Eero Hyry, Yukio Nakamura, Lauri Ojala
In this article we investigate when a complete ideal in a two-dimensional regular local ring is a multiplier ideal of some ideal with an integral multiplying parameter. In particular, we show that this question is closely connected to the Gorenstein property of the blowup along the ideal.
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Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
本文推广了Jouanolou对偶，引入弱Gorenstein环类，为研究爆破代数提供新框架，并用于确定某些理想族的Rees代数的定义方程。
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…
In this paper, we consider monomial curves in ${\mathbb A}_k^3$ parameterized by $t \rightarrow (t^{2q +1}, t^{2q +1 + m}, t^{2q +1 +2 m})$ where $gcd( 2q+1,m)=1$. The symbolic blowup algebras of these monomial curves is Gorenstein (\cite{goto-nis-shim}, \cite{goto-nis-shim-2}). We give a simple proof for the the Gorenstein property for the symbolic blowup algebras of these curves.
We present new algorithms for computing adjoint ideals of curves and thus, in the planar case, adjoint curves. With regard to terminology, we follow Gorenstein who states the adjoint condition in terms of conductors. Our main algorithm yields the Gorenstein adjoint ideal G of a given curve as the intersection of what we call local Gorenstein adjoint ideals. Since the respective local computations do not depend on each other, our approach is inherently parallel. Over the rationals, further parallelization is achieved by a modular version of the algorithm which first computes a number of the cha…
In this paper we explicitly describe the symbolic powers of curves ${\mathcal C}(q,m)$ in ${\mathbb P}^3$ parametrized by $( x^{d+2m}, x^{d+m} y^m, x^{d} y^{2m}, y^{d+2m})$, where $q,m$ are positive integers, $d=2q+1$ and $\gcd(d,m)=1$. The defining ideal of these curves is a set-theoretic complete intersection. We show that the symbolic blowup algebra is Noetherian and Gorenstein. An explicit formula for the resurgence and the Waldschmidt constant of the prime ideal ${\mathfrak p}:={\mathfrak p}_{ { \mathcal C}(q,m) }$ defining the curve ${\mathcal C}(q,m)$ is computed. We also give a formula…
In this paper, we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using characteristic $p$ methods. When $X$ is a normal Gorenstein closed subvariety of a smooth complex variety $A$, we formulate a restriction property of the adjoint ideal sheaf $\adj_X(A)$ of $A$ along $X$ involving the l.c.i. ideal sheaf $\mathcal{D}_X$ of $X$. The proof relies on a modification of generalized test ideals of Hara and Yoshida.
Let $d \geq 2$ and $m\geq 1$ be integers such that $\gcd (d,m)=1.$ Let ${\mathfrak p}$ be the defining ideal of the monomial curve in ${\mathbb A}_{ \Bbbk{k}}^d$ parametrized by $(t^{n_1}, \ldots, t^{n_d})$ where $n_i = d + (i-1)m$ for all $i = 1, \ldots, d$. In this paper, we describe the symbolic powers ${\mathfrak p}^{(n)} $ for all $n \geq 1$. As a consequence we show that the symbolic blowup algebras ${\mathcal R}_s{({\mathfrak p})}$ and $G_{s}({\mathfrak p}) $ are Cohen-Macaulay. This gives a positive answer to a question posed by S.~Goto in \cite{goto}. We also discuss when these blowup…
The class of good semigroups is a class of subsemigroups of $N^h$, that includes the value semigroups of rings associated to curve singularities and their blowups, and allows to study combinatorically the properties of these rings. In this paper we give a characterization of almost symmetric good subsemigroups of $N^h$, extending known results in numerical semigroup theory and in one-dimensional ring theory, and we apply these results to obtain new results on almost Gorenstein one-dimensional analytically unramified rings.
Let $R$ be a ring with identity and $\C(R)$ denote the category of complexes of $R$-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over $R$, denoted $\KPC$, is always well generated and is compactly generated provided $\KPR$ is so. Based on this result, it will be proved that the class of Gorenstein projective complexes is precovering, whenever $R$ is a commutative noetherian ring of finite Krull dimension…
When is the canonical conductor minimal?
研究一维Cohen-Macaulay局部环中导数最小的条件，探讨其与almost Gorenstein环的关系。
Let $\u_{1\times n}$, $\X_{n\times n}$, and $\v_{n\times 1}$ be matrices of indeterminates, $\Adj \X$ be the classical adjoint of $\X$, and $H(n)$ be the ideal $I_1(\u\X)+I_1(\X\v)+I_1(\v\u-\Adj \X)$. Vasconcelos has conjectured that $H(n)$ is a perfect Gorenstein ideal of grade $2n$. In this paper, we obtain the minimal free resolution of $H(n)$; and thereby establish Vasconcelos' conjecture.
There appeared not long ago a Reduction Formula for derived Hochschild cohomology, that has been useful e.g., in the study of Gorenstein maps and of rigidity w.r.t. semidualizing complexes. The formula involves the relative dualizing complex of a ring homomorphism, so brings out a connection between Hochschild homology and Grothendieck duality. The proof, somewhat ad hoc, uses homotopical considerations via a number of noncanonical projective and injective resolutions of differential graded objects. Recent efforts aim at more intrinsic approaches, hopefully upgradable to "higher" contexts--lik…
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