A. V. Jayanthan, J. K. Verma
Formulas are obtained in terms of complete reductions for the bigraded components of local cohomology modules of bigraded Rees algebras of 0-dimensional ideals in 2-dimensional Cohen-Macaulay local rings. As a consequence, cohomological expressions for the coefficients of the Bhattacharya polynomial of such ideals are obtained.
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The Grothendieck-Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the Bhattacharya function and Bhattacharya polynomial of two m-primary ideals I and J in a local ring (A,m) in terms of local cohomology modules of Rees algebras of I and J. The cohomology of a variation of the Kirby-Mehran complex for bigraded Rees algebras is studied which is used to characterize the Cohen-Macaulay property of bigraded Rees algebra of I and J fo…
In this paper we consider multi-graded extended Rees algebras of zero dimensional ideals which are Cohen-Macaulay (CM) with minimal multiplicity. We show that the minimal multiplicity property can occur only for the ordinary extended Rees algebra and the bigraded extended Rees algebra. For the bigraded extended Rees algebra we find necessary conditions for it to be CM with minimal multiplicity. We also produce bigraded Rees algebras which are Cohen-Macaulay with minimal multiplicity.
Let $(R,\m)$ be an analytically unramified Cohen-Macaulay local ring of dimension 2 with infinite residue field and $\ov{I}$ be the integral closure of an ideal $I$ in $R$. Necessary and sufficient conditions are given for $\ov{I^{r+1}J^{s+1}}=a\ov{I^rJ^{s+1}}+b\ov{I^{r+1}J^s}$ to hold ${for all}r \geq r_0{and}s \geq s_0$ in terms of vanishing of $[H^2_{(at_1,bt_2)}(\ov{\mathcal{R}^\prime}(I,J))]_{(r_0,s_0)}$, where $a \in I,b \in J$ is a good joint reduction of the filtration $\{\ov{I^rJ^s}\}.$ This is used to derive a theorem due to Rees on normal joint reduction number zero. The vanishing o…
We show that diagonal subalgebras and generalized Veronese subrings of a bigraded Koszul algebra are Koszul. We give upper bounds for the regularity of sidediagonal and relative Veronese modules and apply the results to symmetric algebras and Rees rings.
The aim of this work is to study the ring-theoretic properties of the diagonals of a Rees algebra, which from a geometric point of view are the homogeneous coordinate rings of embeddings of blow-ups of projective varieties along a subvariety. In Chapter 1 we extend the definitions and results about the biprojective scheme and the Hilbert polynomial of finitely generated bigraded modules defined over standard bigraded k-algebras to finitely generated bigraded S-modules, for S=k[X_1,..., X_n,Y_1,...,Y_r] the polynomial ring bigraded by deg X_i=(1,0), deg Y_j=(d_j,1), d_j \geq 0. We also relate t…
The Hilbert functions and the regularity of the graded components of local cohomology of a bigraded algebra are considered. Explicit bounds for these invariants are obtained for bigraded hypersurface rings.
This paper is a systematic study of the Hilbert polynomial of a bigraded algebra R which are generated by elements of bidegrees (1,0), (d_1,1),...,(d_r,1), where d_1,...,d_r are non-negative integers. The obtained results can be applied to study Rees algebras of homogeneous ideals and their diagonal subalgebras. The Hilbert polynomial is computed explicitly in the following cases: (1) R is a bigraded polynomial ring of the above type; (2) R is the Rees algebra of an ideal generated by a homogeneous regular sequence; (3) R is the Rees algebra of the ideal generated by the maximal minors of a ge…
We compute a minimal bigraded resolution of the Rees Algebra associated to a proper rational parametrization of a monomial plane curve. We describe explicitly both the bigraded Betti numbers and the maps of the resolution in terms of a generalized version of the Euclidean Algorithm. We also explore the relation between pencils of adjoints of the monomial plane curve and elements in a suitable piece of the defining ideal of the Rees Algebra.
In this paper we study the local cohomology of all finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals.
Asymptotic behaviour of bigraded components of local cohomology modules
研究大梯度分量的局部上同调模的渐进行为，探讨其在特定条件下渐近稳定性及相关不变量的性质。
In \cite{rees} Rees gave a characterization for the normal joint reduction number zero of two $\m$-primary ideals in an analytically unramified Cohen-Macaulay local ring of dimension two. Rees' result is a generalization of Zariski's product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees' theorem for the ordinary powers of $\m$-primary ideals $I$ and $J$ in a Cohen-Macaulay local ring of dimension two. Following Rees' approach, we define the modified Koszul homology modules $M^1_{r,s}(a^k,b^k)$ for a joint reduction $(a,b)$ of $I$ …
Almost Complete Intersections: Regularity of powers and Rees algebra
针对几乎完全交理想，给出满足特定条件的分次几乎完全交理想幂的正则度精确公式，研究相关里斯代数的极小双分次自由分解及对角线的Cohen-Macaulay性、Koszul性等性质。
In this paper we study local cohomology of finitely generated bigraded modules over a standard bigraded ring with respect to the irrelevant bigraded ideals and establish a duality theorem. Several applications are considered.
