Projective varieties with projectively equivalent singular 0-dimensional sections

Fix integers m, n such that 1 ≤ m ≤ n ? 3. Let X ? Pⁿ be an integral non-degenerate m-dimensional variety. Assume either char(K) = 0 or char(K) > deg(X). Here we prove that all general 0-dimensional sections of X containing a tangent vector to a smooth point of X are protectively equivalent if and only if n ? m + 1 ≤ deg(X) ≤ n ? m + 2.     

Construction of quotient BCI(BCK)-algebra via a fuzzy ideal

The present paper gives a new construction of a quotient BCI(BCK)-algebraX/μ by a fuzzy ideal μ inX and establishes the Fuzzy Homomorphism Fundamental Theorem. We show that if μ is a fuzzy ideal (closed fuzzy ideal) ofX, thenX/μ is a commutative (resp. positive implicative, implicative) BCK (BCI)-algebra if and only if μ is a fuzzy commutative (resp positive implicative, implicative) ideal ofX. Moreover we prove that a fuzzy ideal of a BCI-algebra is closed if and only if it is a fuzzy subalgebra ofX. We show that if the period of every element in a BCI-algebraX is finite, then any fuzzy ideal ofX is closed. Especially, in a well (resp, finite, associative, quasi-associative, simple) BCI-algebra, any fuzzy ideal must be closed.     

Which compacta are noncommutative ARs?

We give a short answer to the question in the title: dendrits. Precisely we show that the C^*-algebra C(X) of all complex-valued continuous functions on a compactum X is projective in the category C¹ of all (not necessarily commutative) unital C^*-algebras if and only if X is an absolute retract of dimension dimX?1 or, equivalently, that X is a dendrit.     

A family of cohen-macaulay modules over singularities of type xt + y3

Let K be a field with char K ≠ 3 and it two positive integers such that 1 ≤i <t/2,t ≠ 3i. The classification problem for maximal Cohen-Macaulay modules over K[[X,Y]]/(X^t+Y³ ) is complicated if t≥ 6, because there exist parameter families of non-isomorphic maximal Cohen-Macaulay modules [Sc], or [GK], [Yo, Ch.9] and [DG]). Here we describe parameter families of such modules N, such that N/YN is a direct sum of copies of K[[X]]/(X ⁱ)K[[X]]/(X^t-i ).     

Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces

Let K(n) be the nth Morava K-theory at a prime p, and let T(n) be the telescope of a v_n-self map of a finite complex of type n. In this paper we study the K(n)_*-homology of Ω^∞X, the 0th space of a spectrum X, and many related matters.We give a sampling of our results.Let PX be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map

s_n(X):L_T(n)P(X)→L_T(n)Σ^∞(Ω^∞X)₊     

Numerical Semigroup Rings and Almost Prüfer v-Multiplication Domains

Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ? ?₀, D[Γ] be the semigroup ring of Γ over D (and hence D ? D[Γ] ? D[X]), and D + X ⁿ K[X] = {a + X ⁿ g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X ⁿ K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X ⁿ K[X].     

Covering properties on X2⧹Δ, W-sets,and compact subsets of Σ-products

Let Δ ? X¹ be the diagonal. In the first part of this paper, we show that a compact space X is Corson compact (resp., Eberlein compact; compact metric) if and only if X²?Δ is metalindelöf (resp., σ-metacompact; paracompact). In the second part of the paper, we investigate the notion of a W-set in a space X, which is defined in terms of an infinite game. We show that a compact space X is Corson compact if and only if X has a W-set diagonal, and that a compact scattered space X is strong Eberlein compact if and only if each point of X is a W-set in X.     

Locally C1-equivalent algebras

It is shown that a Banach star algebra is a C¹-algebra in an equivalent norm, if each of its commutative closed star subalgebras is a C¹-algebra in an equivalent norm. This theorem has several interesting consequences.     

