Recent zbMATH articles in MSC 13Dhttps://zbmath.org/atom/cc/13D2023-05-31T16:32:50.898670ZWerkzeugNoncover complexes, independence complexes, and domination numbers of hypergraphshttps://zbmath.org/1508.051292023-05-31T16:32:50.898670Z"Kim, Jinha"https://zbmath.org/authors/?q=ai:kim.jinha"Kim, Minki"https://zbmath.org/authors/?q=ai:kim.minkiSummary: Let \(\mathcal{H}\) be a hypergraph on a finite set \(V\). An independent set of \(\mathcal{H}\) is a set of vertices that does not contain an edge of \(\mathcal{H}\). The indepenence complex of \(\mathcal{H}\) is the simplicial complex on \(V\) whose faces are independent sets of \(\mathcal{H}\). A cover of \(\mathcal{H}\) is a vertex subset which meets all edges of \(\mathcal{H}\). The noncover complex of \(\mathcal{H}\) is the simplicial complex on \(V\) whose faces are noncovers of \(\mathcal{H}\). In this extended abstract, we study homological properties of the independence complexes and the noncover complexes of hypergraphs. In particular, we obtain a lower bound on the homological connectivity of independence complexes and an upper bound on the Leray number of noncover complexes. The bounds are in terms of hypergraph domination numbers. Our proof method is applied to compute the reduced Betti numbers of the independence complexes of certain uniform hypergraphs, called tight paths and tight cycles. This extends to hypergraphs known results on graphs.Burch index, summands of syzygies and linearity in resolutionshttps://zbmath.org/1508.130122023-05-31T16:32:50.898670Z"Dao, Hailong"https://zbmath.org/authors/?q=ai:dao.hailong"Eisenbud, David"https://zbmath.org/authors/?q=ai:eisenbud.davidMotivated by the work of the first author et al. [Algebra Number Theory 14, No. 8, 2121--2150 (2020; Zbl 1459.13010)] the authors of the present paper introduce a new invariant, the Burch index of a local ring \((R,\mathfrak{m}, k)\). It indicades that in an infinite minimal resolution of an module \(M\) the matrices contain many elements outside the square of the maximal ideal, or that the syzygies of \(M\) contain \(k\) as a direct summand. Suppose that \(R\) is of depth \(0\) and \(R = S/I\), where \((S,\mathfrak{n},k)\) is a regular local ring. In this special case the Burch index of \(R\) (relative to \(S\)) is \(\operatorname{Burch}_S R = \dim_k \mathfrak{n}/I\mathfrak{n}: (I: \mathfrak{n})\). Then the following is shown: Let \(M\) denote a finitely generated \(R\)-module that is not free. Suppose that \(\operatorname{depth} R = 0\) and \(\operatorname{Burch}_S R \geq 2\). Then the entries of each matrix in a minimal free resolution of \(\operatorname{syz}_5^R(M)\) generate \(\mathfrak{m}\). Moreover, \(k\) is a direct summand of \(\operatorname{syz}_n^R(M)\) for all \(n \geq 7\). Furthermore, the authors discuss a connection between the Burch index of an ideal \(I \subset S\) and the linear entries of matrices in its \(S\)-free resolution. They conclude with the computation of the Burch index for several classes of examples, including ideals in a regular local ring of dimension 2, ideals of general sets of points in the projective plane, ideals with almost linear resolution and ideals of certain fibre products.
Reviewer: Peter Schenzel (Halle)Genera of numerical semigroups and polynomial identities for degrees of syzygieshttps://zbmath.org/1508.130132023-05-31T16:32:50.898670Z"Fel, Leonid G."https://zbmath.org/authors/?q=ai:fel.leonid-gSummary: We derive polynomial identities of arbitrary degree \(n\) for syzygies degrees of numerical semigroups \(S_m=\langle d_1,\ldots ,d_m\rangle\) and show that for \(n\ge m\) they contain higher genera \(G_r = \sum_{s\in{\mathbb{Z}}_>\setminus S_m}s^r\) of \(S_m\). We find a number \(g_m = B_m-m+1\) of algebraically independent genera \(G_r\) and equations, related any of \(g_m+1\) genera, where \(B_m = \sum_{k=1}^{m-1}\beta_k\) and \(\beta_k\) denote the total and partial Betti numbers of non-symmetric semigroups. The number \(g_m\) is strongly dependent on symmetry of \(S_m\) and decreases for symmetric semigroups and complete intersections.
