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In this paper, we prove a result similar to results of Itoh [J. Algebra 150 (1992), pp. 101–117] and Hong-Ulrich [J. Lond. Math. Soc. (2) 90 (2014), pp. 861–878], proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

We resolve an open question posed by Nagpal, Sam and Snowden [Selecta. Math. (N.S.) 22 (2016), pp. 913–937] in 2015 concerning a Gröbner theoretic approach to the noetherianity of the twisted commutative algebra Sym(Sym^2(\mathbf {C}^\infty)). We provide a negative answer to their question by producing an explicit antichain. In doing so, we establish a connection to well-studied posets of graphs under the subgraph and induced subgraph relation. We then analyze this connection to suggest future paths of investigation. [ABSTRACT FROM AUTHOR]

In this paper, we study conditions guaranteeing that a product of ideals defines a Golod ring. We show that for a 3-dimensional regular local ring (or 3-variable polynomial ring) (R, \mathfrak {m}), the ideal I \mathfrak {m} always defines a Golod ring for any proper ideal I \subset R. We also show that non-Golod products of ideals are ubiquitous; more precisely, we prove that for any proper ideal with grade \geqslant 4, there exists an ideal J \subseteq I such that IJ is not Golod. We conclude by showing that if I is any proper ideal in a 3-dimensional regular local ring and \mathfrak {a} \subseteq I a complete intersection, then \mathfrak {a} I is Golod. [ABSTRACT FROM AUTHOR]

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is uncountable, as well as a subalgebra with finitely many distinct initial algebras. [ABSTRACT FROM AUTHOR]

Conca and Herzog proved that any product of ideals of linear forms in a polynomial ring has a linear resolution. The goal of this paper is to establish the same result for any quadric hypersurface. The main tool we develop and use is a flexible version of Derksen and Sidman's approximation systems. [ABSTRACT FROM AUTHOR]

In this paper, we study the strong Lefschetz property of artinian complete intersection ideals generated by products of linear forms. We prove the strong Lefschetz property for a class of such ideals with binomial generators. [ABSTRACT FROM AUTHOR]

Let D be a Dedekind domain with finite residue fields. We provide topological insights into certain classes of ideals of Int(D) lying over a given maximal ideal \mathfrak m of D. We completely determine invertible/divisorial ideals in terms of topological properties of subsets of the \mathfrak m-adic completion of D. Moreover, these results are naturally extended to overrings of Int(D). As an application we provide explicit constructions of divisorial ideals of Int(D) which are not finitely generated. [ABSTRACT FROM AUTHOR]

Say that (x, y, z) is a positive primitive integral Pythagorean triple if x, y, z are positive integers without common factors satisfying x^2 + y^2 = z^2. An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on (3, 4, 5) and (4, 3, 5) generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren's theorem in the context of a one-variable polynomial ring over a field of characteristic \neq 2. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring. [ABSTRACT FROM AUTHOR]

Let K be a field, and let σ be an automorphism of K of finite order. Let K(X; σ) be the quotient ring of the skew polynomial ring K[X; σ]. Then any order in K(X; σ) which contains K and its center is a valuation ring of the center of K(X; σ) is a crossed-product algebra Af, where f is some normalized 2-cocycle. Associated to f is a subgroup H of ‹σ› and a graph. In this paper, we determine the connections between hereditary-ness and maximal order properties of Af and the properties of H, f and the graph of f. [ABSTRACT FROM AUTHOR]

Let R be a finitely generated \mathbb N-graded algebra domain over a Noetherian ring and let I be a homogeneous ideal of R. Given P\in Ass(R/I) one defines the v-invariant v_P(I) of I at P as the least c\in \mathbb N such that P=I:f for some f\in R_c. A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that Ass(R/I^n) is constant for large n. So it makes sense to consider a prime ideal P\in Ass(R/I^n) for all the large n and investigate how v_P(I^n) depends on n. We prove that v_P(I^n) is eventually a linear function of n. When R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [ Asymptotic behaviour of the \text {v}-number of homogeneous ideals , https://arxiv.org/abs/2306.14243, 2023]. [ABSTRACT FROM AUTHOR]

Let A = K[[X1,...,Xn]] and let m = (X1,...,Xn). Let M be a Cohen-Macaulay A-module of codimension p. In this paper we give a necessary and sufficient condition for the associated graded module Gm(M) to have a pure resolution over the polynomial ring Gm(A) ≅ = K[X1,...,Xn]. [ABSTRACT FROM AUTHOR]

Let I be an equidimensional ideal of a ring polynomial R over \mathbb {C} and let J be its generic linkage. We prove that there is a uniform bound of the difference between the F-pure thresholds of I_p and J_p via the generalized Frobenius powers of ideals. This provides evidence that the F-pure threshold of an equidimensional ideal I is less than that of its generic linkage. As a corollary we recover a result on log canonical thresholds of generic linkage by Niu. [ABSTRACT FROM AUTHOR]

We introduce to the context of multigraded modules the methods of modules over categories from algebraic topology and homotopy theory. We develop the basic theory quite generally, with a view toward future applications to a wide class of graded modules over graded rings. The main application in this paper is to study the Betti poset B = B(I, k) of a monomial ideal I in the polynomial ring R = k[x1, ..., xm] over a field k, which consists of all degrees in ℤm of the homogeneous basis elements of the free modules in the minimal free ℤm-graded resolution of I over R. We show that the order simplicial complex of B supports a free resolution of I over R. We give a formula for the Betti numbers of I in terms of Betti numbers of open intervals of B, and we show that the isomorphism class of B completely determines the structure of the minimal free resolution of I, thus generalizing with new proofs the results of Gasharov, Peeva, and Welker in 1999. We also characterize the finite posets that are Betti posets of a monomial ideal. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the polarization of Koszul cycles and use it to study the depth function of powers of edge ideals of whisker graphs. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a cohomological criterion for the splitting of a vector bundle on multiprojective space. The criterion extends a result of Ballico-Malaspina towards a generalization of the Horrocks criterion on multiprojective space. [ABSTRACT FROM AUTHOR]

It has been conjectured by Eisenbud, Green and Harris that if I is a homogeneous ideal in k[x1, . . ., xn] containing a regular sequence f1, . . ., fn of degrees deg(fi) = ai, where 2 = a1 = ·· · = an, then there is a homogeneous ideal J containing x1a1, . . ., xnan with the same Hilbert function. In this paper we prove the Eisenbud-Green-Harris Conjecture when fi splits into linear factors for all i. [ABSTRACT FROM AUTHOR]

We study the differential symmetric signature, an invariant of rings of finite type over a field, introduced in a previous work by the authors in an attempt to find a characteristic-free analogue of the F-signature. We compute the differential symmetric signature for invariant rings k[x1, . . .,xn]G, where G is a finite small subgroup of GL(n, k), and for hypersurface rings k[x1, . . ., xn]/(f) of dimension ≥ 3 with an isolated singularity. In the first case, we obtain the value 1/|G|, which coincides with the F-signature and generalizes a previous result of the authors for the two-dimensional case. In the second case, following an argument by Bruns, we obtain the value 0, providing an example of a ring where differential symmetric signature and F-signature are different. [ABSTRACT FROM AUTHOR]

Let R=k[x,y] be a polynomial ring over a field k of prime characteristic p and let E denote the injective hull of k (which is isomorphic to H^2_{(x,y)}(R)). We prove that E is not an injective object in the category of graded F-modules over R. This answers in the negative a question raised by Lyubeznik-Singh-Walther [J. Eur. Math. Soc. (JEMS) 18 (2016), pp. 2545- 2578]. [ABSTRACT FROM AUTHOR]

Let K be a field and S = K[x1, . . .,xn] be the polynomial ring in n variables over the field K. Let G be a forest with p connected components G1, . . .,Gp and let I = I(G) be its edge ideal in S. Suppose that di is the diameter of Gi, 1 ≤ i ≤ p, and consider d = max{di | 1 ≤ i ≤ p}. Morey has shown that for every t ≥ 1, the quantity max{...} is a lower bound for depth(S/It). In this paper, we show that for every t ≥ 1, the mentioned quantity is also a lower bound for sdepth(S/It). By combining this inequality with Burch's inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem. [ABSTRACT FROM AUTHOR]

Let K be a commutative field of characteristic zero, A be a domain containing K and ∂ be a locally nilpotent K-derivation of A. We give in this paper a description of the differential K-algebra (A, ∂) under the assumptions that the ring of constants A∂ of ∂ is a PID, ∂ is fixed point free and its special fibers are reduced. [ABSTRACT FROM AUTHOR]

We determine the Grothendieck rings of the category of finite-dimensional modules over Queer Lie superalgebras via their rings of characters. In particular, we show that the \mathbb {Q}-span of the ring of characters of the Queer Lie supergroup Q(n) is isomorphic to the ring of Laurent polynomials in x_{1},\ldots,x_{n} for which the evaluation x_{n-1}=-x_{n}=t is independent of t. We thus complete the description of Grothendieck rings for all classical Lie superalgebras. [ABSTRACT FROM AUTHOR]

Poonen proved an analogue for Drinfeld modules of the Mordell-Weil theorem. We shall generalize his results to the case of t-modules which have the height property. Moreover, we also prove a Mordell–Weil theorem for certain t-modules which lack the height property. In particular, let A be the polynomial ring in one variable \theta over the finite field \mathbb {F}_q with the quotient field K. Let (E, \phi) be an iterated extension of the tensor powers of the Carlitz module defined over K. We show that the \mathbb {F}_q[t]-module E(K) is isomorphic to the direct sum of its torsion module, which is finite, with a free \mathbb {F}_q[t]-module of rank \aleph _0. [ABSTRACT FROM AUTHOR]

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show that these counts are interrelated via partial sums. Moreover, we interpret the partitions we study in the language of graded polynomial rings and apply this to the mod 2 Steenrod algebra to compute the free rank of certain homology modules in stable homotopy theory. [ABSTRACT FROM AUTHOR]

In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers \sqrt {D} is the solution to Pell's equation for D. It is well-known that, once an integer solution to Pell's equation exists, we can use it to generate all other solutions (u_n,v_n)_{n\in \mathbb {Z}}. Our object of interest is the polynomial version of Pell's equation, where the integers are replaced by polynomials with complex coefficients. We then investigate the factors of v_n(t). In particular, we show that over the complex polynomials, there are only finitely many values of n for which v_n(t) has a repeated root. Restricting our analysis to \mathbb {Q}[t], we give an upper bound on the number of "new" factors of v_n(t) of degree at most N. Furthermore, we show that all "new" linear rational factors of v_n(t) can be found when n \leq 3, and all "new" quadratic rational factors when n \leq 6. [ABSTRACT FROM AUTHOR]

Given a field k and a finite group G, the Beckmann–Black problem asks whether every Galois field extension F/k with group G is the specialization at some t_0 \in k of some Galois field extension E/k(T) with group G and E \cap \overline {k} = k. We show that the answer is positive for arbitrary k and G, if one waives the requirement that E/k(T) is normal. In fact, our result holds if \operatorname {Gal}(F/k) is any given subgroup H of G and, in the special case H=G, we provide a similar conclusion even if F/k is not normal. We next derive that, given a division ring H and an automorphism \sigma of H of finite order, all finite groups occur as automorphism groups over the skew field of fractions H(T, \sigma) of the twisted polynomial ring H[T, \sigma ]. [ABSTRACT FROM AUTHOR]

Let R be a polynomial ring over a field. We prove an upper bound for the multiplicity of R/I when I is a homogeneous ideal of the form I = J+(F), where J is a Cohen-Macaulay ideal and F ∉ J. The bound is given in terms of two invariants of R/J and the degree of F. We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given. [ABSTRACT FROM AUTHOR]

We use the k[V]-module generator of the dual module of the polynomial ring k[V] over its subring of invariants of a finite group to construct modular invariants and show that it behaves better than the transfer homomorphism. [ABSTRACT FROM AUTHOR]

Given a circle-valued Morse function of a closed oriented manifold, we prove that Reidemeister torsion over a non-commutative formal Laurent polynomial ring equals the product of a certain non-commutative Lefschetz-type zeta function and the algebraic torsion of the Novikov complex over the ring. This paper gives a generalization of the result of Hutchings and Lee on abelian coefficients to the case of skew fields. As a consequence we obtain a Morse theoretical and dynamical description of the higher-order Reidemeister torsion. [ABSTRACT FROM AUTHOR]

The Hardy spaces $H^{2}(D^{2})$ can be viewed as a module over the polynomial ring $C[z_1,z_2]$. Based on a study of {em core operator}, a new equivalence relation for submodules, namely {em congruence}, was introduced. This paper gives a classification of congruent submodules by the rank of core operators. [ABSTRACT FROM AUTHOR]

The main result of this paper is that all antichains are finite in the poset of monomial ideals in a polynomial ring, ordered by inclusion. We present several corollaries of this result, both simpler proofs of results already in the literature and new results. One natural generalization to more abstract posets is shown to be false. [ABSTRACT FROM AUTHOR]

Let G be a simple graph on n vertices and IG denotes parity binomial edge ideal of G in the polynomial ring S = K[x1, . . . , xn, y1, . . . , yn]. We obtain lower bound for the regularity of parity binomial edge ideals of graphs. We then classify all graphs whose parity binomial edge ideals have regularity 3. We classify graphs whose parity binomial edge ideals have pure resolution. [ABSTRACT FROM AUTHOR]

We generalize an improved Lech bound, due to Huneke, Smirnov, and Validashti, from the Hilbert-Samuel multiplicity to the Buchsbaum-Rim multiplicity and mixed multiplicity. We reduce the problem to the graded case and then to the polynomial ring case. There we use complete reductions, studied by Rees, to prove sharper bounds for the mixed multiplicity in low dimensions before proving the general case. [ABSTRACT FROM AUTHOR]

We settle a conjecture of Herzog and Hibi, which states that the function depth, S/Qn, n ≥ 1, where Q is a homogeneous ideal in a polynomial ring S, can be any convergent numerical function. We also give a positive answer to a longstanding open question of Ratliff on the associated primes of powers of ideals. [ABSTRACT FROM AUTHOR]

Given a standard graded polynomial ring over a commutative Noetherian ring A, we prove that the cohomological dimension and the height of the ideals defining any of its Veronese subrings are equal. This result is due to Ogus when A is a field of characteristic zero, and follows from a result of Peskine and Szpiro when A is a field of positive characteristic; our result applies, for example, when A is the ring of integers. [ABSTRACT FROM AUTHOR]

If $P$ is a prime ideal of a polynomial ring $K[x]$, where $K$ is a field, then $P$ is determined by an irreducible polynomial in $K[x]$. The purpose of this paper is to show that any prime ideal of a polynomial ring in $n$-indeterminates over a not necessarily commutative ring $R$ is determined by its intersection with $R$ plus $n$ polynomials. [ABSTRACT FROM AUTHOR]

An overlooked corollary to the main result of the stated paper (Proc. Amer. Math. Soc. {\bf 120} (1994), 989--993) is that any Goldie ring $R$ of Goldie dimension 1 has Artinian classical quotient ring $Q$, hence is a Kerr ring in the sense that the polynomial ring $R[X]$ satisfies the $acc$ on annihilators $(=acc \bot )$. More generally, we show that a Goldie ring $R$ has Artinian $Q$ when every zero divisor of $R$ has essential annihilator (in this case $Q$ is a local ring; see Theorem $1^\prime $). A corollary to the proof is Theorem 2: A commutative ring $R$ has Artinian $Q$ iff $R$ is a Goldie ring in which each element of the Jacobson radical of $Q$ has essential annihilator. Applying a theorem of Beck we show that any $acc \bot $ ring $R$ that has Noetherian local ring $R_p$ for each associated prime $P$ is a Kerr ring and has Kerr polynomial ring $R[X]$ (Theorem 5). [ABSTRACT FROM AUTHOR]

This article studies separating invariants for the ring of multisymmetric polynomials in m sets of n variables over an arbitrary field K. We prove that in order to obtain separating sets it is enough to consider polynomials that depend only on ⌊n/2⌋ + 1 sets of these variables. This improves a general result by Domokos about separating invariants. In addition, for n ≤ 4 we explicitly give minimal separating sets (with respect to inclusion) for all m in case char(K) = 0 or char(K) > n. [ABSTRACT FROM AUTHOR]

An important problem in algebra and related fields (such as algebraic and complex analytic geometry) is to find an explicit, well-structured, minimal set of generators for the module of logarithmic derivations of classes of homogeneous ideals in polynomial rings. In this note we settle the case of the ideal P ⊂ R = K[{Xi,j}] generated by the maximal minors of an (n + 1) × n generic matrix (Xi,j) over an arbitrary field K with n ≥ 2. We also characterize when the derivation module of R/P is Ulrich, and we investigate this property if we replace R/P by determinantal rings arising from simple degenerations of the generic case. [ABSTRACT FROM AUTHOR]

Let K be a field and let S = K[x1 , . . . ,xn] be the polynomial ring in n variables over K. Assume that I ⊂ S is a squarefree monomial ideal. For every integer k ≥ 1, we denote the k-th symbolic power of I by I(k). Recently, Montaño and Núñez-Betancourt (2018), and independently Nguyen and Trung (to appear), proved that for every pair of integers k, i ≥ 1, depth(S(k) ≤ depth(S/I(⌈k/i⌉). We provide an alternative proof for this inequality. Moreover, we re-prove the known results that the sequence {depth (S/I(k))}k=1∞ is convergent and mink depth(S/I(k) = limk→∞ depth(S/I(k) = n-ls(I), where ls(I) denotes the symbolic analytic spread of I. We also determine an upper bound for the index of depth stability of symbolic powers of I. Next, we consider the Stanley depth of symbolic powers and prove that the sequences {sdepth(S/I(k)}k=1∞ and {sdepth(I(k)}k=1∞ are convergent and the limit of each sequence is equal to its minimum. Furthermore, we determine an upper bound for the indices of sdepth stability of symbolic powers. [ABSTRACT FROM AUTHOR]

Schur modules give the irreducible polynomial representations of the general linear group GLt. Viewing the symmetric group Stas a subgroup of GLt, we may restrict Schur modules to St and decompose the result into a direct sum of Specht modules, the irreducible representations of St. We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for St for t large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternate in sign by degree. In particular, this allows us to define a new basis of symmetric functions whose structure constants are stable Kronecker coefficients and which expand with signs alternating by degree into the Schur basis. [ABSTRACT FROM AUTHOR]

Let k be a field, let S be a bigraded k-algebra, and let SΔ denote the diagonal subalgebra of S corresponding to Δ = {(cs,es) |s ∈ Z}. It is known that the SΔ is Koszul for c,e >> 0. In this article, we find bounds for c,e for SΔ to be Koszul when S is a geometric residual intersection. Furthermore, we also study the Cohen-Macaulay property of these algebras. Finally, as an application, we look at classes of linearly presented perfect ideals of height two in a polynomial ring, show that all their powers have a linear resolution, and study the Koszul and Cohen-Macaulay properties of the diagonal subalgebras of their Rees algebras. [ABSTRACT FROM AUTHOR]

Let R be a polynomial ring and let I ⊂ R be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of I. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of I. Applying ideas introduced by Busé, D'Andrea, and the author, we obtain the value of the j-multiplicity of I and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of I. [ABSTRACT FROM AUTHOR]

We show that, under a natural probability distribution, random monomial ideals will almost always have minimal free resolutions of maximal length; that is, the projective dimension will almost always be n, where n is the number of variables in the polynomial ring. As a consequence we prove that Cohen-Macaulayness is a rare property. We characterize when a random monomial ideal is generic/strongly generic, and when it is Scarf--i.e., when the algebraic Scarf complex of M ⊂ S = k[x1,…, xn] gives a minimal free resolution of S/M. It turns out, outside of a very specific ratio of model parameters, random monomial ideals are Scarf only when they are generic. We end with a discussion of the average magnitude of Betti numbers. [ABSTRACT FROM AUTHOR]

We give a complete algebraic characterization of the non-degenerated centers for planar trigonometric Cherkas systems. The main tools used in our proof are the classical results of Cherkas on planar analytic Liénard systems, the characterization of some subfields of the quotient field of the ring of trigonometric polynomials, and results due to Rosenlicht concerning algebraic solutions to transcendental equations. The results obtained are reminiscent of the ones for planar polynomial Cherkas systems, but the proofs are different. [ABSTRACT FROM AUTHOR]

When A = k[x1, . . . ,xn] and G is a small subgroup of GLn(k), Auslander's Theorem says that the skew group algebra A#G is isomorphic to EndAG(A) as graded algebras. We prove a generalization of Auslander's Theorem for permutation actions on (-1)-skew polynomial rings, (-1)-quantum Weyl algebras, three-dimensional Sklyanin algebras, and a certain homogeneous down-up algebra. We also show that certain fixed rings AG are graded isolated singularities in the sense of Ueyama. [ABSTRACT FROM AUTHOR]

We construct an action of the Hecke algebra Hn(q) on a quotient of the polynomial ring F[x1, . . . , xn], where F = Q(q). The dimension of our quotient ring is the number of k-block ordered set partitions of {1, 2, . . . , n}. This gives a quantum analog of a construction of Haglund-Rhoades-Shimozono and interpolates between their result at q = 1 and work of Huang-Rhoades at q = 0. [ABSTRACT FROM AUTHOR]

We give an explicit upper bound for the degree of a tropical basis of a homogeneous polynomial ideal. As an application f-vectors of tropical varieties are discussed. Various examples illustrate differences between Gröbner and tropical bases. [ABSTRACT FROM AUTHOR]

We introduce a family of Artinian Gorenstein algebras, whose combinatorial structure characterizes the ones presented by quadrics. Under certain hypotheses these algebras have non-unimodal Hilbert vector. In particular we provide families of counterexamples to the conjecture that Artinian Gorenstein algebras presented by quadrics should satisfy the weak Lefschetz property. [ABSTRACT FROM AUTHOR]

We characterize the lcm lattices that support a monomial ideal with a pure resolution. Given such a lattice, we provide a construction that yields a monomial ideal with that lcm lattice and whose minimal free resolution is pure. [ABSTRACT FROM AUTHOR]