Homological domino effects and the first Finitistic Dimension Conjecture

Summary Counterexamples to the first Finitistic Dimension Conjecture are constructed. In fact, for any fieldk and any integerm2, there exist finite dimensionalk-algebras such that the little finitistic dimension of ism, while the big finitistic dimension ism+1. Our examples are monomial relation algebras; within this class of algebras the big and little finitistic dimensions cannot differ by more than 1. The analysis of the examples is based on a sharp picture of arbitrary second syzygies over monomial relation algebras.Oblatum 5-VI-1991 & 28-X-1991This research was partially supported by a grant from the National Science Foundation. It is dedicated to Maurice Auslander on the occasion of his sixtyfifth birthday.     

关于有限维数猜想的一些新进展

在Artin代数的表示理论中,有一个著名的有限维数猜想：任意给定一个Artin代数,它的有限维数都是有限的．这个猜想已有45年的历史,至今悬而未决．本文主要综述它的一些历史发展情况,并介绍关于有限维数猜想的一些最新进展．     

关于对偶扩张代数有限维数的注记

没A是一个有限维代数，R为A的对偶扩张代数．本我们讨论R的有限维数findim R of R，证明了，在—般情况下findim R≠2findim A，这就回答了惠昌常教授所提的一个问题．     

Contravariantly Finite Subcategories and Torsion Theories

Contravariantly finite subcategories have been useful in differentaspects of representation theory (Auslander and Reiten, 1989, 1991;Auslander and Smalø, 1981) and they appear very naturally, forexample, the torsion class of a torsion theory is contravariantly finite. Inthis paper we explore further relations between contravariantly finite,resolving, subcategories and torsion theories. We study these connections inthe category of functors that vanishes on projectives. Our resultsgeneralize some theorems we obtained previously in our interpretation ofNakayamas conjecture (Martínez-Villa, 1994).     

Perverse Coherent t-Structures Through Torsion Theories

Bezrukavnikov, later together with Arinkin, recovered Deligne’s work defining perverse t-structures in the derived category of coherent sheaves on a projective scheme. We prove that these t-structures can be obtained through tilting with respect to torsion theories, as in the work of Happel, Reiten and Smalø. This approach allows us to define, in the quasi-coherent setting, similar perverse t-structures for certain noncommutative projective planes.     

Finitistic dimension and restricted injective dimension

We study the relations between finitistic dimensions and restricted injective dimensions. Let R be a ring and T a left R-module with A = End _R T. If _R T is selforthogonal, then we show that rid(T _A ) ? findim(A _A ) ? findim( _R T) + rid(T _A ). Moreover, if R is a left noetherian ring and T is a finitely generated left R-module with finite injective dimension, then rid(T _A ) ? findim(A _A ) ? fin.inj.dim( _R R)+rid(T _A ). Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.     

Finitistic dimension and Igusa–Todorov algebras

The notion of Igusa–Todorov algebras is introduced in connection with the (little) finitistic dimension conjecture, and the conjecture is proved for those algebras. Such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds, e.g., algebras with the representation dimension at most 3, algebras with radical cube zero, monomial algebras and left serial algebras, etc. It is an open question whether all artin algebras are Igusa–Todorov. We provide some methods to construct many new classes of (2-)Igusa–Todorov algebras and thus obtain many algebras such that the finitistic dimension conjecture holds. In particular, we show that the class of 2-Igusa–Todorov algebras is closed under taking endomorphism algebras of projective modules. Hence, if all quasi-hereditary algebras are 2-Igusa–Todorov, then all artin algebras are 2-Igusa–Todorov by [V. Dlab, C.M. Ringel, Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring, Proc. Amer. Math. Soc. 107 (1) (1989) 1–5] and have finite finitistic dimension.     

Torsion Modules, Lattices and P-Points

Answering a long-standing question in the theory of torsionmodules, we show that weakly productively bounded domains arenecessarily productively bounded. (See the Introduction fordefinitions.) Moreover, we prove a twin result for the ideallattice L of a domain equating weak and strong global intersectionconditions for families (X_i)_iI of subsets of L with the propertythat _iI A_i 0 whenever A_iX_i. Finally, we show that for domainswith Krull dimension (and countably generated extensions thereof),these lattice-theoretic conditions are equivalent to productiveboundedness. 1991 Mathematics Subject Classification 03E05,06A23, 13C12, 16U20, 16P60.     