For the entire collection see [Zbl 1506.11002].Quadratic Gorenstein rings and the Koszul property. IIhttps://zbmath.org/1508.130142023-05-31T16:32:50.898670Z"Mastroeni, Matthew"https://zbmath.org/authors/?q=ai:mastroeni.matthew"Schenck, Hal"https://zbmath.org/authors/?q=ai:schenck.hal"Stillman, Mike"https://zbmath.org/authors/?q=ai:stillman.michael-eugeneLet \(I\) be a homogeneous ideal in a standard graded polynomial ring \(S\) over a field. Let \(R = S/I\). We say that \(R\) is quadratic if \(I\) is generated in degree 2. In this paper, which is a continuation of work started by the same authors in an earlier paper, the authors study the relationships between the properties of being quadratic and Gorenstein, being Koszul, and having prescribed regularity. They also consider the codimension (i.e. the number of variables in \(S\)). In the process they extend work of Conca-Rossi-Valla, of Vishik-Finkelberg, of Polishchuk, of Matsuda and of Nagel and the reviewer. The general question addressed is: Is every quadratic Gorenstein algebra of codimension \(c\) and regularity \(r\) Koszul? In the case that \(c=r+1\) they prove that the answer is yes by giving a structure theorem for such algebras. On the negative side they prove that there is a family of quadratic Gorenstein non-Koszul algebras of regularity 4 with Hilbert function \((1,c,2c,c,1)\) for every \(c \geq 7\). Related to this, they construct a quadratic Gorenstein non-Koszul algebra over \(\mathbb Q\) with Hilbert function \((1,6,12,6,1)\) (answering a question of Nagel and the reviewer). Between this paper and previous work of the authors and others, there remain few situations where the above general question remains open.
Reviewer: Juan C. Migliore (Notre Dame)Poincaré series of multiplier and test idealshttps://zbmath.org/1508.130152023-05-31T16:32:50.898670Z"Àlvarez Montaner, Josep"https://zbmath.org/authors/?q=ai:alvarez-montaner.josep"Núñez-Betancourt, Luis"https://zbmath.org/authors/?q=ai:nunez-betancourt.luisLet \(A\) be a (commutative, Noetherian) ring over a field. Assume that \(A\) is local or graded. Denote by \(\mathfrak{m}\) the maximal (homogeneous) ideal. Let \(\mathfrak{a}\) be an \(\mathfrak{m}\)-primary ideal. Geometrically the space \(X:=\mathrm{Spec}(A)\) is singular, and this singularity is tested by \(\mathfrak{a}.\) One way to study the pair \((A,\mathfrak{a})\) is via the associated ideals:
\begin{itemize}
\item the multiplier ideals are defined via the resolution of \(X\) and the relative canonical class;
\item the test ideals are defined via Frobenius isomorphisms.
\end{itemize}
Though these definitions are very different, the two classes of ideals share a lot in common. They both define filtrations \(A\supsetneq\mathfrak{J}_{\alpha_1}\supsetneq \mathfrak{J}_{\alpha_2}\supsetneq\cdots\), and their properties are studied via the Poincaré series of \(\mathfrak{J}\).
The natural question is whether these series are rational functions. This has been known in some particular cases. The authors establish the rationality in the following cases:
\begin{itemize}
\item for multiplier ideals and \(X\)-normal, of any dimension.
\item for test ideals in \(F\)-finite rings.
\end{itemize}
As an auxiliary step (important in its own) they develop the theory of Hilbert functions for filtrations indexed over \(\mathbb{R}\) (rather than over \(\mathbb{Z}\) or \(\mathbb{Q}\)).
Reviewer: Dmitry Kerner (Rehovot)On the Hilbert function of Artinian local complete intersections of codimension threehttps://zbmath.org/1508.130162023-05-31T16:32:50.898670Z"Jelisiejew, Joachim"https://zbmath.org/authors/?q=ai:jelisiejew.joachim"Masuti, Shreedevi K."https://zbmath.org/authors/?q=ai:masuti.shreedevi-k"Rossi, M. E."https://zbmath.org/authors/?q=ai:rossi.maria-evelinaIn the paper under review, the authors give a rigorous characterization of the Hilbert functions of quadratic complete intersections of codimension three. Especially, they show that Hilbert function is admissible for Gorenstein ring if and only if it is admissible for a complete intersection. Then, for a given Hilbert function, they succeed to provide a local complete intersection.
