Recent zbMATH articles in MSC 13Dhttps://zbmath.org/atom/cc/13D2024-03-13T18:33:02.981707ZWerkzeugResidual intersections and linear powershttps://zbmath.org/1528.130082024-03-13T18:33:02.981707Z"Eisenbud, David"https://zbmath.org/authors/?q=ai:eisenbud.david"Huneke, Craig"https://zbmath.org/authors/?q=ai:huneke.craig-l"Ulrich, Bernd"https://zbmath.org/authors/?q=ai:ulrich.berndLet \(S\) be a Noetherian ring and let \(I \subset S\) be an ideal of codimension \(g\). The notion of residual intersections generalizes the notion of linkage. Let \(J = (a_1,\dots,a_s)\) be an ideal contained in \(I\) and define \(K = J:I\). If \(\hbox{codim } K \geq s\) then \(K\) is called the \(s\)-residual intersection of \(I\) with respect to \(J\). If in addition \(\hbox{codim } (I+K) > s\) then the residual intersection if said to be geometric. Under strong hypotheses, residual intersections have nice properties and these have been well studied. The main result of this paper weakens these hypotheses and gives a natural rank 1, self-dual, maximal Cohen-Macaulay (MCM) module over certain residual intersections of certain ideals. More precisely, now we assume \(S\) to be a standard graded polynomial ring over an infinite field \(k\). Let \(I \subset S\) be a non-zero ideal generated by forms of the same degree \(\delta\). Let \(\ell = \ell(I)\) be the analytic spread of \(I\). Let \(J \subset I\) be an ideal generated by \(\ell -1\) general forms in \(I\) of degree \(\delta\). Set \(R = S/(J:I)\), \(\bar R = S/(J:I^\infty)\), \(\bar I = I \bar R\), \(M = M(IR)\) (the latter is a module defined earlier in the paper). If \(R\) is reduced away from \(V(I)\) and all sufficiently high powers of \(I\) are linearly presented then the authors give strong conclusions about \(R, \bar R\) and \(M\), including the rank 1, self-dual, MCM properties mentioned above. They also give several consequences of this result, including a theorem about the ideal of \(2 \times 2\) minors of a \(2 \times n\) generic matrix. Many examples are given to show the breadth of this result.
Reviewer: Juan C. Migliore (Notre Dame)On the first nontrivial strand of syzygies of projective schemes and condition \(\mathrm{ND}(\ell)\)https://zbmath.org/1528.130092024-03-13T18:33:02.981707Z"Ahn, Jeaman"https://zbmath.org/authors/?q=ai:ahn.jeaman"Han, Kangjin"https://zbmath.org/authors/?q=ai:han.kangjin"Kwak, Sijong"https://zbmath.org/authors/?q=ai:kwak.sijongLet $X \subseteq \mathbb{P}^{n+e}$ be a non-degenerate closed subscheme of dimesnion $n$ and codimension $e$ defined over an algebraically closed field $\mathbf{k}$. Although some results of the paper under review hold in this generality, we assume that $X$ is a projective variety and the characteristic of the base field $\mathbf{k}$ is zero for convenience. After Green's pioneering work on syzygies, there has been a great deal of interest in understanding the Betti tables of projective varieties. The Betti table of $X \subseteq \mathbb{P}^{n+e}$ consists of the graded Betti numbers
\[
\beta_{i,j}(X):= \dim_{\mathbf{k}} \operatorname{Tor}_i^R(R/I_{X|\mathbb{P}^r}, \mathbf{k})_{i+j},
\]
where $R:=\mathbf{k}[x_0, \ldots, x_{n+e}]$ is the homogeneous coordinate ring of $\mathbb{P}^{n+e}$. Previously, Han-Kwak proved that
\[
\beta_{i,1}(X) \leq i \binom{e+1}{ i+1}\text{ for }i \geq 1
\]
