共查询到20条相似文献,搜索用时 31 毫秒
1.
Let (S, 𝔫) be an s-dimensional regular local ring with s > 2, and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. As in [2, 3], we examine the leading form ideal I* of I in the associated graded ring G: = gr𝔫(S). Let μ G (I*) = n ≥ 3, and let {ξ1, ξ2,…, ξ n } be a minimal homogeneous system of generators of I* such that ξ1 = f* and ξ2 = g*, and c i : = deg ξ i ≤ deg ξ i+1: = c i+1 for each i ≤ n ? 1. For m ≤ n, we say that K m : = (ξ1,…, ξ m )G is an ideal generated by part of a minimal homogeneous generating set of I*. Let D i : = GCD(ξ1,…, ξ i ) and d i = deg D i for i with 1 ≤ i ≤ m. Let K m be perfect with ht G K m = 2. We prove that the following are equivalent: 1. deg ξ i+1 = deg ξ i + (d i?1 ? d i ) +1, for all i with 3 ≤ i ≤ m ? 1; 2. deg ξ i+1 ≤ deg ξ i + (d i?1 ? d i ) +1, for all i with 3 ≤ i ≤ m ? 1. Furthermore, if these equivalent conditions hold, then K m = I*. Moreover, if e(G/K m ) = e(G/I*), we prove that K m = I*. We illustrate with several examples in the cases where K m is or is not perfect. 相似文献
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It is well known that given an associative algebra or a Lie algebra A, its codimension sequence c n (A) is either polynomially bounded or grows at least as fast as 2 n . In [2] we proved that for a finite dimensional (in general nonassociative) algebra A, dim A = d, the sequence c n (A) is also polynomially bounded or c n (A) ≥ a n asymptotically, for some real number a > 1 which might be less than 2. Nevertheless, for d = 2, we may take a = 2. Here we prove that for d = 3 the same conclusion holds. We also construct a five-dimensional algebra A with c n (A) < 2 n . 相似文献
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Louis J. Ratliff Jr. 《代数通讯》2013,41(3):1058-1073
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Yong-Su Shin 《代数通讯》2013,41(6):2226-2242
We prove that a star-configuration 𝕏 in ?2 is defined by general forms of degrees ≤2 if and only if 𝕏 has generic Hilbert function. We also show that if 𝕏 and 𝕐 are star-configurations in ?2 defined by general forms of degrees ≤2 and σ(𝕏) ≠ σ(𝕐), then the ring R/(I 𝕏 + I 𝕐) has the Weak Lefschetz property. These two results generalize results of Ahn and Shin [3]. Furthermore, we find the Lefschetz element of the graded Artinian ring R/(I 𝕏 + I ?) precisely when 𝕏 and ? are two star-configurations in ?2 defined by general forms F 1,…, F s , and G 1,…, G s , L, respectively, with deg F i = deg G i = 2 for every i ≥ 1, and deg L = 1 with s ≥ 3. 相似文献
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《代数通讯》2013,41(7):3529-3546
Abstract For an ideal I of a Noetherian local ring (R, m ) we consider properties of I and its powers as reflected in the fiber cone F(I) of I. In particular,we examine behavior of the fiber cone under homomorphic image R → R/J = R′ as related to analytic spread and generators for the kernel of the induced map on fiber cones ψ J : F R (I) → F R′(IR′). We consider the structure of fiber cones F(I) for which ker ψ J ≠ 0 for each nonzero ideal J of R. If dim F(I) = d > 0,μ(I) = d + 1 and there exists a minimal reduction J of I generated by a regular sequence,we prove that if grade(G +(I)) ≥ d ? 1,then F(I) is Cohen-Macaulay and thus a hypersurface. 相似文献
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Marilena Crupi 《代数通讯》2013,41(8):2386-2408
Let ? be the family of graded ideals J in the exterior algebra E of a n-dimensional vector space over a field K such that e(E/J) = dim K (E/J) = e, indeg(E/J) = i and H E/J (i) = dim K (E/J) i are fixed integers. It is shown that there exists a unique lexsegment graded ideal J(n, e, i) ? ? whose Betti numbers give an upper bound for the Betti numbers of the ideals of ?. The authors continue the computation of upper bounds for the Betti numbers of graded ideals with given data started in Crupi and Utano (1999). 相似文献
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Wolfgang Müller 《Monatshefte für Mathematik》2011,162(1):69-88
Denote by 0 = λ 0 < λ 1 ≤ λ 2 ≤ . . . the infinite sequence given by the values of a positive definite irrational quadratic form in k variables at integer points. For l ≥ 2 and an (l ?1)-dimensional interval I = I 2×. . .×I l we consider the l-level correlation function \({K^{(l)}_I(R)}\) which counts the number of tuples (i 1, . . . , i l ) such that \({\lambda_{i_1},\ldots,\lambda_{i_l}\leq R^2}\) and \({\lambda_{i_{j}}-\lambda_{i_{1}}\in I_j}\) for 2 ≤ j ≤ l. We study the asymptotic behavior of \({K^{(l)}_I(R)}\) as R tends to infinity. If k ≥ 4 we prove \({K^{(l)}_I(R)\sim c_l(Q)\,{\rm vol}(I)R^{lk-2(l-1)}}\) for arbitrary l, where c l (Q) is an explicitly determined constant. This remains true for k = 3 under the restriction l ≤ 3. 相似文献
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Lingling Fan 《代数通讯》2013,41(3):799-806
