We study primary submodules and primary decompositions from a differential and computational point of view. Our main theoretical contribution is a general structure theory and a representation theorem for primary submodules of an arbitrary finitely generated module over a polynomial ring. We characterize primary submodules in terms of differential operators and punctual Quot schemes. Moreover, we introduce and implement an algorithm that computes a minimal differential primary decomposition for a module.
We study global primary decompositions in the category of sheaves on a scheme which are equivariant under the action of an algebraic group. We show that equivariant primary decompositions exist if the group is connected. As main application we consider the case of varieties which are quotients of a quasi-affine variety by the action of a diagonalizable group and thus admit a homogeneous coordinate ring, such as toric varieties. Comparing these decompositions with primary decompositions of graded modules over the homogeneous coordinate ring, we show that these are equivalent if the action of th…
Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms of the associated primes of select powers of $I$ and of $J$. We record two applications. First, in case the field is algebraically closed, we construct primary decompositions for powers of $I+J$ from primary decompositions for powers of $I$ and $J$. Separately, we attack the persistence problem for associated primes of powers of an ideal in case one of $I$ or…
We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential primary decompositions are unique up to change of bases. Our results generalize the construction of Noetherian operators for primary ideals in the analytic theory of Ehrenpreis-Palamodov, and they offer a concise method for representing affine schemes. The case of modules is also addressed. We implemented an algorithm in Macaulay2 that computes the minimal dec…
Noetherian operators are differential operators that encode primary components of a polynomial ideal. We develop a framework, as well as algorithms, for computing Noetherian operators with local dual spaces, both symbolically and numerically. For a primary ideal, such operators provide an alternative representation to one given by a set of generators. This description fits well with numerical algebraic geometry, taking a step toward the goal of numerical primary decomposition.
Primary decomposition is a very important tool of commutative algebra and geometry. In this paper we generalized some of the existing algorithms of primary decomposition developed by Eisenbud et al. (cf. [EHV]) for free modules and also filled some gaps by providing proofs of important theorems(2.9, 2.12, 2.13) appeared in [EHV]. All these algorithms are programmed and implemented in {\sc Singular}.
We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and the idea of Shimoyama-Yokoyama resp. Eisenbud-Hunecke-Vasconcelos to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in SINGULAR. Examples and timings are given at the end of the article.
In this paper we study primality and primary decomposition of certain ideals which are generated by homogeneous degree $2$ polynomials and occur naturally from determinantal conditions. Normality is derived from these results.
We present an algorithm to compute the primary decomposition of a submodule $\mathcal{N}$ of the free module $\Z[x_1, \ldots, x_n]^m$. For this purpose we use algorithms for primary decomposition of ideals in the polynomial ring over the integers. The idea is to compute first the minimal associated primes of $\mathcal{N}$, i.e. the minimal associated primes of the ideal $\Ann(\Z[x_1, \ldots, x_n]^m /\mathcal{N})$ in $\Z[x_1,\ldots,x_n]$ and then compute the primary components using pseudo-primary decomposition and extraction, following the ideas of Shimoyama-Yokoyama. The algorithms are implem…
In this paper we describe the method which we applied to successfully compute the primary decomposition of a certain ideal coming from applications in combinatorial algebra and algebraic statistics regarding conditional independence statements with hidden variables. While our method is based on the algorithm for primary decomposition by Gianni, Trager and Zacharias, we were not able to decompose the ideal using the standard form of that algorithm, nor by any other method known to us.