Reviewer: Ali Benhissi (Monastir)On the linear strand of edge ideals of some zero-divisor graphshttps://zbmath.org/1508.130172023-05-31T16:32:50.898670Z"Pirzada, S."https://zbmath.org/authors/?q=ai:pirzada.shariefuddin"Rather, Shahnawaz Ahmad"https://zbmath.org/authors/?q=ai:rather.shahnawaz-ahmadAuthors' abstract: Let \(I(G)\) be the edge ideal associated to a simple graph \(G\). Given a prime number
\(p\) and \(n > 1\) an integer, the authors study the \(N\)-graded Betti numbers that appear in the linear strand of the minimal free resolution of \(I(\Gamma(Z_{p_{n}} ))\), where \(\Gamma(Z_{p_{n}} ))\) is the zero-divisor graph of the ring \(Z_{p_{n}}\). In particular, they compute the extremal Betti number of \(I(\Gamma(Z_{p_{n}} ))\). As a consequence, the authors find the Castelnuovo-Mumford regularity and projective dimension of these ideals. Moreover, the authors exhibit formulae that determine all \(N\)-graded Betti numbers in the linear strand of the minimal free resolution of \(I(\Gamma(Z_{p_{n}} ))\) for certain values of \(n\).
Reviewer: Mohammad Javad Nikmehr (Tehran)Relative grade and relative Gorenstein dimension with respect to a semidualizing modulehttps://zbmath.org/1508.130232023-05-31T16:32:50.898670Z"Salimi, Maryam"https://zbmath.org/authors/?q=ai:salimi.maryam"Tavasoli, Elham"https://zbmath.org/authors/?q=ai:tavasoli.elhamLet \(R\) be a commutative Noetherian ring. Let \(C\) be a semidualizing \(R\)-module, and let \(M\) and \(N\) be \(R\)-modules. The paper introduced notions \(\mathrm{grade}_{\mathcal{P}_C}(M,N), \mathrm{grade}_{\mathcal{I}_C}(M,N)\) and showed some properties of these concepts. The authors also provided the notions of \(C\)-perfect and \(G_C\)-perfect modules. Some results about relative grade of tensor and Hom functors with respect to \(C\) were given.
Main results:
{Theorem 3.4:} Let \(C\) be a semidualizing \(R\)-module, and let \(M\) and \(N\) be finitely generated \(R\)-modules. Then the following statements hold:
(i) \(\mathrm{grade}(M,N)=\inf\{\mathrm{depth}_{R_{\mathfrak{p}}} N_{\mathfrak{p}} \mid \mathfrak{p}\in \mathrm{Supp}_R(M)\}\)
(ii) \(\mathrm{grade}_{\mathcal{P}_C}(M,N)=\inf\{\mathrm{depth}_{R_{\mathfrak{p}}} (\mathrm{Hom}_R(C,N))_{\mathfrak{p}} \mid \mathfrak{p}\in \mathrm{Supp}_R(M)\}\)
(iii) \(\mathrm{grade}_{\mathcal{P}_C}(M,N)=\mathrm{grade}(M,\mathrm{Hom}_R(C,N))\)
(iv) \(\mathrm{grade}_{\mathcal{I}_C}(M,N)=\inf\{\mathrm{depth}_{R_{\mathfrak{p}}} ((C\otimes_RN)_{\mathfrak{p}}) \mid \mathfrak{p}\in \mathrm{Supp}_R(M)\}\)
(v) \(\mathrm{grade}_{\mathcal{I}_C}(M,N)=\mathrm{grade}(M,C\otimes_R N).\)
{Theorem 3.12:} Let \(C\) be a semidualizing \(R\)-module, and let \(M\) be a \(G_C\)-perfect \(R\)-module. If \(\mathrm{grade} M = n,\) then the \(R\)-module \(\mathrm{Ext}^n_R(M, C)\) is \(G_C\)-perfect of grade \(n.\)
{Theorem 4.1:} Let \(C\) be a semidualizing \(R\)-module, and let \(M\) and \(N\) be finitely generated \(R\)-modules. Then the following statements hold:
(i) \(\mathrm{depth}_R(\mathrm{Hom}_R(C,N))-\dim_R(M) \le \mathrm{grade}_{\mathcal{P}_C}(M,N).\)
(ii) If \(\mathrm{Supp}_R(M)\subseteq \mathrm{Supp}_R(\mathrm{Hom}_R(C,N)),\) then
\[
\mathrm{grade}_{\mathcal{P}_C}(M,N)\le \dim_R(N)-\dim_R(M).
