Commutative Algebra Research Articles

Rational motivic path spaces and Kim’s relative unipotent section conjecture

We develop the foundations of commutative algebra objects in the category of motives, which we call “motivic dga’s.” Works of White and Cisinski and Déglise provide us with a suitable model structure. This enables us to reconstruct the unipotent fundamental group of a pointed scheme from the associated augmented motivic dga and provides us with a factorization of Kim’s relative unipotent section conjecture into several smaller conjectures with a homotopical flavor.

Developments in Commutative Algebra: In honor of Jürgen Herzog on the occasion of his 80th Birthday

Developments in Commutative Algebra: In honor of Jürgen Herzog on the occasion of his 80th Birthday

Bifold Algebras and Commutants for Enriched Algebraic Theories

Commuting pairs of algebraic structures on a set have been studied by several authors and may be described equivalently as algebras for the tensor product of Lawvere theories, or more basically as certain bifunctors that here we call bifold algebras. The much less studied notion of commutant for Lawvere theories was first introduced by Wraith and generalizes the notion of centralizer clone in universal algebra. Working in the general setting of enriched algebraic theories for a system of arities, we study the interaction of the concepts of bifold algebra and commutant. We show that the notion of commutant arises via a universal construction in a two-sided fibration of bifold algebras over various theories. On this basis, we study special classes of bifold algebras that are related to commutants, introducing the notions of commutant bifold algebra and balanced bifold algebra. We establish several adjunctions and equivalences among these categories of bifold algebras and related categories of algebras over various theories, including commutative, contracommutative, saturated, and balanced algebras. We also survey and develop examples of commutant bifold algebras, including examples that employ Pontryagin duality and a theorem of Ehrenfeucht and Łoś on reflexive abelian groups. Along the way, we develop a functorial treatment of fundamental aspects of bifold algebras and commutants, including tensor products of theories and the equivalence of bifold algebras and commuting pairs of algebras. Because we work relative to a (possibly large) system of arities in a closed category $${\mathscr {V}}$$ , our main results are applicable to arbitrary $${\mathscr {V}}$$ -monads on a finitely complete $${\mathscr {V}}$$ , the enriched theories of Borceux and Day, the enriched Lawvere theories of Power relative to a regular cardinal, and other notions of algebraic theory.

Gelfand–Naimark theorems for ordered -algebras

Abstract The classical Gelfand–Naimark theorems provide important insight into the structure of general and of commutative $C^*$ -algebras. It is shown that these can be generalized to certain ordered $^*$ -algebras. More precisely, for $\sigma $ -bounded closed ordered $^*$ -algebras, a faithful representation as operators is constructed. Similarly, for commutative such algebras, a faithful representation as complex-valued functions is constructed if an additional necessary regularity condition is fulfilled. These results generalize the Gelfand–Naimark representation theorems to classes of $^*$ -algebras larger than $C^*$ -algebras, and which especially contain $^*$ -algebras of unbounded operators. The key to these representation theorems is a new result for Archimedean ordered vector spaces V: If V is $\sigma $ -bounded, then the order of V is induced by the extremal positive linear functionals on V.

Gorenstein‐projective modules over short local algebras

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra A $A$ with radical J $J$ will be said to be short provided J 3 = 0 $J^3 = 0$ . As in the commutative case, we show: If a short local algebra A $A$ has an indecomposable non-projective Gorenstein-projective module M $M$ , then either A $A$ is self-injective (so that all modules are Gorenstein-projective) and then, of course, | J 2 | ⩽ 1 $|J^2| \leqslant 1$ , or else | J 2 | = | J / J 2 | − 1 $|J^2| = |J/J^2| - 1$ and | J M | = | J 2 | | M / J M | $|JM| = |J^2||M/JM|$ . More generally, we focus the attention to semi-Gorenstein-projective and ∞ $\infty$ -torsionfree modules, even to ℧ $\mho$ -paths of length 2, 3 and 4. In particular, we show that the existence of a non-projective reflexive module implies that | J 2 | < | J / J 2 | $|J^2| < |J/J^2|$ and further restrictions. In addition, we consider exact complexes of projective modules with a non-projective image. Again, as in the commutative case, we see that if such a complex exists, then A $A$ is self-injective or satisfies the condition | J 2 | = | J / J 2 | − 1 $|J^2| = |J/J^2| - 1$ . Also, we show that any non-projective semi-Gorenstein-projective module M $M$ satisfies Ext 1 ( M , M ) ≠ 0 $\operatorname{Ext}^1(M,M) \ne 0$ . In this way, we prove the Auslander-Reiten conjecture (one of the classical homological conjectures) for arbitrary short local algebras. Many arguments used in the commutative case actually work in general, but there are interesting differences and some of our results may be new also in the commutative case.