Students studying the Lasker-Noether theorem on primary decomposition of ideals may want to see an example of an ideal (necessarily in a non-Noetherian ring) which does not have a primary decomposition. The most well-known counterexample is alluded to in an exercise from Atiyah and MacDonald's Commutative Algebra text. It involves the ring of continuous real-valued functions on a compact Hausdorff space, and the details require the use of Urysohn's lemma from topology. In this article, we excise the unnecessary connection to topology by finding a purely algebraic counterexample in the power se…
A famous result due to L. S. Levy provides a classification of all finitely generated indecomposable modules over Dedekind-like rings. This motivates us to outline an approach to the classification of indecomposable pseudo-absorbing primary multiplication modules with finite-dimensional top over certain kinds of pullback rings. In this paper, we give a complete classification, up to isomorphism, of all indecomposable pseudo-absorbing primary multiplication modules with finite-dimensional top over a pullback of two valuation domains with the same residue field. We also find a connection between…
It was previously known, by work of Smith-Swanson and of Sharp-Nossem, that the linear growth property of primary decompositions of Frobenius powers of ideals in rings of prime characteristic has strong connections to the localization problem in tight closure theory. The localization problem has recently been settled in the negative by Brenner and Monsky, but the linear growth question is still open. We study growth of primary decompositions of Frobenius powers of dimension one homogeneous ideals in graded rings over fields. If the ring is positively graded we prove that the linear growth prop…
Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. These mesoprimary decompositions are highly combinatorial in nature, and are designed to parallel standard primary decomposition over Noetherean rings. In this paper, we generalize mesoprimary decomposition from binomial ideals to "binomial submodules" of certain graded modules over the corresponding monoid algebra, analogous to the way primary decomposition of ideals over a Noetherean ri…
Let $R$ be a Noetherian ring and $x_1,\ldots,x_t$ a permutable regular sequence of elements in $R$. Then there exists a finite set of primes $Λ$ and natural number $C$ so that for all $n_1,\ldots,n_t$ there exists a primary decomposition $(x_1^{n_1},\ldots,x_t^{n_t})=Q_1\cap \cdots \cap Q_\ell$ so that $\sqrt{Q_i}\in Λ$ and $\sqrt{Q_i}^{C(n_1+\cdots + n_t)}\subseteq Q_i$ for all $1\leq i\leq \ell$.
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to find extra properties of these graded submodules.
We present an alternative method for computing primary decomposition of zero-dimensional ideals over finite fields. Based upon the further decomposition of the invariant subspace of the Frobenius map acting on the quotient algebra in the algorithm given by S. Gao, D. Wan and M. Wang in 2008, we get an alternative approach to compute all the primary components at once. As one example of our method, an improvement of Berlekamp's algorithm by theoretical considerations which computes the factorization of univariate polynomials over finite fields is also obtained.
An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables.
Let $R=\bigoplus_{\underline{n} \in \mathbb{N}^t}R_{\underline{n}}$ be a commutative Noetherian $\mathbb{N}^t$-graded ring, and $L = \bigoplus_{\underline{n}\in\mathbb{N}^t}L_{\underline{n}}$ be a finitely generated $\mathbb{N}^t$-graded $R$-module. We prove that there exists a positive integer $k$ such that for any $\underline{n} \in \mathbb{N}^t$ with $L_{\underline{n}} \neq 0$, there exists a primary decomposition of the zero submodule $O_{\underline{n}}$ of $L_{\underline{n}}$ such that for any $P \in {\rm Ass}_{R_0}(L_{\underline{n}})$, the $P$-primary component $Q$ in that primary decomp…
For finitely generated modules $N \subsetneq M$ over a Noetherian ring $R$, we study the following properties about primary decomposition: (1) The Compatibility property, which says that if $\ass (M/N)=\{P_1, P_2, ..., P_s\}$ and $Q_i$ is a $P_i$-primary component of $N \subsetneq M$ for each $i=1,2,...,s$, then $N =Q_1 \cap Q_2 \cap ... \cap Q_s$; (2) For a given subset $X=\{P_1, P_2, ..., P_r \} \subseteq \ass(M/N)$, $X$ is an open subset of $\ass(M/N)$ if and only if the intersections $Q_1 \cap Q_2\cap ... \cap Q_r= Q_1' \cap Q_2' \cap ... \cap Q_r'$ for all possible $P_i$-primary component…