and the equality holds for some (or each) $1 \leq i \leq e$ if and only if $X \subseteq \mathbb{P}^{n+e}$ is arithmetically Cohen-Macaulay with $2$-linear resolution. One may expect to generalize this result to the first nontrivial strand of the Betti table -- assuming $H^0(\mathbb{P}^{n+e}, \mathscr{I}_{X|\mathbb{P}^{n+e}}(\ell)) = 0$ for $\ell \geq 1$, we seek to find a reasonable upper bound for $\beta_{i, \ell}(X)$. For this purpose, we need an additional condition that is $\operatorname{ND}(\ell)$ condition introduced in this paper. We say that $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{ND}(\ell)$ condition if $H^0(\Lambda, \mathscr{I}_{X \cap \Lambda|\Lambda}(\ell)) = 0$ for every general linear subspace $\Lambda \subseteq \mathbb{P}^{n+e}$ with $\dim \Lambda \geq e$. It is equivalent to $\operatorname{Gin}(I_{X|\mathbb{P}^{n+e}}) \subseteq (x_0, \ldots, x_{e-1})^{\ell+1}$, where $\operatorname{Gin}(I_{X|\mathbb{P}^{n+e}})$ is the generic initial ideal with respect to the degree reverse lexicographic order (Proposition 2.3). Note that $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{ND}(1)$ condition if and only if $X \subseteq \mathbb{P}^{n+e}$ is non-degenerate. This means that $\operatorname{ND}(1)$ condition is automatic while $\operatorname{ND}(\ell)$ condition for $\ell \geq 2$ is nontrivial. In Section 4, relevant examples and some questions on $\operatorname{ND}(\ell)$ condition are presented. The first main result of the paper is Theorem 1.1: If $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{ND}(\ell)$ condition, then
\[
\beta_{i, \ell}(X) \leq \binom{i+\ell-1}{ \ell} \binom{e + \ell}{i + \ell}\text{ for }i \geq 1
\]
and the equality holds for some (or each) $1 \leq i \leq e$ if and only if $X \subseteq \mathbb{P}^{n+e}$ is arithmetically Cohen-Macaulay with $(\ell+1)$-linear resolution.
Next, recall that $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{N}_{d,p}$ condition if $\beta_{i,j}(X) = 0$ for $i \leq p$ and $j \geq d$. A well-known result of Eisenbud-Green-Hulek-Popescu says that if $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{N}_{2,e}$, then $X$ is arithmetically Cohen-Macaulay with $2$-linear resolution. To generalized this result, we also need $\operatorname{ND}(\ell)$ condition. More precisely, the second main result of the paper (Theorem 1.2) states ``if $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{N}_{d,e}$ condition and $\operatorname{ND}(d-1)$ condition, then $X$ is arithmetically Cohen-Macaulay with $d$-linear resolution.''
Reviewer: Jinhyung Park (Daejeon)The facet ideals of chessboard complexeshttps://zbmath.org/1528.130102024-03-13T18:33:02.981707Z"Jiang, Chengyao"https://zbmath.org/authors/?q=ai:jiang.chengyao"Zhao, Yakun"https://zbmath.org/authors/?q=ai:zhao.yakun"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.11|wang.hong.12"Zhu, Guangjun"https://zbmath.org/authors/?q=ai:zhu.guangjunGraph complexes have provided an important link between combinatorics and algebra, topology, and geometry. The two most important graph complexes are the matching complex and independence complex. The chessboard complex is the matching complex of a complete bipartite graph $K_{m,n}$. In this paper the authors describe the irreducible decomposition of the facet ideal of the chessboard complex, with $n\geq m$. They also provide some lower bounds for the depth and regularity of the facet ideal.