The Cyclical Compactness in Banach C∞(Q)-Modules

In this paper we study the class of laterally complete commutative unital regular algebras A over arbitrary fields. We introduce a notion of passport Γ(X) for a faithful regular laterally complete Amodules X, which consist of uniquely defined partition of unity in the Boolean algebra of all idempotents in A and of the set of pairwise different cardinal numbers. We prove that A-modules X and Y are isomorphic if and only if Γ(X) = Γ(Y). Further we study Banach A-modules in the case A = C∞(Q) or A = C∞(Q)+i ·C∞(Q). We establish the equivalence of all norms in a finite-dimensional (respectively, σ-finite-dimensional) A-module and prove an A-version of Riesz Theorem, which gives the criterion of a finite-dimensionality (respectively, σ-finite-dimensionality) of a Banach A-module.

Rational stabilization and maximal ideal spaces of commutative Banach algebras

Rational stabilization and maximal ideal spaces of commutative Banach algebras

Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results

We present a survey and new results on the construction and Gelfand theory of commutative Toeplitz algebras over the standard weighted Bergman and Hardy spaces over the unit ball in mathbb {C}^n. As an application we discuss semi-simplicity and the spectral invariance of these algebras. The different function Hilbert spaces are dealt with in parallel in successive chapters so that a direct comparison of the results is possible. As a new aspect of the theory we define commutative Toeplitz algebras over spaces of functions in infinitely many variables and present some structural results. The paper concludes with a short list of open problems in this area of research.

Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras

In 1971 R. L. Carpenter proved that every derivation on a semisimple commutative Frechet algebra with identity is continuous. The concept of almost derivations on Frechet algebras is introduced in this article. Also, R. L. Carpenter result motivates us to ask an open question: Is every almost derivation on semisimple commutative Frechet algebras continuous?. Moreover, a partial answer to this open question is derived in the sense that every almost derivation T on semisimple commutative Frechet Q-algebras A, with an additional condition on A, is continuous. Furthermore, an example is provided to illustrate our main result.

Complementary decompositions of monomial ideals and involutive bases

Complementary decompositions of monomial ideals—also known as Stanley decompositions—play an important role in many places in commutative algebra. In this article, we discuss and compare several algorithms for their computation. This includes a classical recursive one, an algorithm already proposed by Janet and a construction proposed by Hironaka in his work on idealistic exponents. We relate Janet’s algorithm to the Janet tree of the Janet basis and extend this idea to Janet-like bases to obtain an optimised algorithm. We show that Hironaka’s construction terminates, if and only if the monomial ideal is quasi-stable. Furthermore, we show that in this case the algorithm of Janet determines the same decomposition more efficiently. Finally, we briefly discuss how these results can be used for the computation of primary and irreducible decompositions.

On a family of vertex operator superalgebras

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight-32 homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra V is simple, then V(2) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and V(32) is naturally a V(2)-module equipped with a V(2)-valued symmetric bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that A is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that U is an A-module equipped with a symmetric A-valued bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra L(A,U) and a simple vertex operator superalgebra LL(A,U)(ℓ,0) for every nonzero number ℓ such that LL(A,U)(ℓ,0)(2)=A and LL(A,U)(ℓ,0)(32)=U.

Peter Vámos, 1940–2020

Peter Vámos was born at Szekesfehervar, Hungary, on 6 November 1940. As a young man, he developed a distaste for the communist regime in Hungary, and his participation in anti-communist activities had repercussions that caused him to flee to England, leaving his close relatives behind. He arrived in Sheffield in the mid-1960s. He was able to enrol as a postgraduate student at Sheffield University under the supervision of Douglas Northcott, who nurtured Peter's love of algebra and encouraged him and David Sharpe to turn their notes from a seminar on ‘Injective modules’ into a book. In 1968, Peter was awarded the degree of PhD, discovered the first example of a non-representable matroid of rank 4 on a set of 8 elements (this matroid is the subject of a Wikipedia article entitled ‘Vámos matroid’) and successfully applied for a Junior Research Fellowship in Pure Mathematics at Sheffield University. In this way, Peter began to build a career in academia in the UK. He served as Lecturer (later Reader) in Pure Mathematics at Sheffield University, and then in 1983 succeeded David Rees as Professor of Pure Mathematics at Exeter University. The main focus of his resesarch was in commutative algebra, and in that subject Peter is well known for his substantial contribution to the 1976 answer to a question raised more than 20 years earlier by Irving Kaplansky, namely: which commutative rings have the property that all their finitely generated modules are direct sums of cyclic modules?

Feasibility criteria for high-multiplicity partitioning problems

For fixed weights w 1 , ⋯ , w n , and for d > 0 , we let B denote a collection of d ⋅ n balls, with d balls of weight w i for each i = 1 , ⋯ , n . We consider the problem of assigning the balls to n bins with capacities C 1 , ⋯ , C n , in such a way that each bin is assigned d balls, without exceeding its capacity. When d ≫ 0 , we give sufficient criteria for the feasibility of this problem, which coincide up to explicit constants with the natural set of necessary conditions. Furthermore, we show that our constants are optimal when the weights w i are distinct. The feasibility criteria that we present here are used elsewhere (in commutative algebra) to study the asymptotic behavior of the Castelnuovo–Mumford regularity of symmetric monomial ideals.