All rings are commutative with $1\neq0$, and all modules are unital. The purpose of this paper is to investigate the concept of $2$-absorbing primary submodules generalizing $2$-absorbing primary ideals of rings. Let $M$ be an $R$-module. A proper submodule $N$ of an $R$-module $M$ is called a $2$-absorbing primary submodule of $M$ if whenever $a,b\in R$ and $m\in M$ and $abm\in N$, then $am\in M$-$rad(N)$ or $bm\in M$-$rad(N)$ or $ab\in(N:_RM)$. It is shown that a proper submodule $N$ of $M$ is a $2$-absorbing primary submodule if and only if whenever $I_1I_2K\subseteq N$ for some ideals $I_1…
In this paper, we introduce the expansion function $δ$ on an $L$-module $M$. We define and investigate a $δ$-primary element in an $L$-module $M$. Its characterizations and many of its properties are obtained. $δ_0$-primary and $δ_1$-primary elements of an $L$-module $M$ are related with 2-absorbing, 2-absorbing primary elements of an $L$-module $M$ to obtain their special properties. The element $δ_1(N)\in M$ is related to $rad(N)\in M$, the radical element of $M$ to obtain its properties where $N\in M$. We define a $δ_L$-primary element in an $L$-module $M$ where $δ_L$ is an expansion functi…
Let $R$ be a $G$-graded ring and M be a $G$-graded $R$-module. We define the graded primary spectrum of $M$, denoted by $\mathcal{PS}_G(M)$, to be the set of all graded primary submodules $Q$ of M such that $(Gr_M(Q):_R M)=Gr((Q:_R M))$. In this paper, we define a topology on $\mathcal{PS}_G(M)$ having the Zariski topology on the graded prime spectrum $Spec_G(M)$ as a subspace topology, and investigate several topological properties of this topological space.
Let $R$ be a commutative Noetherian ring, $E$ a non-zero finitely generated $R$-module and $I$ an ideal of $R$. One purpose of this paper is to show that the sequences $\Ass_RE/ \widetilde{I_E^n}$ and $\Ass_R\widetilde{I^n _E}/\widetilde{I^{n+1}_E}$, $n = 1,2, \dots$, of associated prime ideals are increasing and eventually stabilize. This extends the main result of Mirbagheri and Ratliff \cite[Theorem 3.1]{MR}. In addition, a characterization concerning the set $\widetilde{A^*}(I,E)$ is included. A second purpose of this paper is to prove that $I$ has linear growth primary decompositions for …
In this paper we study how prime filtrations and squarefree Stanley decompositions of squarefree modules over the polynomial ring and the exterior algebra behave with respect to Alexander duality.
We characterize the fixed divisor of a polynomial $f(X)$ in $\mathbb{Z}[X]$ by looking at the contraction of the powers of the maximal ideals of the overring ${\rm Int}(\mathbb{Z})$ containing $f(X)$. Given a prime $p$ and a positive integer $n$, we also obtain a complete description of the ideal of polynomials in $\mathbb{Z}[X]$ whose fixed divisor is divisible by $p^n$ in terms of its primary components.
We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and it has numerous applications within and beyond pure mathematics. The ideals defining toric varieties are precisely the binomial prime ideals. Our main results concern primary decomposition: If $I$ is a binomial ideal then the radical, associated primes, and isolated primary components of $I$ are again binomial, and $I$ admits primary decompositions in terms …
Given a local Noetherian ring $(R, {\mathfrak m})$ of dimension $d>0$ and infinite residue field, we study the invariants $($dimension and multiplicity$)$ of the Sally module $S_J(I)$ of any ${\mathfrak m}$-primary ideal $I$ with respect to a minimal reduction $J$. As a by-product we obtain an estimate for the Hilbert coefficients of ${\mathfrak m}$ that generalizes a bound established by J. Elias and G. Valla in a local Cohen-Macaulay setting. We also find sharp estimates for the multiplicity of the special fiber ring ${\mathcal F}(I)$, which recover previous bounds established by C. Polini, …
In this paper, we give a generalization for weakly primary submodules called $I$-primary submodule and we study some properties of it. We give some characterizations of $I$-primary submodules. Also we establish the situation of $I$-primary submodules in module localizations and decomposition modules.
Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ to be weakly $S$-primary if there exists $s\in S$ such that whenever $a\in R$ and $m\in M$ with $0\neq am\in N$, then either $sa\in\sqrt{(N:_{R}M)}$ or $sm\in N$. We present various properties and characterizations of this concept (especially in finitely generated faithful multiplication modules). Moreover, the behavior of this structure under module homomorphisms, localizations, q…
Let $R$ be a commutative ring with identity, $S$ a multiplicatively closed subset of $R$, and $M$ be an $R$-module. In this paper, we study and investigate some properties of $S$-primary submodules of $M$. Among the other results, it is shown that this class of modules contains the family of primary (resp. $S$-prime) submodules properly.
The first aim of this work is to characterize when the lattice of all submodules of a module is a direct product of two lattices. In particular, which decompositions of a module $M$ produce these decompositions: the \emph{lattice decompositions}. In a first \textit{étage} this can be done using endomorphisms of $M$, which produce a decomposition of the ring $\textrm{End}_R(M)$ as a product of rings, i.e., they are central idempotent endomorphisms. But since not every central idempotent endomorphism produces a lattice decomposition, the classical theory is not of application. In a second step w…
We introduce the notion of E-depth of graded modules over polynomial rings to measure the depth of certain Ext modules. First, we characterize graded modules over polynomial rings with (sufficiently) large E-depth as those modules whose (sufficiently) partial general initial submodules preserve the Hilbert function of local cohomology modules supported at the irrelevant maximal ideal, extending a result of Herzog and Sbarra on sequentially Cohen-Macaulay modules. Second, we describe the cone of local cohomology tables of modules with sufficiently high E-depth, building on previous work of the …
Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings. These features are captured by the more refined theory of mesoprimary decomposition of congruences, introduced here complete with witnesses and associated prime objects. The combinatorial theory of mesoprimary decomposition lifts to arbitrary binomial ideals in monoid algebras. The resulting binomial mesoprimary decomposition is a new type of intersection decomposition for binomial ideals that enjoys computational efficiency and independence from ground fiel…
The package Binomials contains implementations of specialized algorithms for binomial ideals, including primary decomposition into binomial ideals. The current implementation works in characteristic zero. Primary decomposition is restricted to binomial ideals with trivial coefficients to avoid computations over the algebraic numbers. The basic ideas of the algorithms go back to Eisenbud and Sturmfels' seminal paper on the subject. Two recent improvements of the algorithms are discussed and examples are presented.
Over any partially ordered abelian group whose positive cone is closed in an appropriate sense and has finitely many faces, modules that satisfy a weak finiteness condition admit finite primary decompositions. This conclusion rests on the introduction of basic notions in the relevant generality, such as closedness of partially ordered abelian groups, faces and their coprimary modules, and finiteness conditions as well local and global support functors for modules over partially ordered groups.
