Let $K$ be a field, $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be a standard bigraded polynomial ring ... more Let $K$ be a field, $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be a standard bigraded polynomial ring and $M$ a finitely generated bigraded $S$-module. In this paper we study sequentially Cohen--Macaulayness of $M$ with respect to $Q=(y_1,\ldots,y_n)$. We characterize the sequentially Cohen--Macaulayness of $L\tensor_KN$ with respect to $Q$ as an $S$-module when $L$ and $N$ are non-zero finitely generated graded modules over $K[x_1, \dots, x_m]$ and $K[y_1, \dots, y_n]$, respectively. All hypersurface rings that are sequentially Cohen--Macaulay with respect to $Q$ are classified.
Journal of Algebra and Its Applications, Sep 7, 2015
Let S = K[x1,…,xm, y1,…,yn] be the standard bigraded polynomial ring over a field K, and M a fini... more Let S = K[x1,…,xm, y1,…,yn] be the standard bigraded polynomial ring over a field K, and M a finitely generated bigraded S-module. In this paper we study the generalized Cohen–Macaulayness and sequentially generalized Cohen–Macaulayness of M with respect to Q = (y1,…,yn). We prove that if I ⊆ S be a monomial ideal with cd (Q, S/I) ≤ 2, then S/I is sequentially generalized Cohen–Macaulay with respect to Q.
Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text... more Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text], and [Formula: see text] a finitely generated [Formula: see text]-module. We consider the idealization [Formula: see text] of [Formula: see text] over [Formula: see text]. The goal of this paper is to investigate algebraic properties of [Formula: see text] that are related to those of [Formula: see text] and [Formula: see text]. Specifically, we provide characterizations for the Cohen–Macaulayness, sequentially Cohen–Macaulayness, generalized Cohen–Macaulayness, and graded maximal depth property of [Formula: see text] with respect to [Formula: see text], in terms of the corresponding properties for [Formula: see text] and [Formula: see text] with respect to [Formula: see text].
Let $K$ be a field, $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be a standard bigraded polynomial ring ... more Let $K$ be a field, $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be a standard bigraded polynomial ring and $M$ a finitely generated bigraded $S$-module. In this paper we study sequentially Cohen--Macaulayness of $M$ with respect to $Q=(y_1,\ldots,y_n)$. We characterize the sequentially Cohen--Macaulayness of $L\tensor_KN$ with respect to $Q$ as an $S$-module when $L$ and $N$ are non-zero finitely generated graded modules over $K[x_1, \dots, x_m]$ and $K[y_1, \dots, y_n]$, respectively. All hypersurface rings that are sequentially Cohen--Macaulay with respect to $Q$ are classified.
Journal of Algebra and Its Applications, Sep 7, 2015
Let S = K[x1,…,xm, y1,…,yn] be the standard bigraded polynomial ring over a field K, and M a fini... more Let S = K[x1,…,xm, y1,…,yn] be the standard bigraded polynomial ring over a field K, and M a finitely generated bigraded S-module. In this paper we study the generalized Cohen–Macaulayness and sequentially generalized Cohen–Macaulayness of M with respect to Q = (y1,…,yn). We prove that if I ⊆ S be a monomial ideal with cd (Q, S/I) ≤ 2, then S/I is sequentially generalized Cohen–Macaulay with respect to Q.
Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text... more Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text], and [Formula: see text] a finitely generated [Formula: see text]-module. We consider the idealization [Formula: see text] of [Formula: see text] over [Formula: see text]. The goal of this paper is to investigate algebraic properties of [Formula: see text] that are related to those of [Formula: see text] and [Formula: see text]. Specifically, we provide characterizations for the Cohen–Macaulayness, sequentially Cohen–Macaulayness, generalized Cohen–Macaulayness, and graded maximal depth property of [Formula: see text] with respect to [Formula: see text], in terms of the corresponding properties for [Formula: see text] and [Formula: see text] with respect to [Formula: see text].
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