Asymptotic behaviour of bigraded components of local cohomology modules
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| Format: | Preprint |
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2026
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| author | Bhattacharyya, Rajsekhar Puthenpurakal, Tony J. Roy, Sudeshna Singh, Jyoti |
| author_facet | Bhattacharyya, Rajsekhar Puthenpurakal, Tony J. Roy, Sudeshna Singh, Jyoti |
| contents | Let $C$ be a commutative Noetherian ring containing a field $K$ of characteristic zero. Let $R=C[X_1, \ldots, X_n, Y_1, \ldots, Y_m]$ be a polynomial ring over $C$ with $\mathrm{bideg}~ c=(0,0)$ for all $c \in C$, $\mathrm{bideg}~ X_i=(1,0)$ and $\mathrm{bideg}~ Y_j=(0,1)$ for $i=1, \ldots, n$ and $j=1, \ldots, m$. Let $I$ be a bihomogeneous ideal in $R$. In this article, we study asymptotic behaviour of bigraded pieces of the local cohomology module $H^i_I(R)$. Moreover, under the extra assumption that $C$ is regular, we investigate the asymptotic stability of invariants associated to its bigraded components. Consequently, we obtain certain properties of components of the bigraded local cohomology module $H^i_I(R)$, where $C=K$ is a field and $I$ is a binomial edge ideal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_21253 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic behaviour of bigraded components of local cohomology modules Bhattacharyya, Rajsekhar Puthenpurakal, Tony J. Roy, Sudeshna Singh, Jyoti Commutative Algebra Primary 13D45, 14B15, Secondary 13N10, 32C36 Let $C$ be a commutative Noetherian ring containing a field $K$ of characteristic zero. Let $R=C[X_1, \ldots, X_n, Y_1, \ldots, Y_m]$ be a polynomial ring over $C$ with $\mathrm{bideg}~ c=(0,0)$ for all $c \in C$, $\mathrm{bideg}~ X_i=(1,0)$ and $\mathrm{bideg}~ Y_j=(0,1)$ for $i=1, \ldots, n$ and $j=1, \ldots, m$. Let $I$ be a bihomogeneous ideal in $R$. In this article, we study asymptotic behaviour of bigraded pieces of the local cohomology module $H^i_I(R)$. Moreover, under the extra assumption that $C$ is regular, we investigate the asymptotic stability of invariants associated to its bigraded components. Consequently, we obtain certain properties of components of the bigraded local cohomology module $H^i_I(R)$, where $C=K$ is a field and $I$ is a binomial edge ideal. |
| title | Asymptotic behaviour of bigraded components of local cohomology modules |
| topic | Commutative Algebra Primary 13D45, 14B15, Secondary 13N10, 32C36 |
| url | https://arxiv.org/abs/2603.21253 |