Reviewer: Monica La Barbiera (Messina)Regularity of powers of edge ideals of Cohen-Macaulay weighted oriented forestshttps://zbmath.org/1528.130112024-03-13T18:33:02.981707Z"Kumar, Manohar"https://zbmath.org/authors/?q=ai:kumar.manohar"Nanduri, Ramakrishna"https://zbmath.org/authors/?q=ai:nanduri.ramakrishnaA weighted oriented graph is a graph \(D = (V(D), E(D),w)\), where \(V(D)\) is its vertex set, \(E(D) = \{(x, y) \ | \ \text{there is an edge from } x \text{ to } y\}\) is its edge set and \(w\) is its weight function which assigns a weight \(w(x)\) to each vertex \(x\) of \(D\). The underlying graph \(G\) of \(D\) is a simple graph with \(V(G) = V(D)\) and \(E(G) = \{\{x, y\} \ |\ (x, y)\in E(D)\}\). A vertex \(x\) of \(D\) is called a sink if there is no vertex \(y\) with \((x,y)\in E(D)\). Let \(V(D) = \{x_1, \dots , x_n\}\) and \(R = K[x_1, \dots , x_n]\), a polynomial ring over a field \(K\). Then the edge ideal of \(D\) is defined as the ideal
\[
I (D) = \langle x_i x_j^{w(x_j)} \ | \ (x_i , x_j) \in E(D)\rangle.
\]
In the paper under review, the regularity of powers of \(I(D)\) is computed where \(D\) is a Cohen-Macaulay weighted oriented forest. In fact firstly the result is gained for the first power and then inductively it is shown:
Let \(D\) be a weighted oriented unmixed forest (equivalently Cohen-Macaulay forest). Let \(\{\{x_1, y_1\}, \dots , \{x_r , y_r \}\}\) be a perfect matching in the underlying graph \(G\) of \(D\). Suppose \(y_1, \dots, y_r\) are sinks. Then for any \(k \geq 1\),
\begin{align*}
\mathrm{reg}(I (D)^k ) = \max \{ &(k - 1)(\max \{ w(y_{i_j})\} + 1) + \sum_{j=1}^s w(y_{i_j}) + 1 :\\
& \text{none of the edges } \{x_{i_j} , y_{i_j} \} \text{ being adjacent}\}.
\end{align*}
Recall that two edges are said to be adjacent if there exists an edge between them.
Reviewer: Fahimeh Khosh-Ahang Ghasr (Ilam)Elementary construction of minimal free resolutions of the Specht ideals of shapes \((n-2,2)\) and \((d,d,1)\)https://zbmath.org/1528.130122024-03-13T18:33:02.981707Z"Shibata, Kosuke"https://zbmath.org/authors/?q=ai:shibata.kosuke"Yanagawa, Kohji"https://zbmath.org/authors/?q=ai:yanagawa.kohjiLet \(\lambda\) be a partition of \(n\), \(T\) a Young tableau of shape \(\lambda\) filled in bijectively with the integers \(1,2,\dots,n\), and \(R=K[x_1,\dots,x_n]\) the \(n\)-variable polynomial algebra over a field \(K\) of characteristic zero. Denote by \(f_T\in R\) the product of the linear forms \(x_i-x_j\) where \(i,j\) are contained in the same column of \(T\) and \(i\) is above \(j\). Note that the symmetric group \(S_n\) acts on the algebra \(R\) by permutation of the variables, and the subspace of \(R\) spanned by the \(f_T\) as \(T\) ranges over the Young tableaux of shape \(\lambda\) is isomorphic to the Specht module labelled by \(\lambda\). The ideal \(I_{\lambda}\) of \(R\) spanned by this subspace is called a \textit{Specht ideal}. The authors construct a minimal free resolution of \(R/I_{\lambda}\) in the cases when \(\lambda=(n-2,2)\) and \(\lambda=(d,d,1)\). The construction is explicit, and the paper uses only the basic theory of Specht modules.
