A matrix of linear forms which is annihilated by a vector of indeterminates

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• 作者：Andrew R. Kustin^a ; ^{kustin@math.sc.edu" class="auth_mail" title="E-mail the corresponding author} ; Claudia Polini^b ; ^{cpolini@nd.edu" class="auth_mail" title="E-mail the corresponding author} ; Bernd Ulrich^c ; ^{ulrich@math.purdue.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 13D02 ; secondary ; 13C40 ; 13A30
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：469
• 期：Complete
• 页码：120-187
• 全文大小：884 K

文摘

Let R=k[T₁,…,T_f]$R=k[T_1,\dots ,T_{\mathfrak{f}}]$ be a standard graded polynomial ring over the field k   and Ψ be an f×g$\mathfrak{f}\times \mathfrak{g}$ matrix of linear forms from R  , where 1≤g<f$1\le \mathfrak{g}<\mathfrak{f}$. Assume $\left[\begin{array}{ccc}\hfill T_1\hfill & \hfill \cdots \hfill & \hfill T_{\mathfrak{f}}\hfill \end{array}\right]\mathrm{\Psi }$ is 0 and that $\mathrm{grade}{I}_{\mathfrak{g}}(\mathrm{\Psi })$ is exactly one short of the maximum possible grade. We resolve $\bar{R}=R/I_{\mathfrak{g}}(\mathrm{\Psi })$, prove that $\bar{R}$ has a g$\mathfrak{g}$-linear resolution, record explicit formulas for the h  -vector and multiplicity of $\bar{R}$, and prove that if f−g$\mathfrak{f}-\mathfrak{g}$ is even, then the ideal I_g(Ψ)$I_{\mathfrak{g}}(\mathrm{\Psi })$ is unmixed. Furthermore, if f−g$\mathfrak{f}-\mathfrak{g}$ is odd, then we identify an explicit generating set for the unmixed part, I_g(Ψ)^unm$I_{\mathfrak{g}}{(\mathrm{\Psi })}^{\text{unm}}$, of I_g(Ψ)$I_{\mathfrak{g}}(\mathrm{\Psi })$, resolve R/I_g(Ψ)^unm$R/I_{\mathfrak{g}}{(\mathrm{\Psi })}^{\text{unm}}$, and record explicit formulas for the h  -vector of R/I_g(Ψ)^unm$R/I_{\mathfrak{g}}{(\mathrm{\Psi })}^{\text{unm}}$. (The rings R/I_g(Ψ)$R/I_{\mathfrak{g}}(\mathrm{\Psi })$ and R/I_g(Ψ)^unm$R/I_{\mathfrak{g}}{(\mathrm{\Psi })}^{\text{unm}}$ automatically have the same multiplicity.) These results have applications to the study of the blow-up algebras associated to linearly presented grade three Gorenstein ideals.