Resolutions of determinantal ideals: The submaximal minors

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

We give a characterization theorem for the E(A)-normalized subgroups of G2 (A), where A is any commutative ring. This is the last of the simple Chevalley-Demazure group-schemes for which such a theorem is lacking. INTRODUCTION In this paper we prove a theorem (3.6) which characterizes the E(A)-normalized subgroups of G2(A) for an arbitrary commutative ring A. The theorem brings to completion the larger project of finding characterization theorems for E(A)normalized subgroups of simple Chevalley-Demazure group-schemes G(A) of rank > 2 over arbitrary commutative rings. The bulk of t;he project was done by Vaserstein in his essential paper [16]. There he proved the standard characterization theorem, given by a ladder corresponding to an ideal, for all commutative rings for all G of rank > 2 except those of types B2= C2 and G2. Vaserstein also suggested in that paper that previous work by Abe [1] and Abe-Suzuki [3] could be improved to yield more. This was accomplished by Abe in [2], wherein he showed that E(A)normalized subgroups of G(A) could be characterized by ladders determined by pairs of ideals, with the exception that for G of type B2 = C2 or G2 rings A having residue class fields of two elements must be excluded. Our paper [8] subsequently succeeded in removing this restriction for G of type C2 (and so also for G of type B2) by recognizing that the presence of residue fields of two elements required lowering the bottom of the ladder. This left only the case G of type G2 to be finished; the present paper does so by applying the lower bottom rung idea of [8] (see also [7]) to this case. We now give a more explicit description of our results. Let A be an arbitrary commutative ring and denote by G2(A) the group of type G2 defined over A. The main theorem (3.6) is that a subgroup N of G2(A) is E(A)-normalized if and only if C(J, J') C N C G(J, J') for a pair of ideals J, J' satisfying the condition J3 C J' C J. Here, J3 denotes the ideal of A generated by all elements of the form x3 and 3x for all x in J, so that (J, J') is an admissible pair in the sense of Abe. The bracketing groups C(J, J') and G(J, J') are defined in Section 1 below. In essence, C(J, J') is the mixed commutator group [E(A), E(J, J')], where E(J, J') is the E(A)-normalized subgroup generated by all elements x,(j) with a long and Received by the editors April 1, 1997 and, in revised form, May 22, 1997. 1991 Mathematics Sutbject Classification. Primary 20H05; Secondary 20G35. Research partially supported by NSA grant MDA 904-94-H-2008 and NSF grant DMS-9622899. (?)1999 American Mathematical Society

Working in a polynomial ring , where is an arbitrary commutative ring with 1, we consider the th Veronese subalgebras , as well as natural ‐submodules inside . We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple ‐equivariant minimal free ‐resolutions for the quotient ring and for these modules . These also lead to elegant descriptions of for all and for any pair of these modules .

The so called dense pairings were studied mainly by Radford in his work on coreflexive coalegbras over fields. They were generalized in a joint paper with Gómez-Torricillas and Lobillo to the so called rational pairings over a commutative ground ring R to study the interplay between the comodules of an R-coalgebra C and the modules of an R-algebra A that admits an R-algebra morphism κ : A → C*. Such pairings, satisfying the so called α-condition, were called in the author's dissertation measuring α-pairings and can be considered as the corner stone in his study of duality theorems for Hopf algebras over commutative rings. In this paper we lay the basis of the theory of rational modules of corings extending results on rational modules for coalgebras to the case of arbitrary ground rings. We apply these results mainly to categories of entwined modules (e.g., Doi-Koppinen modules, alternative Doi-Koppinen modules) generalizing results of Doi, Koppinen, Menini et al.

Block-form control of linear time-invariant discrete-time systems defined over commutative rings

In this paper we prove a sandwich classification theorem for subgroups of the classical symplectic group over an arbitrary commutative ring $R$ that contain the elementary block-diagonal (or subsystem) subgroup $\operatorname{Ep}(ν, R)$ corresponding to a unitary equivalence realation $ν$ such that all self-conjugate equivalence classes of $ν$ are of size at least 4 and all not-self-conjugate classes of $ν$ are of size at least 5. Namely, given a subgroup $H$ of $\operatorname{Sp}(2n, R)$ such that $\operatorname{Ep}(ν, R) \le H$ we show that there exists a unique exact major form net of ideals $(σ, Γ)$ over $R$ such that $\operatorname{Ep}(σ, Γ) \le H \le N_{\operatorname{Sp}(2n,R)}(\operatorname{Sp}(σ, Γ))$. Further, we describe the normalizer $N_{\operatorname{Sp}(2n,R)}(\operatorname{Sp}(σ, Γ))$ in terms of congruences.

It was shown by C. Wickham [12] that “for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g.” In this note, we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham's theorem to arbitrary (not necessary finite) commutative rings.

This is a survey of results describing the parabolic subgroups of Chevalley groups over rings of various types. For Chevalley groups of classical types over an arbitrary commutative ring, a description of parabolic subgroups is obtained which may be considered definitive. Some errors in the results of V. M. Levchuk in Mat. Zametki, Volume 31, No. 4 (1982) are pointed out.