Reviewer: Matyas Domokos (Budapest)On \(w_\infty\)-Warfield cotorsion modules and Krull domainshttps://zbmath.org/1528.130132024-03-13T18:33:02.981707Z"Pu, Yongyan"https://zbmath.org/authors/?q=ai:pu.yongyan"Zhao, Wei"https://zbmath.org/authors/?q=ai:zhao.wei.8"Tang, Gaohua"https://zbmath.org/authors/?q=ai:tang.gaohua"Wang, Fanggui"https://zbmath.org/authors/?q=ai:wang.fanggui"Xiao, Xuelian"https://zbmath.org/authors/?q=ai:xiao.xuelianAuthors' abstract: Let \(R\) be a commutative domain with \(1\) and \(Q\)(\(\ne R\)) its field of quotients. In this note an \(R\)-module \(M\) is called \(w_{\infty}\)-Warfield cotorsion if \(M\in\mathcal{WC}\cap\mathcal{P}^{\perp}_{w_{\infty}}\), where \(\mathcal{WC}\) denotes the class of all Warfield cotorsion \(R\)-modules and \(\mathcal{P}_{w_{\infty}}\) the class of all \(w_{\infty}\)-projective \(R\)-modules. It is shown that \(R\) is a PVMD if and only if all \(w\)-cotorsion \(R\)-modules are \(w_{\infty}\)-Warfield cotorsion, and that \(R\) is a Krull domain if and only if every \(w\)-Matlis cotorsion strong \(w\)-module over \(R\) is a \(w_{\infty}\)-Warfield cotorsion \(w\)-module.
Reviewer: François Couchot (Caen)Dominant local rings and subcategory classificationhttps://zbmath.org/1528.130142024-03-13T18:33:02.981707Z"Takahashi, Ryo"https://zbmath.org/authors/?q=ai:takahashi.ryoSummary: We introduce a new notion of commutative noetherian local rings, which we call dominant. We explore fundamental properties of dominant local rings and compare them with other local rings. We also provide several methods to get a new dominant local ring from a given one. Finally, we classify resolving subcategories of the module category \(\mathsf{mod}\,R\) and thick subcategories of the derived category \(\mathsf{D}^b(R)\) and the singularity category \(\mathsf{D}^{sg}(R)\) for a local ring \(R\) whose certain localizations are dominant local rings. Our results recover and refine all the known classification theorems described in this context.Some results on top local cohomology moduleshttps://zbmath.org/1528.130152024-03-13T18:33:02.981707Z"Aghapournahr, Moharram"https://zbmath.org/authors/?q=ai:aghapournahr.moharram"Bahmanpour, Kamal"https://zbmath.org/authors/?q=ai:bahmanpour.kamalLet \(R\) be a commutative noetherian ring, \(\mathfrak{a}\) an ideal of \(R\), and \(M\) an \(R\)-module. For any \(i \geq 0\), the \(i\)th local cohomology module of \(M\) with respect to \(\mathfrak{a}\) is given by
\[
H^{i}_{\mathfrak{a}}(M) \cong \underset{n\in \mathbb{N}}\varinjlim \mathrm{Ext}^{i}_{R}(R/ \mathfrak{a}^{n},M).
\]
Moreover, the cohomological dimension of \(M\) with respect to \(\mathfrak{a}\) is defined by \[
\mathrm{cd}(\mathfrak{a},M)= \sup \{i\geq 0\text{ such that }H^{i}_{\mathfrak{a}}(M)\neq 0\}.