Let $R$ be a (possibly noncommutative) ring and let $\mathcal C$ be a class of finitely generated (right) $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set $\mathcal V (\mathcal C)$ of isomorphism classes of modules is a commutative semigroup with operation induced by the direct sum. This semigroup encodes all possible information about direct sum decompositions of modules in $\mathcal C$. If the endomorphism ring of each module in $\mathcal C$ is semilocal, then $\mathcal V (\mathcal C)$ is a Krull monoid. Although this fact was observed nea…
A direct sum decomposition theory is developed for direct summands (and complements) of modules over a semiring $R$, having the property that $v+w = 0$ implies $v = 0$ and $w = 0$. Although this never occurs when $R$ is a ring, it always does holds for free modules over the max-plus semiring and related semirings. In such situations, the direct complement is unique, and the decomposition is unique up to refinement. Thus, every finitely generated projective module is a finite direct sum of summands of $R$ (assuming the mild assumption that $1$ is a finite sum of orthogonal primitive idempotents…
Let $M$ be a finitely generated module of dimension $d$ over a Noetherian local ring $(R,\m)$ and $\q $ the parameter ideal generated by a system of parameters $\x = (x_1,..., x_d)$ of $M$. For each positive integer $n$, set $$Λ_{d,n}=\{α=(α_1,...,α_d)\in\Bbb{Z}^d|α_i\geqslant 1, \forall 1\leqslant i\leqslant d \text{and} \sum\limits_{i=1}^dα_i=d+n-1\}$$ and $\qa = (x_1^{α_1},...,x_d^{α_d})$. Then we prove in this note that $M$ is a sequentially Cohen-Macaulay module if and only if there exists a certain system of parameters $\x$ such that the equality $\q^nM=\pd$ holds true for all $n$. As an…
This paper is concerned with tight closure in a commutative Noetherian ring $R$ of prime characteristic $p$, and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal $I$ of $R$ has linear growth of primary decompositions, then tight closure (of $I$) commutes with localization at the powers of a single element. It is shown in this paper that, provided $R$ has a weak test element, linear growth of primary decompositions for other sequences of ideals of $R$ that approximate, in a certain sense, the sequence of Frobenius p…
We present Binomials, a package for the computer algebra system Macaulay2, which specializes well known algorithms to binomial ideals. These come up frequently in algebraic statistics and commutative algebra, and it is shown that significant speedup of computations like primary decomposition is possible. While central parts of the implemented algorithms go back to Eisenbud and Sturmfels (1996), we also discuss a new algorithm for computing the minimal primes of a binomial ideal. All decompositions make significant use of combinatorial structure found in binomial ideals, and to demonstrate the …
We study random walks on contingency tables with fixed marginals, corresponding to a (log-linear) hierarchical model. If the set of allowed moves is not a Markov basis, then there exist tables with the same marginals that are not connected. We study linear conditions on the values of the marginals that ensure that all tables in a given fiber are connected. We show that many graphical models have the positive margins property, which says that all fibers with strictly positive marginals are connected by the quadratic moves that correspond to conditional independence statements. The property pers…
Let $R$ be a commutative ring with identity. For a finitely generated $R$-module $M$, the notion of associated prime submodules of $M$ is defined. It is shown that this notion inherits most of essential properties of the usual notion of associated prime ideals. In particular, it is proved that for a Noetherian multiplication module $M$, the set of associated prime submodules of $M$ coincides with the set of $M$-radicals of primary submodules of $M$ which appear in a minimal primary decomposition of the zero submodule of $M$. Also, Anderson's theorem [{\bf 2}] is extended to minimal prime submo…
A full subcategory of modules over a commutative ring $R$ is wide if it is abelian and closed under extensions. Hovey \cite{wide} gave a classification of wide subcategories of finitely presented modules over regular coherent rings in terms of certain specialisation closed subsets of $\Spec(R)$. We use this classification theorem to study K-theory and Krull-Schmidt type decompositions for wide subcategories. It is shown that the K-group, in the sense of Grothendieck, of a wide subcategory $\W$ of finitely presented modules over a regular coherent ring is isomorphic to that of the thick subcate…
Stanley decompositions of multigraded modules $M$ over polynomials rings have been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition, called the Stanley depth. Stanley conjectured that the Stanley depth of a module $M$ is always at least the (classical) depth of $M$. In this paper we introduce a weaker type of decomposition, which we call Hilbert decomposition, since it only depends on the Hilbert function of $M$, and an analogous notion of depth, called Hilbert depth. Since Stanley decompositions are Hilbert decompositions, the la…
Boij-Söderberg theory shows that the Betti table of a graded module can be written as a liner combination of pure diagrams with integer coefficients. Using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules.