We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence we are able to prove a generalisation of a result of Reid-Roberts-Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.
Let $M$ be a finitely generated bigraded module over the standard bigraded polynomial ring $S=K[x_1,...,x_m, y_1,...,y_n]$, and let $Q=(y_1,...,y_n)$. The local cohomology modules $H^k_Q(M)$ are naturally bigraded, and the components $H^k_Q(M)_j=\Dirsum_iH^k_Q(M)_{(i,j)}$ are finitely generated graded $K[x_1,...,x_m]$-modules. In this paper we study the regularity of $H^k_Q(M)_j$, and show in several cases that $\reg H^k_Q(M)_j$ is linearly bounded as a function of $j$.
In this paper, we obtain a generalization, in dimension $3$, of a theorem of David Rees about joint reductions of the bigraded filtration $\{ \overline{I^rJ^s}\}$ of complete ${\mathfrak m}$-primary ideals and vanishing of the second normal Hilbert coefficient $\overline{e}_2(IJ)$ where $R$ is a two-dimensional Cohen-Macaulay analytically unramified local ring with maximal ideal $\mathfrak m.$ This generalization is obtained as a consequence of a formula for the third local cohomology module of the extended Rees algebras of the $\mathbb Z^3$-graded filtration $\{\overline{I^rJ^sK^t}\}$ with su…
Let $K$ be a field and $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be the standard bigraded polynomial ring over $K$. In this paper, we explicitly describe the structure of finitely generated bigraded "sequentially Cohen--Macaulay" $S$-modules with respect to $Q=(y_1,\ldots,y_n)$. Next, we give a characterization of sequentially Cohen--Macaulay modules with respect to $Q$ in terms of local cohomology modules. Cohen--Macaulay modules that are sequentially Cohen--Macaulay with respect to $Q$ are considered.
Quasi-Gorenstein extended Rees algebras associated with filtrations
本文研究诺特局部环上Hilbert滤链对应的扩张Rees代数的拟Gorenstein性质，利用局部上同调模的Matlis对偶的Cohen-Macaulay性给出变形条件，并借助局部上同调长度给出拟Gorenstein性质的刻画。
Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce information about the defining equations of the Rees algebra of $I$. Let $\mathcal{K}$ be the kernel of the canonical map $α: \text{Sym}(I) \rightarrow \text{Rees}(I)$ from the symmetric algebra of $I$ onto the Rees algebra of $I$. We prove that $\mathcal{K}$ can be described as the solution set of a system of differential equations, that the whole bigraded stru…
Let $k$ be a field of characteristic zero, and $R=k[x_1, \ldots, x_d]$ with $d \geq 3$ be a polynomial ring in $d$ variables. Let $\m=(x_1, \ldots, x_d)$ be the homogeneous maximal ideal of $R$. Let $\mathcal{K}$ be the kernel of the canonical map $α: \sym(I) \rightarrow \R(I)$, where $\sym(I)$ (resp. $\R(I)$) denotes the symmetric algebra (resp. the Rees algebra) of an ideal $I$ in $R$. We study $\mathcal{K}$ when $I$ is a height two perfect ideal minimally generated by $d+1$ homogeneous elements of same degree and satisfies $G_d$, that is, the minimal number of generators of the ideal $I_{\m…
We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality.
In this paper we show that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology of the Rees algebra and the associated graded ring for the good filtrations case. We obtain relationships between the Castelnuovo-Mumford regularity of the fiber cone, associated graded ring, Rees algebra and reduction number for the good filtrations case.
We consider a rational map $ϕ: \mathbb{P}_k^{m} \dashrightarrow \mathbb{P}_k^n$ that is a parameterization of an $m$-dimensional variety. Our main goal is to study the $(m-1)$-dimensional fibers of $ϕ$ in relation to the $m$-th local cohomology modules of the Rees algebra of its base ideal.
