Quadratic forms of projective spaces over rings

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules—unmixed—is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.

For an element $x$ of a commutative ring $R$, let $\Gamma_x(R)$ be the graph whose vertices are the elements of $R$ that divide $x$ such that distinct vertices $r$ and $s$ are adjacent if and only if $rs=x$. If $x$ is a nonzero nonirreducible nonunit of an integral domain $R$, then $\Gamma_x^\mathcal{C}(R)$ is the graph whose vertices are the associate classes of divisors of $x$ that are neither units nor associates of $x$ such that distinct vertices $A$ and $B$ are adjacent if and only if $rs$ divides $x$ for some (and hence every) $r\in A$ and $s\in B$. The graphs $\Gamma_x(R)$ are considered when $R$ is a local Artinian principal ideal ring, and $\Gamma_x^\mathcal{C}(R)$ is examined when $R$ is a valuation domain. For example, it is shown that a finite local ring $R$ is a principal ideal ring if and only if there exists $x\in R$ such that every connected component of $\Gamma_x(R)$ is a star graph of a prescribed cardinality. Moreover, it is proved that an integral domain $R$ is a discrete valuation ring if and only if its collection of graphs $\Gamma_x^\mathcal{C}(R)$ consists precisely of a single-vertex graph, along with every (up to isomorphism) graph that is realizable as the compressed $0$-divisor graph of a local Artinian principal ideal ring. Certain graphs associated with partially ordered abelian groups have an essential role in the work.

One-dimensional rings of finite Cohen-Macaulay type

We introduce the notion of hyperbolic equivalence for quadric bundles and quadratic forms on vector bundles and show that hyperbolic equivalent quadric bundles share many important properties: they have the same Brauer data; moreover, if they have the same dimension over the base, they are birational over the base and have equal classes in the Grothendieck ring of varieties. Furthermore, when the base is a projective space we show that two quadratic forms are hyperbolic equivalent if and only if their cokernel sheaves are isomorphic up to twist, their fibers over a fixed point of the base are Witt equivalent, and, in some cases, certain quadratic forms on intermediate cohomology groups of the underlying vector bundles are Witt equivalent. For this we show that any quadratic form over $\mathbb{P}^n$ is hyperbolic equivalent to a quadratic form whose underlying vector bundle has many cohomology vanishings; this class of bundles, called VLC bundles in the paper, is interesting by itself.

13 - Matrices and related results

Hermitian forms and locally toroidal regular polytopes

LetQ i, i=1, …,t, be real nondegenerate indefinite quadratic forms ind variables. We investigate under what conditions the closure of the set $$\left\{ {\left( {Q_1 \left( {\bar x} \right),...,Q_t \left( {\bar x} \right)} \right):\bar x \in \mathbb{Z}^d - \left\{ {\bar 0} \right\}} \right\}$$ contains (0, …, 0). As a corollary, we deduce several results on the magnitude of the set Δ ofg ∈ GL(d, ℝ) such that the closure of the set $$\left\{ {\left( {Q_1 \left( {g\bar x} \right),...,Q_t \left( {g\bar x} \right)} \right):\bar x \in \mathbb{Z}^d - \left\{ {\bar 0} \right\}} \right\}$$ contains (0, …, 0). Special cases are described when, depending on the mutual position of the hypersurfaces {Q i=0},i=1, …,t, the set Δ has full Haar measure or measure zero and Hausdorff dimensiond 2−(d−2)/2.

We study the parameters of codes spanned by quadratic and Hermitian forms in projective space. The resulting codes have high minimum distance and few distinct weights. An example is a new ternary [91.9,54] code.

In this paper, we considered linear block codes over Rq=Fq+uFq+vFq+uvFq, u2=v2=0,uv=vu where q=pm, m∈N . First we looked at the structure of the ring. It was shown that Rq is neither a finite chain ring nor a principal ideal ring but is a local ring. We then established a generator matrix for the linear block codes and equipped it with a homogeneous weight function. Field codes were then constructed as images of these codes by using a basis of Rq over Fq . Bounds on the minimum Hamming distance of the image codes were then derived. A code meeting such bounds is given as an example.

