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In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the length of all such modules is bounded. [ABSTRACT FROM AUTHOR]

Let \mathcal {A} denote a central hyperplane arrangement of rank n in affine space \mathbb {K}^n over a field \mathbb {K} of characteristic zero and let l_1,\ldots, l_m\in R≔\mathbb {K}[x_1,\ldots,x_n] denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial f≔l_1\cdots l_m\in R. The focus of the paper is on the ideal J_f\subset R generated by the partial derivatives of f. We conjecture that J_f is a minimal reduction of the ideal \mathbb {I}\subset R generated by the (m-1)-fold products of distinct forms among l_1,\ldots, l_m. We prove this conjecture for an almost generic \mathcal {A} (i.e., any n-1 among the defining linear forms are linearly independent). In this case we obtain a stronger version of a result by Dimca and Papadima, and we confirm the conjecture unconditionally for n=3. We also conjecture that J_f is an ideal of linear type (i.e., the respective symmetric and Rees algebras coincide). We prove this conjecture for n=3. In the sequel we explain the tight relationship between the two ideals J_f, \mathbb {I}\subset R; in particular, we show that in the generic case (J_f)^{\text {sat}}=\mathbb I. As a consequence, we can provide a simpler proof of a conjectured result of Yuzvinsky, proved by Rose and Terao, on the vanishing of the depth of R/J_f. [ABSTRACT FROM AUTHOR]

Let B be an integral domain over a field K of characteristic 0. The derivation δ of B[Yd] = B[y1, . . . ,yd] is elementary if δ (B) = 0 and δ (yi) ∈ B, i = 1, . . . ,d. Then for d ≥ 2 the elements uij = δ (yi)yj-δ (yj)yi, 1 ≤ i < j ≤ d, belong to the algebra B[Yd]δ of constants of δ, and it is a natural question whether the B-algebra B[Yd]δ is generated by all uij. In this paper we consider the special case of B = K[Xd] = K[x1, . . . ,xd]. If δ (yi) = xi, i = 1, . . . ,d, this is the Nowicki conjecture from 1994, which was confirmed in several papers based on different methods. The case δ (yi) = xini, ni > 0, i = 1, . . . ,d, was handled by Khoury in the first proof of the Nowicki conjecture given by him in 2004. As a consequence of the proof of Kuroda in 2009, if δ (yi) = ƒi(xi), for any nonconstant polynomials ƒi(xi), i = 1, . . . ,d, then B[Yd]δ = K[Xd,Yd]δ is generated by Xd and Ud = uij = ƒi(xi)yj-yiƒj(xj) mid 1 ≤ i < j ≤ d. In the present paper we have found a presentation of the algebra K[Xd,Yd]δ = K[Xd,Ud mid R = S = 0],d ≥ 4, R = r(i,j,k,l)mid 1 ≥ i < j < k < l \≤ d, S = s(i,j,k)mid 1 ≤ i < j < k ≤ d, and a basis of K[Xd,Yd]δ as a vector space. As a corollary we have shown that the defining relations R ∩ S form the reduced Gröbner basis of the ideal which they generate with respect to a specific admissible order. This is an analogue of the result of Makar-Limanov and the author in their proof of the Nowicki conjecture in 2009. The algebras K[Xd,Yd]δ, d < 4, are also described. [ABSTRACT FROM AUTHOR]