Graded Betti Numbers Research Articles

Stable behavior of Frobenius powers over a general hypersurface

The main goal of this paper is to prove, in positive characteristic p p , stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general choices of hypersurface in three variables whose degree has the opposite parity to that of p p . We also find some of the structure of the matrix factorization giving the resolution. We achieve this by developing a method for obtaining the degrees of the generators of the defining ideal of an c \mathfrak {c} -compressed Gorenstein Artinian graded algebra from its socle degree, where c \mathfrak {c} is a Frobenius power of the homogeneous maximal ideal. As an application, we also obtain the Hilbert–Kunz function of the hypersurface ring, as well as the Castelnuovo–Mumford regularity of the quotients by Frobenius powers of the homogeneous maximal ideal.

Frobenius-Poincaré function and Hilbert-Kunz multiplicity

We generalize the notion of Hilbert-Kunz multiplicity of a graded triple ( M , R , I ) (M,R,I) in characteristic p > 0 p>0 by proving that for any complex number y y , the limit lim n → ∞ ( 1 p n ) dim ⁡ ( M ) ∑ j = − ∞ ∞ λ ( ( M I [ p n ] M ) j ) e − i y j / p n \begin{equation*} \underset {n \to \infty }{\lim }(\frac {1}{p^n})^{\operatorname {dim}(M)}\sum \limits _{j= -\infty }^{\infty }\lambda \left ( (\frac {M}{I^{[p^n]}M})_j\right )e^{-iyj/p^n} \end{equation*} exists. We prove that the limiting function in the complex variable y y is entire and name this function the Frobenius-Poincaré function. We establish various properties of Frobenius-Poincaré functions including its relation with the tight closure of the defining ideal I I ; and relate the study Frobenius-Poincaré functions to the behaviour of graded Betti numbers of R I [ p n ] \frac {R}{I^{[p^n]}} as n n varies. Our description of Frobenius-Poincaré functions in dimension one and two and other examples raises questions on the structure of Frobenius-Poincaré functions in general.

Some remarks on the Kp,1$\mathcal {K}_{p,1}$ theorem

AbstractLet be a non‐degenerate projective irreducible variety of dimension , degree , and codimension over an algebraically closed field of characteristic 0. Let be the th graded Betti number of . Green proved the celebrating ‐theorem about the vanishing of for high values for and potential examples of nonvanishing graded Betti numbers. Later, Nagel–Pitteloud and Brodmann–Schenzel classified varieties with nonvanishing . It is clear that when there is an ‐dimensional variety of minimal degree containing , however, this is not always the case as seen in the example of the triple Veronese surface in .In this paper, we completely classify varieties with nonvanishing such that does not lie on an ‐dimensional variety of minimal degree. They are exactly cones over smooth del Pezzo varieties, whose Picard number is .

Linear strand of edge ideals of zero divisor graphs of the ring ℤn

For a simple graph G with edge ideal I(G), we study the N -graded Betti numbers in the linear strand of the minimal free resolution of I ( Γ ( Z n ) ) , where Γ ( Z n ) is the zero divisor graph of the ring Z n . We present sharp bounds for the Betti numbers of Γ ( Z n ) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of Γ ( Z n ) for n = p 2 q and pqr, with primes p < q < r .

Graded Betti Numbers of Some Bipartite Graphs

Graded Betti Numbers of Some Bipartite Graphs

Asymptotic syzygies for products of projective spaces

We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld we give an explicit range in which the graded Betti numbers of Pn1×Pn2 embedded by OPn1×Pn2(d1,d2) are non-zero.

Barile–Macchia resolutions

We construct cellular resolutions for monomial ideals via discrete Morse theory. In particular, we develop an algorithm to create homogeneous acyclic matchings and we call the cellular resolutions induced from these matchings Barile–Macchia resolutions. These resolutions are minimal for edge ideals of weighted oriented forests and (most) cycles. As a result, we provide recursive formulas for graded Betti numbers and projective dimension. Furthermore, we compare Barile–Macchia resolutions to those created by Batzies and Welker and some well-known simplicial resolutions. Under certain assumptions, whenever the above resolutions are minimal, so are Barile–Macchia resolutions.