The variety of the asymptotic values of a real polynomial etale map

A polynomial map F: R² → R² is said to satisfy the Jacobian condition if ∀(X, Y)ϵ R², J(F)(X, Y) ≠ 0. The real Jacobian conjecture was the assertion that such a map is a global diffeomorphism. Recently the conjecture was shown to be false by S. Pinchuk. According to a theorem of J. Hadamard any counterexample to the conjecture must have asymptotic values. We give the structure of the variety of all the asymptotic values of a polynomial map F: R² → R² that satisfies the Jacobian condition. We prove that the study of the asymptotic values of such maps can be reduced to those maps that have only X- or Y-finite asymptotic values. We prove that a Y-finite asymptotic value can be realized by F along a rational curve of the type (X^{− k}, A₀ + A₁ X + … + A_{N − 1} X^{N − 1} + YX^N), where X → 0, Y is fixed and K, N > 0 are integers. More precisely we prove that the coordinate polynomials P(U, V) of F(U, V) satisfy finitely many asymptotic identities, namely, identities of the following type, P(X^{− k}, A₀ + A₁ X + … + A_{N − 1} X^{N − 1} + YX^N) = A(X, Y)ϵ R[X, Y], which ‘capture’ the whole set of asymptotic values of F.     

Strong NMV-algebras, commutative basic algebras and naBL-algebras

The aim of the paper is to investigate the relationship among NMV-algebras, commutative basic algebras and naBL-algebras (i.e., non-associative BL-algebras). First, we introduce the notion of strong NMV-algebra and prove that

1. a strong NMV-algebra is a residuated l-groupoid (i.e., a bounded integral commutative residuated lattice-ordered groupoid)
2. a residuated l-groupoid is commutative basic algebra if and only if it is a strong NMV-algebra.

Secondly, we introduce the notion of NMV-filter and prove that a residuated l-groupoid is a strong NMV-algebra (commutative basic algebra) if and only if its every filter is an NMV-filter. Finally, we introduce the notion of weak naBL-algebra, and show that any strong NMV-algebra (commutative basic algebra) is weak naBL-algebra and give some counterexamples.     

On Quadratic Integral Polynomials with only Finitely Many Roots in any Commutative Finite-Dimensional Algebra

The monic quadratic polynomials f with integer coefficients such that each commutative finite-dimensional algebra over a field contains only finitely many roots of f are determined as the polynomials of the form f = X ² + (2m + 1)X + m ² + m, where ${m \in \mathbb{Z}}$ .     

Lifting Defects for Nonstable K0-theory of Exchange Rings and C*-algebras

The assignment (nonstable K₀-theory), that to a ring R associates the monoid V(?R?) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor:

1. There is no functor Γ, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that V °?Γ?? id.
2. There is no functor Γ, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that V °?Γ?? id.
3. There is a {0,1}³-indexed commutative diagram  ${\vec{D}}$ of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0.

By using categorical tools (larders, lifters, CLL) from a recent book from the author with P. Gillibert, we deduce that there exists a unital exchange ring of cardinality  $\aleph_3$ (resp., an $\aleph_3$ -separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(?R?) is the positive cone of a dimension group but it is not isomorphic to V(?B?) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.     

Derivations and right ideals of algebras

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let ρ be a nonzero right ideal of R and let f(X₁,…,X_t) be a nonzero polynomial over K with constant term 0 such that μR≠0 for some coefficient μ of f(X₁,…,X_t). Suppose that d:R→R is a nonzero derivation. It is proved that if rankd(f(x₁,…,x_t))?m for all x₁,…,x_t∈ρ and for some positive integer m, then either ρ is generated by an idempotent of finite rank or d=ad(b) for some b∈End(V_D) of finite rank. In addition, if f(X₁,…,X_t) is multilinear, then b can be chosen such that rank(b)?2(6t+13)m+2.     

The Algebra of Flows in Graphs

We define a contravariant functorKfrom the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graphX, an abelian groupB, and a nonnegative integerj, an element of Hom(K^j(X), B) is a coherent family ofB-valued flows on the set of all graphs obtained by contracting some (j − 1)-set of edges ofX; in particular, Hom(K¹(X), ) is the familiar (real) “cycle-space” ofX. We show thatK ^· (X) is torsion-free and that its Poincaré polynomial is the specializationt^{n − k}T_X(1/t, 1 + t) of the Tutte polynomial ofX(hereXhasnvertices andkcomponents). Functoriality ofK ^· induces a functorial coalgebra structure onK ^· (X); dualizing, for any ringBwe obtain a functorialB-algebra structure on Hom(K ^· (X), B). WhenBis commutative we present this algebra as a quotient of a divided power algebra, leading to some interesting inequalities on the coefficients of the above Poincaré polynomial. We also provide a formula for the theta function of the lattice of integer-valued flows inX, and conclude with 10 open problems.     