For a commutative local ring R, we know that every ideal is a direct sum of local modules if and only if the unique maximal ideal of R is of the form , where and are principal ideal rings and for all γ’s. This motivates us to determine a commutative ring, not necessarily local, in which every (proper) ideal is a direct sum of local modules. We prove that such a ring is exactly a finite direct product of local rings R with the unique maximal ideal in the above mentioned form. Moreover, we characterize more precisely certain commutative rings such as Noetherian rings, reduced rings, Rickart rings, semi-Artinian rings and self-injective rings, in which every (proper) ideal is a direct sum of local modules.

Let M be a finitely generated module of rank r and finite projective dimension over a commutative Noetherian local ring S. We show that M has a free direct summand of rank r if and only if its Buchsbaum–Eisenbud invariant J(M) is a principal ideal. As an application, we consider the problem of finding rigid triples. Recall that the module M is said to be rigid if for each i ≥ 0 and each finitely generated S-module N the condition implies for every j ≥ i. A triple of positive integers b = (b 0, b 1, b 2) is said to be rigid if for every commutative Noetherian local ring S one has that every S-module with minimal free resolution of the form 0 → S b 2 → S b 1 → S b 0 → 0 is rigid. We use our criterion for free direct summands of maximal rank to derive a necessary and sufficient condition for a triple b to be rigid in terms of the rigidity of the universal modules of type b over complete regular local rings. In particular, we exhibit the rigidity of any triple of the form (k, m + 1,m), thus providing a new class of rigid modules.

There are close relations between Bernoulli numbers and quadratic fields or quadratic forms. In order to explain those, we give a survey of the ideal theory of quadratic fields and quadratic forms. Since Gauss , it is well known that there is a deep relation between the ideal theory of quadratic fields (i.e. quadratic extensions of the rational number field) and integral quadratic forms. This is obvious for specialists, but textbooks which explain this in detail are rare. In most textbooks, they treat the correspondence of ideals only in the case of maximal orders (the ring of all integers) for which the description is simple and easy. This is very disappointing. We cannot see the whole picture of the theory of quadratic forms by such a restricted treatment and sometimes it causes misunderstanding. We need the full description of this kind of relation when we explain later the relation between L-functions of prehomogeneous vector spaces and the Bernoulli numbers. So it would be a good chance to try to explain the full relation here.

The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions

Given a symmetric bilinear form F on a finite dimensional real vector space V we attempt to find ‘nice’ bases where the matrix representation of the form is as simple as possible, in the same sort of way as we did in the Gram-Schmidt orthogonalization process for inner products. The theorem which says that this is possible and which describes all the possible matrix representations one can get is called Sylvester’s law of inertia. (Sylvester’s law of inertia has nothing to do with inertia: the story goes that Sylvester said, ‘If Isaac Newton can have a law of inertia, then so can I’, and consequently gave this theorem the name it has been known by ever since.) This ‘law’ is exactly analogous to Theorem 2.33 which classified all finite dimensional vector spaces using dimension; it turns out that symmetric bilinear forms are classified by the dimension of the underlying vector space and two further numbers, called the rank and the signature of the form.

Let X,X 1,X 2,… be a sequence of i.i.d. random vectors taking values in a d-dimensional real linear space ℝ d . Assume that E X=0 and that X is not concentrated in a proper subspace of ℝ d . Let G denote a mean zero Gaussian random vector with the same covariance operator as that of X. We investigate the distributions of non-degenerate quadratic forms ℚ[S N ] of the normalized sums S N =N −1/2(X 1+⋯+X N ) and show that provided that d≥9 and the fourth moment of X exists. The bound ?(N −1) is optimal and improves, e.g., the well-known bound ?(N − d /( d +1)) due to Esseen (1945). The result extends to the case of random vectors taking values in a Hilbert space. Furthermore, we provide explicit bounds for Δ N and for the concentration function of the random variable ℚ[S N ].