The projective coinvariant algebra, Young invariants and bigraded coordinate rings of Segre embeddings
本文研究了射影线 n 重自乘积的塞格雷嵌入坐标环的双分量结构，通过双分量细化计算了硬币代数特征，并探讨了杨不变量与多重集单词生成函数的关系。
Some characterizations of Gorenstein Rees Algebras
本文利用扩展Rees代数的分次典范模，在不假设基环为Cohen-Macaulay的条件下刻画了Rees代数的Gorenstein性质，并应用于Kawasaki算术Cohen-Macaulay化。
Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $G$ be the associated graded ring of $A$ \wrt \ $I$ and let $\R = A[It,t^{-1}]$ be the extended Rees ring of $A$ with respect to $I$. Notice $t^{-1}$ is a non-zero divisor on $\R$ and $\R/t^{-1}\R = G$. So we have \textit{Bockstein operators} $β^i \colon H^i_{G_+}(G)(-1) \rt H^{i+1}_{G_+}(G)$ for $i \geq 0$. Since $β^{i+1}(+1)\circ β^i = 0$ we have \textit{Bockstein cohomology} modules $BH^i(G)$ for $i = 0,\ldots,d$. In this paper we show that certain natural conditions on…
Let S be a standard N^r-graded algebra over a local ring A, and let M be a finitely generated Z^r-graded S-module. We characterize the Cohen-Macaulayness of M in terms of the vanishing of certain sheaf cohomology modules. As a consequence, we apply our result to study the Cohen-Macaulayness of multi-Rees modules (also called Rees modification). Our work extends previous studies on the Cohen-Macaulayness of multi-Rees algebras.
The main focus is the generic freeness of local cohomology modules in a graded setting. The present approach takes place in a quite nonrestrictive setting, by solely assuming that the ground coefficient ring is Noetherian. Under additional assumptions, such as when the latter is reduced or a domain, the outcome turns out to be stronger. One important application of these considerations is to the specialization of rational maps and of symmetric and Rees powers of a module.
We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and the special fiber ring have the same image in the symmetric algebra. We show that this image is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible grade. An i…
Using vanishing of graded components of local cohomology modules of the Rees algebra of the normal filtration of an ideal, we give bounds on the normal reduction number. This helps to get necessary and sufficient conditions in Cohen-Macaulay local rings of dimension $d\geq 3$, for the vanishing of the normal Hilbert coefficients $\overline{e}_k(I)$ for $k\leq d,$ in terms of the normal reduction number.
Consider a rational projective plane curve C parameterized by three homogeneous forms h1,h2,h3 of the same degree d in the polynomial ring R=k[x,y] over the field k. Extracting a common factor, we may harmlessly assume that the ideal I=(h1,h2,h3)R has height two. Let phi be a homogeneous minimal Hilbert-Burch matrix for the row vector [h1,h2,h3]. So, phi is a 3 by 2 matrix of homogeneous forms from R; the entries in column m have degree dm, with d1 \le d2 and d1+d2=d. The Rees algebra $cal R$ of I is the subring k[h1t,h2t,h3t] of the polynomial ring k[t]. The bi-projective spectrum of $cal R$ …
Bockstein cohomology of Maximal Cohen-Macaulay modules over Gorenstein isolated singularities
本文研究了Gorenstein孤立奇点上极大Cohen-Macaulay模的Bockstein上同调，证明了在特定理想条件下其相伴分次模的Bockstein上同调具有有限长度。
Free divisors form a celebrated class of hypersurfaces which has been extensively studied in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors and investigate central aspects of the blowup algebras of their Jacobian ideals. For instance, for all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay -- a desirable feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property with tool…
Let S=K[x_1,..., x_n], let A,B be finitely generated graded S-modules, and let m=(x_1,...,x_n). We give bounds for the Castelnuovo-Mumford regularity of the local cohomology of Tor_i(A,B) under the assumption that the Krull dimension of Tor_1(A,B) is at most 1. We apply the results to syzygies, Groebner bases, products and powers of ideals, and to the relationship of the Rees and Symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some power equal to a power of m; and if the first (roughly) (n-1)/2 steps of the resolution of I are linear, then I^…
The homological shift algebra of a monomial ideal
本文引入单项式理想的同调移位代数，证明线性幂单项式理想的该代数是Rees代数上有限生成双分次模且不变量渐近良好，确定了若干类理想使其同调移位代数对大$k$有线性分解，并证明线性幂理想的同调移位代数对大$k$是Golod的。
Let A be a Noetherian local ring with the maximal ideal m and d=dimA. Let Q be a parameter ideal in A. Let I=Q:m. The problem of when the equality I^2=QI holds true is explored. When A is a Cohen-Macaulay ring, this problem was completely solved by A. Corso, C. Huneke, C. Polini, and W. Vasconcelos, while nothing is known when A is not a Cohen-Macaulay ring. The present purpose is to show that within a huge class of Buchsbaum local rings A the equality I^2=QI holds true for all parameter ideals Q. The result will supply theorems of K. Yamagishi, S. Goto and K. Nishida with ample examples of id…
Localized Persistent Commutative Algebra
该研究针对Stanley-Reisner环发展了基于坐标素理想局部上同调的局部化持续交换代数，证明了相关分解、公式与稳定性定理，推广了理论至任意坐标素理想。
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