Let R be an associative ring with identity. An element a ∈ R is called strongly clean if a = e + u with e 2 = e ∈ R, u a unit of R, and eu = ue. A ring R is called strongly clean if every element of R is strongly clean. Strongly clean rings were introduced by Nicholson [7]. It is unknown yet when a matrix ring over a strongly clean ring is strongly clean. Several articles discussed this topic when R is local or strongly π-regular. In this note, necessary conditions for the matrix ring 𝕄 n (R) (n > 1) over an arbitrary ring R to be strongly clean are given, and the strongly clean property of 𝕄2(RC 2) over the group ring RC 2 with R local is obtained. 相似文献
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For a square-free monomial ideal I ? S = k[x 1, x 2,…, x n ], we introduce the notion of quasi-linear quotients. By using the quasi-linear quotients, we give a new algebraic criterion for the shellability of a pure simplicial complex Δ over [n]. Also, we provide a new criterion for the Cohen–Macaulayness of the face ring of a pure simplicial complex Δ. Moreover, we show that the face ring of the spanning simplicial complex (defined in [2]) of an r-cyclic graph is Cohen–Macaulay. 相似文献
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Jiaqun Wei 《代数通讯》2013,41(7):2456-2465
Let R be an exchange ring. In this article, we show that the following conditions are equivalent: (1) R has stable range not more than n; (2) whenever x ∈ R n is regular, there exists some unimodular regular w ∈ n R such that x = xwx; (3) whenever x ∈ R n is regular, there exist some idempotent e ∈ R and some unimodular regular w ∈ R n such that x = ew; (4) whenever x ∈ R n is regular, there exist some idempotent e ∈ M n (R) and some unimodular regular w ∈ R n such that x = we; (5) whenever a( n R) + bR = dR with a ∈ R n and b,d ∈ R, there exist some z ∈ R n and some unimodular regular w ∈ R n such that a + bz = dw; (6) whenever x = xyx with x ∈ R n and y ∈ n R, there exist some u ∈ R n and v ∈ n R such that vxyu = yx and uv = 1. These, by replacing unimodularity with unimodular regularity, generalize the corresponding results of Canfell (1995, Theorem 2.9), Chen (Chen 2000, Theorem 4.2 and Proposition 4.6, Chen 2001, Theorem 10), and Wu and Xu (1997, Theorem 9), etc. 相似文献
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Pieter C. Allaart 《随机分析与应用》2013,31(3):531-554
Abstract Let X 1, X 2,… be any sequence of [0,1]-valued random variables. A complete comparison is made between the expected maximum E(max j≤n Y j ) and the stop rule supremum sup t E Y t for two types of discounted sequences: (i) Y j = b j X j , where {b j } is a nonincreasing sequence of positive numbers with b 1 = 1; and (ii) Y j = B 1… B j?1 X j , where B 1, B 2,… are independent [0,1]-valued random variables that are independent of the X j , having a common mean β. For instance, it is shown that the set of points {(x, y): x = sup t E Y {(x, y): x=sup t E Y and y = E(max j≤n Y j ), for some sequence X 1,…,X n and Y j = b j X j }, is precisely the convex closure of the union of the sets {(b j x, b j y): (x, y) ∈ C j }, j = 1,…,n, where C j = {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ x[1 + (j ? 1)(1 ? x 1/(j?1))]} is the prophet region for undiscounted random variables given by Hill and Kertz [8]. As a special case, it is shown that the maximum possible difference E(max j≤n β j?1 X j ) ? sup t E(β t?1 X t ) is attained by independent random variables when β ≤ 27/32, but by a martingale-like sequence when β > 27/32. Prophet regions for infinite sequences are given also. 相似文献
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《代数通讯》2013,41(5):2381-2401
Abstract Let 𝒪 be a discrete valuation ring whose residue field 𝒪/𝔭 is finite and has odd characteristic. Let l be a positive integer. Set R = 𝒪/𝔭 l and let R = R[θ] be the ring obtained by adjoining to R a square root of a non-square unit. Consider the involution σ of R that fixes R elementwise and sends θ to ? θ. Let V be a free R-module of rank n > 0 endowed with a non-degenerate hermitian form ( , ) relative to σ. Let U n (R) be the subgroup of GL(V) that preserves ( , ). Let SU n (R) be the subgroup of all g ∈ U n (R) whose determinant is equal to one. Let Ψ be the Weil character of U n (R). All irreducible constituents of Ψ are determined. An explicit character formula is given for each of them. In particular, all character degrees are computed. For n > 2 the corresponding results are also obtained for the restriction of Ψ to SU n (R). 相似文献
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Let Λ be an Artin algebra over a commutative Artinian ring, k. If M is a finitely generated left Λ -module, we denote by Ω (M) the kernel of η M : P M → M a minimal projective cover. We prove that if M and N are finitely generated left Λ -modules and Ext Λ 1 (M, M) = 0, Ext Λ 1 (N, N) = 0, then M? N if and only if M/rad M? N/rad N and Ω (M)? Ω (N). Now if k is an algebraically closed field and (d i ) i?? is a sequence of nonnegative integers almost all of them zero, then we prove that the family of objects X ? b (Λ), the bounded derived category of Λ, with Hom b (Λ)(X,X[1]) = 0 and dim k H i (X) = d i for all i ? ?, has only a finite number of isomorphism classes (see Huisgen-Zimmermann and Saorín, 2001). 相似文献