Suppose that G is a finite group and k field of characteristic $$p>0$$ . We consider the complete cohomology ring $${\mathcal {E}}_M^* = \sum _{n \in {\mathbb Z}} \widehat{{\text {Ext}}}^n_{kG}(M,M)$$ show has two distinguished ideals $$I^* \subseteq J^* {\mathcal {E}}_M^*$$ such $$I^*$$ bounded above in degrees, {E}}_M^*/J^*$$ below degree $$J^*/I^*$$ eventually periodic with terms dimension. ...

ly isomorphic as rings are conjugate in B. 146 BRIAN CONRAD (APPENDIX BY W. R. MANN) Remark A.10. The order ( A mn A A ) in (2) is the intersection of the maximal orders M2(A) and γnM2(A)γ−1 n = ( A mn m−n A ) , where γn = ( 0 πn 1 0 ) . This example is called a standard Eichler order. Proof. For the first part, it suffices to show that the set R of elements of B integral over R is an order. Th...

The strong ring R is a commutative ring generated by finite abstract simplicial complexes. To every G ∈ R belongs a Hodge Laplacian H = H(G) = D = (d + d∗)2 determining the cohomology and a unimodular connection operator L = L(G). The sum of the matrix entries of g = L−1 is the Euler characteristic χ(G). For any A,B ∈ R the spectra of H satisfy σ(H(A × B)) = σ(H(A)) + σ(H(B)) and the spectra of...

Given matrices A and B shift equivalent over a dense subring R of R, with A primitive, we show that B is strong shift equivalent over R to a primitive matrix. This result shows that the weak form of the Generalized Spectral Conjecture for primitive matrices implies the strong form. The foundation of this work is the recent result that for any ring R, the group NK1(R) of algebraic K-theory class...

A graded K-algebra R has property Np if it is generated in degree 1, has relations in degree 2 and the syzygies of order ≤ p on the relations are linear. The GreenLazarsfeld index of R is the largest p such that it satisfies the property Np. Our main results assert that (under a mild assumption on the base field) the c-th Veronese subring of a polynomial ring has Green-Lazarsfeld index ≥ c+1. T...

Using Fourier–Mukai transformations, we prove some results about the ring of unipotent vector bundles on elliptic curves in positive characteristics. This ring was determined by Atiyah in characteristic zero, who showed that it is a polynomial ring in one variable. It turns out that the situation in characteristic p > 0 is completely different and rather bizarre: the ring is nonnoetherian and c...

0. Introduction. 1. Let G be a group of linear transformations on a finite dimensional real or complex vector space X. Assume X is completely reducible as a G-module. Let 5 be the ring of all complexvalued polynomials on X, regarded as a G-module in the obvious way, and let J C 5 be the subring of all G-invariant polynomials on X. Now let J be the set of all ƒ £ J having zero constant term and ...

Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (∀∃∀∃)(F = 0) where the ∀quantifiers run over a total of 8 variables, and where F is a polynomial. We show that for a large class of number fields, including Q, for every ε > 0, there exists a set of primes S of natural density exceeding 1− ε, such that Z can be defined as a subse...

The matrix representation of the set ∆(d), d = (d1, d2, d3), of the integers which are unrepresentable by d1, d2, d3 is found. The diagrammatic procedure of calculation of the generating function Φ ( d; z ) for the set ∆(d) is developed. The Frobenius number F ( d ) , genus G ( d ) and Hilbert series H(d; z) of a graded subring for non–symmetric and symmetric semigroups S ( d ) are found. The u...

Given a finite ring A which is a free left module over a subring R of A, two types of R-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality preserving maps from codes over A to codes over R. Both types of bases are generalizations of similar concepts for fields. Many illustrative examples are given to shed l...