometry, and is non-singular.

This extends to a map dnW n! n 1 between the Z2 vector spaces whose bases are all such conjugacy classes in SO.n/ and then SO.n 1/. Using orbifold graphs, we prove dW ! is a differential and defines a homology, H . We develop a map sW ! C 1 for a subcomplex of groups which admit orientation-reversing automorphisms. We then look at examples and algebraic properties of d and s , including that d is a derivation. We prove that the natural map between the set of diffeomorphism classes of closed, locally oriented n‐orbifolds and n maps into kerdn and that this map is onto kerdn for n 4. We relate d to orbifold cobordism and surgery and show that quotients to a map between oriented orbifold cobordism and H . 57R18, 57R90; 55N32, 57R65

In this paper we give a detailed exposition of some of the results announced in [18]. The primary motivation for this work is Witt's observation [31, Satz 7] that if F is a field of characteristic #72, his ring 1V(F) of classes of anisotropic quadratic forms may be written as Z[G]/K where G is an abelian group of exponent two (actually G =F*/F*2), Z[G] is the integral group ring of G, and K is an ideal of Z[G] generated by elements of the form gl + g2-g3g4 and 1 + g5 with gi in G. Of course these elements can be described more explicitly, but for our purposes the only information we need about K is that any homomorphism of Z[G] -> Z sends K to 0 or to an ideal of the form 2nZ. In this introduction we shall call any ideal of Z[G] with this property admissible. In [26], Pfister proved certain structure theorems for IV(F) usilg his theory of multiplicative forms. His proofs were simplified in [11, 22, 23, 29, 30]. Harrison [11] and Leicht and Lorenz [22] gave important complements to Pfister's results concerning the ideal theory of WV(F). The main goal of our paper is to understand and generalize these structure theorems using standard techniques of commutative algebra. In [28] and [3], Scharlau and Belskii have introduced and studied Witt and Witt-Grothendieck rings for profinite groups. Their definitions generalize the usual notions of Wit.t rings, WV(F), and Witt-Grothendieck rings, K (F), of quadratic forms over fields of characteristic #72. These rings also have the form Z[G]/K with G an abelian group of exponent two and K an admissible ideal of Z[G]. Here we are mainly iinterested in another generalization of 1V(F) and K(F), namely the Witt rings, IV(C, J), and the Witt-Grothendieck rings, K(C,J) of classes of hermitian forms over a connected commutative semilocal ring C with involution J. Since J may be the identity these include the Witt and Witt-Grothendieck rings of classes of symmetric bilinear forms

Adjoint functors and equivalences of subcategories

It is shown that each commutative Artin local ring having each of its ideals generated by two elements is the homomorphic image of a one-dimensional local complete intersection ring which also has each of its ideals generated by two elements. It is indicated how this can be applied to show that the property that each ideal is projective over its endomorphism ring does not pass to homomorphic images, and in determining the commutative group rings with the two-generator property.