On twin and anti-twin words in the support of the free Lie algebra

Let ${\mathcal{L}}_{K}(A)$ be the free Lie algebra on a finite alphabet A over a commutative ring K with unity. For a word u in the free monoid A ^? let $\tilde{u}$ denote its reversal. Two words in A ^? are called twin (resp. anti-twin) if they appear with equal (resp. opposite) coefficients in each Lie polynomial. Let l denote the left-normed Lie bracketing and ?? be its adjoint map with respect to the canonical scalar product on the free associative algebra K??A??. Studying the kernel of ?? and using several techniques from combinatorics on words and the shuffle algebra , we show that, when K is of characteristic zero, two words u and v of common length n that lie in the support of ${\mathcal{L}}_{K}(A)$ ??i.e., they are neither powers a ⁿ of letters a??A with exponent n>1 nor palindromes of even length??are twin (resp. anti-twin) if and only if u=v or $u = \tilde{v}$ and n is odd (resp. $u =\tilde{v}$ and n is even).     

On the property of kelley in the hyperspace and Whitney continua

In this paper, we introduce the notion of property [K]¹ which implies property [K], and we show the following: Let X be a continuum and let ω be any Whitney map for C(X). Then the following are equivalent. (1) X has property [K]¹. (2) C(X) has property [K]¹. (3) The Whitney continuum ω⁻¹(t) (0⩽t<ω(X)) has property [K]¹.As a corollary, we obtain that if a continuum X has property [K]¹, then C(X) has property [K] and each Whitney continuum in C(X) has property [K]. These are partial answers to Nadler's question and Wardle's question ([10, (16.37)] and [11, p. 295]).Also, we show that if each continuum X_n (n=1,2,3,…) has property [K]¹, then the product ∏X_n has property [K]¹, hence C(∏X_n) and each Whitney continuum have property [K]¹. It is known that there exists a curve X such that X has property [K], but X×X does not have property [K] (see [11]).     

Commutative Abelian Galois extensions of a Banach algebra

Let A be a commutative unital Banach algebra with connected maximal ideal space X. We show that the Gelfand transform induces an isomorphism between the group of commutative Galois extensions of A with given finite Abelian Galois group, and the corresponding group of extensions of C(X). This result is applied, when X is sufficiently nice, to construct a separable projective finitely generated faithful Banach A-algebra whose maximal ideal space is a given finitely fibered covering space of X.     

The central simple modules of Artinian Gorenstein algebras

Let A be a standard graded Artinian K-algebra, with char K=0. We prove the following.

1.

A has the Weak Lefschetz Property (resp. Strong Lefschetz Property) if and only if has the Weak Lefschetz Property (resp. Strong Lefschetz Property) for some linear form z of A.

2.

If A is Gorenstein, then A has the Strong Lefschetz Property if and only if there exists a linear form z of A such that all central simple modules of (A,z) have the Strong Lefschetz Property.

As an application of these theorems, we give some new classes of Artinian complete intersections with the Strong Lefschetz Property.     

Structure des algebres de valuation discrete

We define a topology τ_e, on anyC-algebra of discrete valuation, generalizing the topology of coefficientwise convergence on C[[X]] studied by G. R. Allan. We give a necessary and sufficient condition for τ_e to be complete and prove that the completion provides an algebra of discrete valuation. We also prove that if aC-algebra of discrete valuation is Fréchet andm-convex for τ_e then it is isomorphic to (C[[X]], τ_e) and then τ_e is the uniqueF-algebra topology in A. We prove that a commutative, unital Fréchet l.m.c.a. that is aC-algebra of valuation is in fact aC-algebra of discrete valuation and so is embeddable in (C[[X]], τ_e). Whence a result of H. Bouloussa.     

Smooth approximations

We prove, among other things, that a Lipschitz (or uniformly continuous) mapping f:X→Y can be approximated (even in a fine topology) by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a separable C(K) space (resp. super-reflexive space). Further, we show how smooth approximation of Lipschitz mappings is closely related to a smooth approximation of C¹-smooth mappings together with their first derivatives. As a corollary we obtain new results on smooth approximation of C¹-smooth mappings together with their first derivatives.