\]
(iii) \(\mathrm{depth}_R(C\otimes_RN)-\dim_R(M) \le \mathrm{grade}_{\mathcal{I}_C}(M,N).\)
(iv) If \(\mathrm{Supp}_R(M)\subseteq \mathrm{Supp}_R(C\otimes_RN),\) then
\[
\mathrm{grade}_{\mathcal{I}_C}(M,N)\le \dim_R(N)-\dim_R(M).
\]
Reviewer: Tri Nguyen (Biên Hòa)Koszul complexes over Cohen-Macaulay ringshttps://zbmath.org/1508.130242023-05-31T16:32:50.898670Z"Shaul, Liran"https://zbmath.org/authors/?q=ai:shaul.liranSummary: We prove a Cohen-Macaulay version of a result by \textit{L. L. Avramov} and \textit{E. S. Golod} [Math. Notes 9, 30--32 (1971; Zbl 0222.13014)] and \textit{A. Frankild} and \textit{P. Jørgensen} [Isr. J. Math. 135, 327--353 (2003; Zbl 1067.13013)] about Gorenstein rings, showing that if a noetherian ring \(A\) is Cohen-Macaulay, and \(a_1, \ldots, a_n\) is any sequence of elements in \(A\), then the Koszul complex \(K(A; a_1, \ldots, a_n)\) is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring \(A\), by finding a Cohen-Macaulay DG-ring \(B\) such that \(\operatorname{H}^0(B) = A\), and using the Cohen-Macaulay structure of \(B\) to deduce results about \(A\). As application, we prove that if \(f : X \to Y\) is a morphism of schemes, where \(X\) is Cohen-Macaulay and \(Y\) is nonsingular, then the homotopy fiber of \(f\) at every point is Cohen-Macaulay. As another application, we generalize the miracle flatness theorem. Generalizations of these applications to derived algebraic geometry are also given.Minuscule Schubert varieties of exceptional typehttps://zbmath.org/1508.140552023-05-31T16:32:50.898670Z"Filippini, Sara Angela"https://zbmath.org/authors/?q=ai:filippini.sara-angela"Torres, Jacinta"https://zbmath.org/authors/?q=ai:torres.jacinta"Weyman, Jerzy"https://zbmath.org/authors/?q=ai:weyman.jerzy-mThis article is discussing the study of exceptional minuscule Schubert varieties from the perspective of the theory of minimal free resolutions. The authors obtain the defining ideals of the intersections of these Schubert varieties with the big open cell, as well as their resolutions. The main findings of the study are summarized in the following Theorem.
{Theorem} (Theorem 1.1 in the article). Let \(G\) be a reductive group of exceptional type. Let \(P \subset G\) a standard parabolic sub-group stabilizing a minuscule fundamental weight \(\omega_i\). Denote by \(V (\omega_i)\) the fundamental \(G\)-representation over \(\mathbb C\) of highest weight \(\omega_i\), and let \(v_{\omega_i} \in V (\omega_i)\) be a highest weight vector. Let \(B \subset P\) be the Borel subgroup containing \(P\) and let \(B^-\) be the opposite Borel subgroup. In particular, the intersection \(B\cap B^-\) coincides with the maximal torus \(T\). Let \(U = B^- \cdot v_{\omega_i}\) be the opposite big open cell \(U \subset \mathbb P(V(\omega_i))\). Then for any given minuscule Schubert variety \(X_\sigma \subset G/P\), the intersection \(Y_\alpha = X_\alpha\cap U\) is as described in Sections 3 and 4. In particular, for \(\sigma\) of type \(E_6\) or as described in Section 4, \(Y_\sigma\) is a complete intersection in one of the following:
\begin{itemize}
\item[a.] \(Y_\sigma\) is a complete intersection -- the minimal free resolution of its coordinate ring is a Koszul complex.