Ideals of submaximal minors of sparse symmetric matrices

We study the ideal of submaximal minors of a sparse generic symmetric matrix. This ideal is generated by all (n−1)-minors of a symmetric n×n matrix whose entries in the upper triangle are distinct variables or zeros, and the zeros are only allowed at off-diagonal places. The surviving off-diagonal entries are encoded in a simple graph G with n vertices. We prove that the minimal free resolution of this ideal is obtained from the case without any zeros via a simple pruning procedure, extending methods of Boocher. This allows us to compute all graded Betti numbers in terms of n and a single invariant of G. Moreover, it turns out that these ideals are always radical and have Cohen–Macaulay quotients if and only if G is either connected or has no edges at all. The key input are some new Gröbner basis results with respect to non-diagonal term orders associated to G.

Algebraic invariants of interval subdivided complexes

We study ring-theoretic properties of the Stanley–Reisner rings of simplicial complexes that undergo the interval subdivisions. In particular, we establish results concerning computational description of dimension, Hilbert series, multiplicity, local cohomology, depth and regularity of the Stanley–Reisner ring of interval subdivided simplicial complex. We also prove results regarding the non-vanishing of graded Betti numbers and bounds on Betti numbers for these Stanley–Reisner rings.

Mapping cones of monomial ideals over exterior algebras

Let K be a field, V a finite dimensional K-vector space and E the exterior algebra of V. We analyze iterated mapping cone over E. If I is a monomial ideal of E with linear quotients, we show that the mapping cone construction yields a minimal graded free resolution F of I via the Cartan complex. Moreover, we provide an explicit description of the differentials in F when the ideal I has a regular decomposition function. Finally, we get a formula for the graded Betti numbers of a new class of monomial ideals including the class of strongly stable ideals.

The regularity of almost all edge ideals

A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. An example is the well-known result saying that almost all triangle-free graphs are bipartite. The “almost” is crucial, without it such theorems do not hold. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool is the critical graphs introduced relatively recently by Balogh and Butterfield, who proved that almost all graphs not containing a critical subgraph have common structural characteristics analogous to being bipartite.For a graph G, let IG denote its edge ideal, the monomial ideal generated by xixj for every edge ij of G. In this paper we study the graded Betti numbers of IG, which are combinatorial invariants that measure the complexity of a minimal free resolution of IG. The Betti numbers of the form βi,2i+2 constitute the “main diagonal” of the Betti table. It is well known that for edge ideals any Betti number to the left of this diagonal is always zero. We identify a certain “parabola” inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let βi,j be a parabolic Betti number on the r-th row of the Betti table, for r≥3. We prove that almost all graphs G with βi,j(IG)=0 can be partitioned into r−2 cliques and one independent set. In particular, for almost all graphs G with βi,j(IG)=0, the regularity of IG is r−1.

Linear strand of edge ideals of comaximal graphs of commutative rings

For an edge ideal I(G) of a simple graph G , we study the N -graded Betti numbers that appear in the linear strand of the minimal free resolution of I ( Γ ( Z n ) ) , where Γ ( Z n ) is the comaximal graph of the integral modulo ring Z n . We show that the extremal Betti number of I ( Γ ( Z n ) ) is ϕ ( n ) , where ϕ ( n ) is the Euler’s totient function and thereby we obtain a large class of edge ideals with even extremal Betti numbers. We find the regularity (Castelnuovo-Mumford) and the projective dimension of these ideals. Moreover, we exhibit explicit formulae that determine all the N -graded Betti numbers in the linear strand of the minimal free resolution of I ( Γ ( Z n ) ) for certain values of n .