This paper is a continuation of RZhMat 1980, 5A439, where there was introduced the subgroup γ(δ) of the Chevalley group G(Φ,R) of type Φ over a commutative ring R that corresponds to a net δ, i.e., to a set б=(б∝),∝∈Φ, of ideals б∝ of R such that б∝бβ⊑б∝+β whenever α,Β,α+Β ∃Φ. It is proved that if the ring R is semilocal, then Γ(б) coincides with the group γ0δ considered earlier in RZhMat 1976, 10A151; 1977, 10A301; 1978, 6A476. For this purpose there is constructed a decomposition of γ(δ) into a product of unipotent subgroups and a torus. Analogous results are obtained for sub-radical nets over an arbitrary commutative ring.

Let M be a nonsingular quadratic module over a commutative ring R. The primary goal of this chapter is the proof of the following fact: If M is free of finite rank, then C(M) has a special element, or, equivalently, the Arf algebra A(M) of M is isomorphic to R[X](X2 - aX - b) for some a and b in R with a2 + 4b ∈ R*. This has the important consequence that the results developed in Chapters 7 and 8 for C(M) hold for any free M. In addition, we will establish that the discriminant a2 + 4b of X2 - aX - b is the classical discriminant of the quadratic module M (except for a factor 2 if rank M is odd). It was established in Chapter 9D that the Arf algebra A(M) is a separable quadratic algebra. The structural features of this algebra will play a decisive role in the proofs. Throughout, R is an arbitrary commutative ring.

Polynomial representations of general linear groups and modules over Schur algebras are compared. We work over an arbitrary commutative ring and show that Schur–Weyl duality is the key for an equivalence between both categories.

Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications. As they are also of paramount importance in Algebraic Geometry, polynomial rings have been intensively studied. On the other hand, rings of formal power series have been extensively used in “algebroid geometry” and have many properties which are parallel to those of polynomial rings. In the first section of this chapter we shall define formal power series rings and we shall show that the main properties of polynomial rings which have been derived in previous chapters (see, in particular, Vol. I, Ch. I, §§ 16–18) hold also for formal power series rings. In the later sections of this chapter we shall give deeper properties of polynomial rings and, whenever possible, the parallel properties of power series rings.

Let R be a commutative ring with identity and $${\mathbb {N}}_0$$ be the additive monoid of nonnegative integers. We say that a function $$t : {\mathbb {N}}_0 \times {\mathbb {N}}_0 \rightarrow R$$ is a twist function on R if t satisfies the following three properties for all $$n, m, q \in {\mathbb {N}}_0$$ : (i) $$t(0,q) = 1$$ , (ii) $$t(n,m) = t(m,n)$$ , and (iii) $$t(n,m) \cdot t(n + m, q) = t (n, m + q) \cdot t(m, q)$$ . Let $$R[\![X]\!]$$ (resp., R[X]) be the set of power series (resp., polynomials) with coefficients in R. For $$f = \sum _{n=0}^{\infty } a_nX^n$$ and $$g = \sum _{n=0}^{\infty } b_nX^n \in R[\![X]\!]$$ , let $$f+g = \sum _{n=0}^{\infty } (a_n+b_n)X^n$$ , $$f*_tg = \sum _{n=0}^{\infty }(\sum _{i+j = n}t(i,j)a_ib_j)X^n$$ . Then, $$R^t[\![X]\!]:= (R[\![X]\!], +, *_t)$$ and $$R^t[X] := (R[X], +, *_t)$$ are commutative rings with identity that contain R as a subring. In this paper, we study ring-theoretic properties of $$R^t[\![X]\!]$$ and $$R^t[X]$$ with focus on divisibility properties including UFDs and GCD-domains. We also show how these two rings are related to the usual power series and polynomial rings.

Due to the theoretical work and computer implementations of, for instance, Barakat, Quadrat and Robertz and their coauthors the theory of finitely generated (f.g.) modules over non-commutative regular noetherian rings of partial differential operators with variable coefficients like the Weyl algebras and over other similar rings has become constructive in recent years. In particular these authors compute the purity or grade filtration of a f.g. module by homological means and discuss its significance for the associated behavior. Pommaret and Quadrat noted this significance already in 1999. In this note it is shown that over an arbitrary commutative noetherian ring of operators the purity filtration of a finitely generated module can be easily computed by means of the the primary decomposition of its zero submodule and indeed, with smaller complexity, by inductively computing equidimensional parts. In most books on Constructive Commutative Algebra the connection between this primary decomposition, the equidimensional parts and the purity filtration is implicitly stated for cyclic modules. For many commutative rings of operators the standard signal modules are injective cogenerators. In this case the purity filtration of the module gives rise to a corresponding filtration of the dual behavior, and the primary decomposition induces additional sum representations of the pure dimensional factors of this filtration. For non-commutative rings of operators the standard signal modules are in general neither injective nor cogenerators, and for such signal modules the usefulness of the purity filtration of the module for the determination of the behavior and its structural properties is not obvious. It is also shown by a counter-example that dimensional purity of the module or behavior does not imply dimensional purity of the initial conditions according to Riquier of the associated homogeneous Cauchy problem.

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring k. One considers a split reductive group scheme G acting on a k-algebra A and leaving invariant a subalgebra R. Let U be the unipotent radical of a split Borel subgroup scheme. If R U = A U then the conclusion is that A is integral over R.