\item[b.] in the codimension three variety of submaximal Pfaffians of a skew symmetric matrix.
\item[c.] in the variety of pure spinors in \(V(\omega_4,D_5)\).
\item[d.] in a variety of complexes.
\item[e.] in a Huneke-Ulrich Gorenstein ideal of codimension 5 and deviation 2.
\item[f.] in the variety defined by the vanishing of the \(2\times 2\) minors of a \(2\times 3\) generic matrix.
\item[g.] in the variety defined by the vanishing of \(4\times 4\) Pfaffians of a \(6\times 6\) skew-symmetric matrix.
\end{itemize}
The paper is organized in three sections, with the first section dedicated to preliminaries. The second and third sections are dedicated to \(E_6\) and \(E_7\), respectively, and they determine the defining equations for all Schubert varieties intersected with the big open cell, and compute their Hilbert polynomials. The authors manipulate the given equations by restricting the fundamental representation corresponding to P to the Levi subgroup of G of type D5 to calculate the minimal free resolutions of their defining ideals and Hilbert functions. The authors also obtain generators of an ideal whose associated resolution is precisely \(R_\alpha\).
Reviewer: Chenyu Bai (Paris)Syzygies in equivariant cohomology in positive characteristichttps://zbmath.org/1508.550062023-05-31T16:32:50.898670Z"Allday, Christopher"https://zbmath.org/authors/?q=ai:allday.christopher"Franz, Matthias"https://zbmath.org/authors/?q=ai:franz.matthias"Puppe, Volker"https://zbmath.org/authors/?q=ai:puppe.volkerIn previous papers the authors studied the Borel-equivariant cohomology of spaces with a torus \(T=(S^1)^r\) action and their relation with the Atiyah-Bredon sequence and syzygies (a notion from commutative algebra) [\textit{C. Allday} et al., Trans. Am. Math. Soc. 366, No. 12, 6567--6589 (2014; Zbl 1304.55005) and Algebr. Geom. Topol. 14, No. 3, 1339--1375 (2014; Zbl 1321.55006)]. Coefficients were taken in a field of characteristic zero.
This paper develops the theory where the coefficients are now taken in a field \(\Bbbk\) of characteristic \(p>0\) for not only actions of tori but also \(p\)-tori \(G=(\mathbb{Z}_p)^r\subset T\). The general strategy is to deduce the results about \(G\)-equivariant cohomology from those about \(T\)-equivariant cohomology.
The authors characterize the exactness of the Chang-Skjelbred sequence (under a mild hypothesis on the \(G\)-space \(X\))
\[
0\to H^*_G(X; \Bbbk)\to H^*_G(X_0; \Bbbk)\to H^{*+1}_G(X_1, X_0; \Bbbk),
\]
where \(X_0=X^G\) is the fixed point set and \(X_1\subset X\) is the union of all orbits with at most \(p\) elements. Since the polynomial ring \(R=H^*(BT; \Bbbk)\) injects into \(H^*(BG; \Bbbk)\), \(H^*_G(X; \Bbbk)\) naturally becomes an \(R\)-module. \textit{T. Chang} and \textit{T. Skjelbred} [Ann. Math. (2) 100, 307--321 (1974; Zbl 0249.57023)] proved that the sequence is exact if \(H^*_G(X; \Bbbk)\) is free over \(R\). This gives a powerful tool to compute \(H^*_G(X; \Bbbk)\) out of the equivariant 1-skeleton \(X_1\) effectively used in GKM-theory.
For \(p\)-tori, reflexive \(R\)-modules are exactly the second syzygies and the authors prove that the Chang-Skjlebred sequence is exact if and only if \(H^*_G(X; \Bbbk)\) is a reflexive \(R\)-module.
Reviewer: Alastair Darby (Suzhou)Non-abelian quantum statistics on graphshttps://zbmath.org/1508.818502023-05-31T16:32:50.898670Z"Maciążek, Tomasz"https://zbmath.org/authors/?q=ai:maciazek.tomasz"Sawicki, Adam"https://zbmath.org/authors/?q=ai:sawicki.adamSummary: We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space \(X\). The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of \(X\) which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.