\]
The authors determine the set of all attached prime ideals of the top local cohomology module \(H^{\mathrm{cd}(\mathfrak{a},M)}_{\mathfrak{a}}(M)\) of any finitely generated \(R\)-module \(M\) with respect to the ideal \(\mathfrak{a}\) of \(R\) in terms of certain elements of \(\mathrm{Supp}_{R}(M)\). As a consequence, they show that for any pair of finitely generated \(R\)-modules \(M\) and \(N\) with \(\mathrm{Supp}_{R}(N) \subseteq\mathrm{Supp}_{R}(M)\), if \(\mathrm{cd}(\mathfrak{a},M)= \mathrm{cd}(\mathfrak{a},N)=c \geq 0\), then \(\mathrm{Att}_{R}\left(H^{c}_{\mathfrak{a}}(N)\right)\subseteq \mathrm{Att}_{R}\left(H^{c}_{\mathfrak{a}}(M)\right)\) and \(\sqrt{\mathrm{ann}_{R}\left(H^{c}_{\mathfrak{a}}(M)\right)}\subseteq \sqrt{\mathrm{ann}_{R}\left(H^{c}_{\mathfrak{a}}(N)\right)}\). Furthermore, in the special case that \(R\) is a local ring, they prove some similar results concerning the associated prime ideals of Matlis dual functors of top local cohomology modules.
Reviewer: Hossein Faridian (Clemson)An extension of \(S\)-Noetherian rings and moduleshttps://zbmath.org/1528.130162024-03-13T18:33:02.981707Z"Jara, Pascual"https://zbmath.org/authors/?q=ai:jara.pascualLet \(A\) be a commutative unitary ring and \(S\) a multiplicatively closed subset of \(A\). The ideas in [\textit{E. Hamann} et al., Pac. J. Math. 135, No. 1, 65--79 (1988; Zbl 0627.13007)] were extended by \textit{D. D. Anderson} and \textit{T. Dumitrescu} [Commun. Algebra 30, No. 9, 4407--4416 (2002; Zbl 1060.13007)] who defined and studied the so-called \(S\)-Noetherian rings: \(A\) is \(S\)-Noetherian if for every ideal \(I\) of \(A\), there exists a finitely generated subideal \(J\) of \(I\) such that \(I/J\) is annihilated by some \(s\in S\). A module variant of this definition was also given. Subsequently, there have been a lot of activity concerning \(S\)-Noetherianity (see the bibliography of the present paper).
In the present paper, the author extends the S-Noetherian concept using the more abstract notion of hereditary torsion theory instead of \(S\). Let \(\sigma\) be a hereditary torsion theory in Mod-\(A\) given by a Gabriel filter of ideals \(L(\sigma)\), that is, a non-empty filter of ideals such that an ideal \(J\) belongs to \(L(\sigma)\) provided there exists \(I\in L(\sigma)\) such that \((J : x) \in L(\sigma)\) for every \(x \in I\). The author defines \(A\) to be a totally \(\sigma\)-Noetherian ring if for every ideal \(I\) of \(A\), there exists a finitely generated subideal \(J\) of \(I\) such that \(I/J\) is annihilated by some \(H\in L(\sigma)\). A module variant of this definition is also given. Besides proving most of the results in [loc. cit.] in this new setup, the paper contains many new results. As an example we mention the Hilbert-like basis theorem. Let \(\sigma\) be a finite type hereditary torsion theory in Mod-\(A\) (that is, every \(I\in L(\sigma)\) contains some finitely generated \(J\in L(\sigma)\)) such that \(\cap_{n\geq 1}H^n\in L(\sigma)\) for each \(H\in L(\sigma)\). If \(A\) is totally \(\sigma \)-Noetherian, then the polynomial ring \(A[X]\) is totally \(\sigma'\)-Noetherian, where \(\sigma'\) is the induced torsion theory on \(A[X]\) (i.e. \(L(\sigma')=\{ I[X]\ |\ I\in L(\sigma) \}\)).
Reviewer: Tiberiu Dumitrescu (Bucureşti)Corrigendum to: ``The variety of polar simplices''https://zbmath.org/1528.140542024-03-13T18:33:02.981707Z"Ranestad, Kristian"https://zbmath.org/authors/?q=ai:ranestad.kristian"Schreyer, Frank-Olaf"https://zbmath.org/authors/?q=ai:schreyer.frank-olafSummary: We point out an important error in our paper [ibid. 18, 469--505 (2013; Zbl 1281.14035)] and provide the necessary corrections.