Spectral operation in locally convex algebras

We show that if A is a spectrally bounded algebra, then all functions operate spectrally on A if and only if SpAx is finite for every x ∈ A. We also prove that if A is a commutative Q-l.m.c.a, then all functions operate spectrally on A if and only if A/RadA is algebraic. Furthermore, if A is a semi-simple commutative Q-l.m.c.a. which is a Baire space, all functions operate spectrally on A if and only if it is isomorphic to Cn for some n ∈ N. A structure result concerning semi-simple commutative complete m-convex algebras of countable dimension is also given.

Monogenic Functions with Values in Commutative Algebras of the Second Rank with Unit and the Generalized Biharmonic Equation with Double Characteristic

Monogenic Functions with Values in Commutative Algebras of the Second Rank with Unit and the Generalized Biharmonic Equation with Double Characteristic

Killing metrized commutative nonassociative algebras associated with Steiner triple systems

With each Steiner triple system there is associated a one-parameter family of commutative, nonassociative, nonunital algebras that are by construction exact, meaning that the trace of every multiplication operator vanishes, and these algebras are shown to be Killing metrized, meaning the Killing type trace-form is nondegenerate and invariant (Frobenius), and simple, except for certain parameter values. The definition of these algebras resembles that of the Matsuo algebra of the Steiner triple system, but they are different. For a Hall triple system, the associated algebra is a primitive axial algebra for a Z/2Z-graded fusion law.

Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules \[ F^2 \xrightarrow{\partial^2} F^1 \xrightarrow{\partial^1} F^0 \] such that $M \cong \ker{\partial^1}/\mathrm{im}{\partial^2}$. It runs in time $O(\sum_i |F^i|^3)$ and requires $O(\sum_i |F^i|^2)$ memory, where $|F^i|$ denotes the size of a basis of $F^i$. We observe that, given the presentation computed by our algorithm, the bigraded Betti numbers of $M$ are readily computed. Our algorithm has been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our implementation outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.

Some relational structures with polynomial growth and their associated algebras II. Finite generation.

The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every nonnegative integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $\mathbb{K}.\mathcal A$ introduced by P. J. Cameron. In a previous paper, we studied the relationship between the properties of a relational structure $R$ and those of its age algebra, particularly when $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups or the rings of quasisymmetric polynomials. The main theorem of this paper characterizes combinatorially when the age algebra is finitely generated in this setting. For tournaments, this boils down to the profile being bounded. We further investigate how far the well known algebraic properties of invariant rings and quasisymmetric polynomials extend to age algebras; notably, we explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra. For a homogeneous structure with a profile bounded by a polynomial, Cameron conjectured in the early eighties that the profile is asymptotically polynomial; Macpherson further conjectured that the age algebra is finitely generated. This was proven recently by Falque and the second author. The combined results support the conjecture that---assuming finite kernel---profiles bounded by a polynomial are asymptotically polynomial, and give hope for a complete characterization of when the age algebra is finitely generated.

Representation homology of simply connected spaces

Let G $G$ be an affine algebraic group defined over a field k $k$ of characteristic 0. We study the derived moduli space of G $G$ -local systems on a pointed connected CW complex X $X$ trivialized at the basepoint of X $X$ . This derived moduli space is represented by an affine DG scheme R Loc G ( X , * ) $ \bm{\mathrm{R}}\mathrm{Loc}_G(X,\ast )$ : we call the (co)homology of the structure sheaf of R Loc G ( X , * ) $ \bm{\mathrm{R}}\mathrm{Loc}_G(X,\ast )$ the representation homology of X $X$ in G $G$ and denote it by HR * ( X , G ) $ \mathrm{HR}_\ast (X,G)$ . The 0-dimensional homology, HR 0 ( X , G ) $ \mathrm{HR}_0(X,G)$ , is isomorphic to the coordinate ring of the G $G$ -representation variety Rep G [ π 1 ( X ) ] $ {\rm {Rep}}_G[\pi _1(X)]$ of the fundamental group of X $X$ — a well-known algebro-geometric invariant that plays a role in many areas of topology. The higher representation homology is much less studied. In particular, when X $X$ is simply connected, HR 0 ( X , G ) $ \mathrm{HR}_0(X,G)$ is trivial but HR ∗ ( X , G ) $ \mathrm{HR}_*(X,G)$ is still an interesting rational invariant of X $X$ that depends on the Lie algebra of G $G$ . In this paper, we use Quillen's rational homotopy theory to compute the representation homology of an arbitrary simply connected space (of finite rational type) in terms of its Lie and Sullivan algebraic models. When G $G$ is reductive, we also compute HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ , the G $G$ -invariant part of representation homology, and study the question when HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ is free of locally finite type as a graded commutative algebra. This question turns out to be related to the so-called Strong Macdonald Conjecture, a celebrated result in representation theory proposed (as a conjecture) by Feigin and Hanlon in the 1980s and proved by Fishel, Grojnowski and Teleman in 2008. Reformulating the Strong Macdonald Conjecture in topological terms, we give a simple characterization of spaces X $X$ for which HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ is a graded symmetric algebra for any complex reductive group G $G$ .

A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra

A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra