Stable Ideals Research Articles

On finitely stable domains, II

Among other results, we prove the following: A locally Archimedean stable domain satisfies accp. A stable domain R is Archimedean if and only if every nonunit of R belongs to a height-one prime ideal of the integral closure R ′ of R in its quotient field (this result is related to Ohm’s theorem for Prüfer domains). An Archimedean stable domain R is one-dimensional if and only if R ′ is equidimensional (generally, an Archimedean stable local domain is not necessarily one-dimensional). An Archimedean finitely stable semilocal domain with stable maximal ideals is locally Archimedean, but generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.

Betti numbers of symmetric shifted ideals

We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.

Alexander duality for the alternative polarizations of strongly stable ideals

We will define the Alexander duality for strongly stable ideals. More precisely, for a strongly stable ideal with for all its dual is a strongly stable ideal with for all This duality has been constructed by Fløystad et al.in a different manner, so we emphasis applications here. For example, we will describe the Hilbert series of the local cohomologies using the irreducible decomposition of I (through the Betti numbers of ).

Generalized Newton complementary duals of monomial ideals

Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphism between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal.

A nonstable 𝐶*-algebra with an elementary essential composition series

A C ∗ C^* -algebra A \mathcal {A} is said to be stable if it is isomorphic to A ⊗ K ( ℓ 2 ) \mathcal {A} \otimes \mathcal {K}(\ell _2) . Hjelmborg and Rørdam have shown that countable inductive limits of separable stable C ∗ C^* -algebras are stable. We show that this is no longer true in the nonseparable context even for the most natural case of an uncountable inductive limit of an increasing chain of separable stable and AF ideals: we construct a GCR, AF (in fact, scattered) subalgebra A \mathcal {A} of B ( ℓ 2 ) \mathcal {B}(\ell _2) , which is the inductive limit of length ω 1 \omega _1 of its separable stable ideals I α \mathcal {I}_\alpha ( α > ω 1 \alpha >\omega _1 ) satisfying I α + 1 / I α ≅ K ( ℓ 2 ) \mathcal {I}_{\alpha +1}/\mathcal {I}_\alpha \cong \mathcal {K}(\ell _2) for each α > ω 1 \alpha >\omega _1 , while A \mathcal {A} is not stable. The sequence ( I α ) α ≤ ω 1 (\mathcal {I}_\alpha )_{\alpha \leq \omega _1} is the GCR composition series of A \mathcal {A} which in this case coincides with the Cantor–Bendixson composition series as a scattered C ∗ C^* -algebra. A \mathcal {A} has the property that all of its proper two-sided ideals are listed as I α \mathcal {I}_\alpha ’s for some α > ω 1 \alpha >\omega _1 , and therefore the family of stable ideals of A \mathcal {A} has no maximal element. By taking A ′ = A ⊗ K ( ℓ 2 ) \mathcal {A}’=\mathcal {A}\otimes \mathcal {K}(\ell _2) we obtain a stable C ∗ C^* -algebra with analogous composition series ( J α ) α > ω 1 (\mathcal {J}_\alpha )_{\alpha >\omega _1} whose ideals J α \mathcal {J}_\alpha are isomorphic to I α \mathcal {I}_\alpha for each α > ω 1 \alpha >\omega _1 . In particular, there are nonisomorphic scattered C ∗ C^* -algebras whose GCR composition series ( I α ) α ≤ ω 1 (\mathcal {I}_\alpha )_{\alpha \leq \omega _1} satisfy I α + 1 / I α ≅ K ( ℓ 2 ) \mathcal {I}_{\alpha +1}/\mathcal {I}_\alpha \cong \mathcal {K}(\ell _2) for all α > ω 1 \alpha >\omega _1 , for which the composition series differs first at α = ω 1 \alpha =\omega _1 .

T-Spread strongly stable monomial ideals*

We introduce the concept of t-spread monomials and t-spread strongly stable ideals. These concepts are a natural generalization of strongly stable and squarefree strongly stable ideals. For the study of this class of ideals we use the t-fold stretching operator. It is shown that t-spread strongly stable ideals are componentwise linear. Their height, their graded Betti numbers and their generic initial ideal are determined. We also consider the toric rings whose generators come from t-spread principal Borel ideals.

On almost revlex ideals with Hilbert function of complete intersections

In this paper, we investigate the behavior of almost reverse lexicographic ideals with the Hilbert function of a complete intersection. More precisely, over a field K, we give a new constructive proof of the existence of the almost revlex ideal $$J\subset K[x_1,\ldots ,x_n]$$ , with the same Hilbert function as a complete intersection defined by n forms of degrees $$d_1\le \cdots \le d_n$$ . Properties of the reduction numbers for an almost revlex ideal have an important role in our inductive and constructive proof, which is different from the more general construction given by Pardue in 2010. We also detect several cases in which an almost revlex ideal having the same Hilbert function as a complete intersection corresponds to a singular point in a Hilbert scheme. This second result is the outcome of a more general study of lower bounds for the dimension of the tangent space to a Hilbert scheme at stable ideals, in terms of the number of minimal generators.

Correspondence between trace ideals and birational extensions with application to the analysis of the Gorenstein property of rings

Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace module is given. The notion of anti-stable rings is introduced, exploring their basic properties.

Resolutions of co-letterplace ideals and generalizations of Bier spheres

We give the resolutions of co-letterplace ideals of posets in a completely explicit, very simple form. This generalizes and simplifies a number of linear resolutions in the literature, among them the Eliahou-Kervaire resolutions of strongly stable ideals generated in a single degree. Our method is based on a general result of K. Yanagawa using the canonical module of a Cohen-Macaulay Stanley-Reisner ring. We discuss in detail how the canonical module may effectively be computed, and from this derive directly the resolutions. A surprising consequence is that we obtain a large class of simplicial spheres comprehensively generalizing Bier spheres.

Strongly stable ideals and Hilbert polynomials

The \texttt{StronglyStableIdeals} package for \textit{Macaulay2} provides a method to compute all saturated strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. A description of the main method and auxiliary tools is given.

On finitely stable domains, I

Among other results, we prove the following: A locally Archimedean stable domain satisfies accp. A stable domain R is Archimedean if and only if every nonunit of R belongs to a height-one prime ideal of the integral closure R ′ of R in its quotient field (this result is related to Ohm’s theorem for Prufer domains). An Archimedean stable domain R is one-dimensional if and only if R ′ is equidimensional (generally, an Archimedean stable local domain is not necessarily one-dimensional). An Archimedean finitely stable semilocal domain with stable maximal ideals is locally Archimedean, but generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.

Extensions of almost faithful prime ideals in virtually nilpotent mod-p Iwasawa algebras

Let $G$ be a nilpotent-by-finite compact $p$-adic analytic group for some $p>2$, and $H = \mathbf{FN}_p(G)$ its finite-by-(nilpotent $p$-valuable) radical. Fix a finite field $k$ of characteristic $p$, and write $kG$ for the completed group ring of $G$ over $k$. We show that almost faithful $G$-stable prime ideals $P$ of $kH$ extend to prime ideals $PkG$ of $kG$.

Bar code for monomial ideals

The aim of this paper is to count 0-dimensional stable and strongly stable ideals in 2 and 3 variables, given their (constant) affine Hilbert polynomial p, by means of a bijection between these ideals and some integer partitions of p, which can be counted via determinantal formulas. This will be achieved by the Bar Code, a bidimensional diagram that allows to represent any finite set of terms M and desume many properties of the corresponding monomial ideal I, if M is an order ideal.

Poset Ideals of P-Partitions and Generalized Letterplace and Determinantal Ideals

For any finite poset P, we have the poset of isotone maps $\text {Hom}(P,\mathbb {N})$ , also called $P^{\text {op}}$ -partitions. To any poset ideal $\mathcal {J}$ in $\text {Hom}(P,\mathbb {N})$ , finite or infinite, we associate monomial ideals: the letterplace ideal $L(\mathcal {J},P)$ and the Alexander dual co-letterplace ideal $L(P,\mathcal {J})$ , and study them. We derive a class of monomial ideals in $\Bbbk [x_{p}, p \in P]$ called P-stable. When P is a chain, we establish a duality on strongly stable ideals. We study the case when $\mathcal {J}$ is a principal poset ideal. When P is a chain, we construct a new class of determinantal ideals which generalizes ideals of maximal minors and whose initial ideals are letterplace ideals of principal poset ideals.

Noncommutative Cantor–Bendixson derivatives and scattered C⁎-algebras

We analyze the sequence obtained by consecutive applications of the Cantor–Bendixson derivative for a noncommutative scattered C⁎-algebra A, using the ideal IAt(A) generated by the minimal projections of A. With its help, we present some fundamental results concerning scattered C⁎-algebras, in a manner parallel to the commutative case of scattered compact or locally compact Hausdorff spaces and superatomic Boolean algebras. It also allows us to formulate problems which have motivated the “cardinal sequences” programme in the classical topology, in the noncommutative context. This leads to some new constructions of noncommutative scattered C⁎-algebras and new open problems. In particular, we construct a type IC⁎-algebra which is the inductive limit of stable ideals Aα, along an uncountable limit ordinal λ, such that Aα+1/Aα is ⁎-isomorphic to the algebra of all compact operators on a separable Hilbert space and Aα+1 is σ-unital and stable for each α<λ, but A is not stable and where all ideals of A are of the form Aα. In particular, A is a nonseparable C⁎-algebra with no ideal which is maximal among the stable ideals. This answers a question of M. Rørdam in the nonseparable case. All the above C⁎-algebras Aαs and A satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers [21,22].

Spectral Metric Spaces on Extensions of C*-Algebras

We construct spectral triples on C*-algebraic extensions of unital C*-algebras by stable ideals satisfying a certain Toeplitz type property using given spectral triples on the quotient and ideal. Our construction behaves well with respect to summability and produces new spectral quantum metric spaces out of given ones. Using our construction we find new spectral triples on the quantum 2- and 3-spheres giving a new perspective on these algebras in noncommutative geometry.

Letterplace and co-letterplace ideals of posets

To a natural number n, a finite partially ordered set P and a poset ideal J in the poset Hom(P,[n]) of isotonian maps from P to the chain on n elements, we associate two monomial ideals, the letterplace ideal L(n,P;J) and the co-letterplace ideal L(P,n;J). These ideals give a unified understanding of a number of ideals studied in monomial ideal theory in recent years. By cutting down these ideals by regular sequences of variable differences we obtain: multichain ideals and generalized Hibi type ideals, initial ideals of determinantal ideals, strongly stable ideals, d-partite d-uniform ideals, Ferrers ideals, edge ideals of cointerval d-hypergraphs, and uniform face ideals.

Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas

Abstract We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen–Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo–Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind–Mertens formula for the content of the product of two polynomials.

On the functoriality of marked families

The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow associating to each ideal $J$ of this type a scheme $\MFScheme {J}$, called a $J$-marked scheme. In this paper, we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that, for all strongly stable ideals $J$, the marked schemes $\MFScheme {J}$ can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on $J$. Finally, we generalize Lederer's result about Gr\"obner strata of zero-dimensional ideals, proving that Gr\"obner strata of any ideals are locally closed subschemes of Hilbert schemes.

Regular CW-Complexes and Poset Resolutions of Monomial Ideals

We use the natural homeomorphism between a regular CW-complex X (a space with Closure-finiteness and the Weak topology) and the order complex Δ(PX) of its face poset PX to establish a canonical isomorphism between the cellular chain complex of X and the result of applying the poset construction of [5] to PX. This provides the missing unpublished ingredient to the proof of Theorem 4.3 in [6] that stable monomial ideals have minimal cellular resolutions. As another application, we study the class of monomial ideals whose lcm-lattices are CW-posets (these include, among others, edge ideals of complete bipartite graphs), and we show that they also have minimal cellular resolutions.