Hook Lengths and 3-Cores

Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured that a(n) = 0 if and only if b(n) = 0, where integers a(n) and b(n) are defined by

Recurrence, Dimension and Entropy

Let (_A, T) be a topologically mixing subshift of finite typeon an alphabet consisting of m symbols and let :_A R^d be a continuousfunction. Denote by (x) the ergodic limit when the limit exists. Possible ergodic limits arejust mean values dµ for all T-invariant measures. Forany possible ergodic limit , the following variational formulais proved: where h_µ denotes the entropy of µ and h_top denotestopological entropy. It is also proved that unless all pointshave the same ergodic limit, then the set of points whose ergodiclimit does not exist has the same topological entropy as thewhole space _A     

Lengths on semigroups and groups

We give a characterization of those semigroups with topology arising from a collection of left sub-invariant pseudometrics or quasimetrics. We also characterize those with topology arising from sub-invariant pseudometrics or quasimetrics.     

两跳蚤移行路径长度比较

讨论有关文献在谈到曲线弧长公式时所提出的一个关于两跳蚤移行路径长度的问题.通过理论推导并借助图形分析显示,只是在一些离散的特殊时刻,两跳蚤移行的摆线长度才相等.     

Chain Lengths in the Tamari Lattice

We find the size of the largest union of two or three chains in the Tamari lattice.AMS Subject Classification: 06A07.     

Fractal Percolation,Porosity, and Dimension

We study the porosity properties of fractal percolation sets \(E\subset \mathbb {R}^d\). Among other things, for all \(0<\varepsilon <\tfrac{1}{2}\), we obtain dimension bounds for the set of exceptional points where the upper porosity of E is less than \(\tfrac{1}{2}-\varepsilon \), or the lower porosity is larger than \(\varepsilon \). Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton–Watson process.     

Zero-Divisors,Torsion Elements,and Unions of Annihilators

Let R be a commutative ring and M be a nonzero R-module. Now Z(M) = {r ∈ R | rm = 0 for some 0 ≠ m ∈ M} is a union of prime ideals of R and T(M) = {m ∈ M | rm = 0 for some 0 ≠ r ∈ R} is a union of prime submodules of M if M ≠ T(M). We investigate representations of Z(M) and T(M) as unions of primes each of which is a union of annihilators.     

Finitistic Dimensions of Artin Algebras with Two Simples and a Conjecture of Marczinzik

Let Λ be an Artin algebra with a unique non-injective indecomposable projective module. In this situation, Marczinzik conjectured that the dominant dimension of Λ agrees with its finitistic dimension. In this paper, we give a proof of a stronger statement. As a byproduct, we obtain excellent control over the finitistic dimensions of Artin algebras with two simples and positive dominant dimension, and also establish the Gorenstein symmetry conjecture for all algebras under consideration.

     

Dimension,Graph and Hypergraph Coloring

There is a natural way to associate with a poset P a hypergraph H _P, called the hypergraph of incomparable pairs, so that the dimension of P is the chromatic number of H _P. The ordinary graph G _P of incomparable pairs determined by the edges in H _P of size 2 can have chromatic number substantially less than H _P. We give a new proof of the fact that the dimension of P is 2 if and only if G _P is bipartite. We also show that for each t 2, there exists a poset P _t for which the chromatic number of the graph of incomparable pairs of P _t is at most 3 t – 4, but the dimension of P _t is at least (3 / 2) ^{t – 1}. However, it is not known whether there is a function f: NN so that if P is a poset and the graph of incomparable pairs has chromatic number at most t, then the dimension of P is at most f(t).     

New Extremal Type I Codes of Lengths 40, 42, and 44

It is known that it is possible to construct a generator matrix for a self-dual code of length 2n+2 from a generator matrix of a self-dual code of length 2n. With the aid of a computer, we construct new extremal Type I codes of lengths 40, 42, and 44 from extremal self-dual codes of lengths 38, 40, and 42 respectively. Among them are seven extremal Type I codes of length 44 whose weight enumerator is 1+224y ⁸+872y ¹⁰+·. A Type I code of length 44 with this weight enumerator was not known to exist previously.