Characterization of projective varieties beyond varieties of minimal degree and del Pezzo varieties

Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question “which varieties are the most basic and simplest except the above two kinds of varieties in view of geometry and syzygies?”, we give an upper bound of the graded Betti numbers in the quadratic strand and characterize the extremal cases.The extremal varieties of dimension n, codimension e, and degree d are exactly characterized by the following two types:(i)Varieties with d=e+2, depthX=n, and Green-Lazarsfeld index a(X)=0,(ii)Arithmetically Cohen-Macaulay varieties with d=e+3. This is a generalization of G. Castelnuovo, G. Fano, and E. Park's results on the number of quadrics and an extension of the characterizations of varieties of minimal degree and del Pezzo varieties in view of linear syzygies of quadrics due to K. Han and S. Kwak ([6,8,30,16]).In addition, we show that every variety X that belongs to (i) or (ii) is always contained in a unique rational normal scroll Y as a divisor. Also, we describe the divisor class of X in Y.

Positivity and nonstandard graded Betti numbers

AbstractA foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of this paper is to determine the extent to which this principle extends to the nonstandard ‐graded case. In this setting, the classical arguments break down, and the results become much more nuanced. We introduce a new notion of Castelnuovo–Mumford regularity and employ exterior algebra techniques to control the shapes of nonstandard ‐graded minimal free resolutions. Our main result reveals a unique feature in the nonstandard ‐graded case: the possible degrees of the syzygies of a graded module in this setting are controlled not only by its regularity, but also by its depth. As an application of our main result, we show that given a simplicial projective toric variety and a module over its coordinate ring, the multigraded Betti numbers of are contained in a particular polytope when satisfies an appropriate positivity condition.

The graded Betti numbers of truncation of ideals in polynomial rings

The graded Betti numbers of truncation of ideals in polynomial rings

Vector-spread monomial ideals and Eliahou–Kervaire type resolutions

We introduce the class of vector-spread monomial ideals. This notion generalizes that of t-spread ideals introduced by Ene, Herzog and Qureshi. In particular, we focus on vector-spread strongly stable ideals, we compute their Koszul cycles and describe their minimal free resolution. As a consequence the graded Betti numbers and the Poincaré series are determined. Finally, we consider a generalization of algebraic shifting theory for such a class of ideals.

Correction to: Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers [J. ALGEBRAIC COMBIN. 27 (2008), NO. 2, 215-245

Correction to: Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers [J. ALGEBRAIC COMBIN. 27 (2008), NO. 2, 215-245

On the linear strand of edge ideals of some zero-divisor graphs

Let I(G) be the edge ideal associated to a simple graph G. Given a prime number p and an integer, we study the -graded Betti numbers that appear in the linear strand of the minimal free resolution of , where is the zero-divisor graph of the ring . In particular, we compute the extremal Betti number of . As a consequence, we find the Castelnuovo-Mumford regularity and projective dimension of these ideals. Moreover, we exhibit formulae that determine all -graded Betti numbers in the linear strand of the minimal free resolution of for certain values of n.

Koszul homology of quotients by edge ideals

We show that the Koszul homology algebra of a quotient of a polynomial ring by the edge ideal of a forest is generated by the lowest linear strand. This provides a large class of Koszul algebras whose Koszul homology algebras satisfy this property. We obtain this result by constructing the minimal graded free resolution of a quotient by such an edge ideal via the so called iterated mapping cone construction and using the explicit bases of Koszul homology given by Herzog and Maleki. Using these methods we also recover a result of Roth and Van Tuyl on the graded Betti numbers of quotients by edge ideals of trees.

Minimal free resolutions of fiber products

We consider a local (or standard graded) ring R R with ideals I ′ \mathcal {I}’ , I \mathcal {I} , J ′ \mathcal {J}’ , and J \mathcal {J} satisfying certain Tor-vanishing constraints. We construct free resolutions for quotient rings R / ⟨ I ′ , I J , J ′ ⟩ R/\langle \mathcal {I}’, \mathcal {I}\mathcal {J}, \mathcal {J}’\rangle , give conditions for the quotient to be realized as a fiber product, and give criteria for the construction to be minimal. We then specialize this result to fiber products over a field k k and recover explicit formulas for Betti numbers, graded Betti numbers, and Poincaré series.