This paper lays out a foundation for a theory of supertropical algebraic geometry, relying on commutative $ν$-algebra. To this end, the paper introduces $\mathfrak{q}$-congruences, carried over $ν$-semirings, whose distinguished ghost and tangible clusters allow both quotienting and localization. Utilizing these clusters, $\mathfrak{g}$-prime, $\mathfrak{g}$-radical, and maximal $\mathfrak{q}$-congruences are naturally defined, satisfying the classical relations among analogous ideals. Thus, a foundation of systematic theory of commutative $ν$-algebra is laid. In this framework, the underlying spaces for a theoretic construction of schemes are spectra of $\mathfrak{g}$-prime congruences, over which the correspondences between $\mathfrak{q}$-congruences and varieties emerge directly. Thereby, scheme theory within supertropical algebraic geometry follows the Grothendieck approach, and is applicable to polyhedral geometry.
This paper is a sequel of [IKR1], where we defined supervaluations on a commutative ring $R$ and studied a dominance relation $ϕ\geq ψ$ between supervaluations $ϕ$ and $ψ$ on $R$, aiming at an enrichment of the algebraic tool box for use in tropical geometry. A supervaluation $ϕ:R \to U$ is a multiplicative map from $R$ to a supertropical semiring $U$, cf. [IR1], [IR2], [IKR1], with further properties, which mean that $ϕ$ is a sort of refinement, or covering, of an m-valuation (= monoid valuation) $v: R \to M$. In the most important case, that $R$ is a ring, m-valuations constitute a mild gene…
We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geometric approach to tropical geometry, viewing tropical varieties as sets of roots of (supertropical) polyn…
The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows: * The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible. * There exists an adjoint matrix $\adj{A}$ such that the matrix $A \adj{A}$ behaves much like the identity matrix (times $|A|$). * Every matrix $A$ is a supertropical root of its Hamilton-Cayley polynomial $f_A$. If these roots are distinct, then $A$ is conjugate (in a certain supertropical sense) to a diagonal matrix. * The tropical d…
The objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of ``ghost surpasses.''Special attention is paid to the various notions of ``base,'' which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas the number of elements in a d-base may vary according to the d-base, it is shown that when an s-base exists, it is unique up to permutation and multiplication by scalars, and can be identified with a set of ``critical'' elements. Linear functionals and…
The only invertible matrices in tropical algebra are diagonal matrices, permutation matrices and their products. However, the pseudo-inverse $A^\nabla$, defined as $\frac{adj(A)}{det(A)}$, with $det(A)$ being the tropical permanent (also called the tropical determinant) of a matrix $A$, inherits some classical algebraic properties and has some surprising new ones. Defining $B$ and $B'$ to be tropically similar if $B' =A^\nabla BA$, we examine the characteristic (max-)polynomials of tropically similar matrices as well as those of pseudo-inverses. Other miscellaneous results include a new proof …
In this paper we present two intrinsic algebraic definitions of tropical variety motivated by the classical Zariski correspondence, one utilizing the algebraic structure of the coordinate semiring of an affine supertropical algebraic set, and the second based on the layered structure. We tie them to tropical geometry, especially in connection with the dimension of an affine variety.
Tropical mathematics often is defined over an ordered cancellative monoid $\tM$, usually taken to be $(\RR, +)$ or $(\QQ, +)$. Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. In this paper we describe an alternative structure, more compatible with valuation theory, studied by the authors over the past few years, that permits fuller use of algebraic theory especially in underst…
The method of direct computation of universal (fibred) product in the category of commutative associative algebras of finite type with unity over a field is given and proven. The field of coefficients is not supposed to be algebraically closed and can be of any characteristic. Formation of fibred product of commutative associative algebras is an algebraic counterpart of gluing algebraic schemes by means of some equivalence relation in algebraic geometry. If initial algebras are finite-dimensional vector spaces the dimension of their product obeys Grassmann-like formula. Finite-dimensional case…
In this paper, we study the André-Quillen homology of simplicial commutative $\ell$-algebras, $\ell$ a field, having certain vanishing properties. When $\ell$ has non-zero characteristic, we obtain an algebraic version of a theorem of J.-P. Serre and Y. Umeda that characterizes such simplicial algebras having bounded homotopy groups. We further discuss how this theorem fails in the rational case and, as an application, indicate how the algebraic Serre theorem can be used to resolve a conjecture of D. Quillen for algebras of finite type over Noetherian rings, which have non-zero characteristic.
This paper deals with the graded commutative rings in which every homogeneous prime ideal is contained in a unique homogeneous maximal ideal called Gelfand graded ring. The purpose is to establish some topological and algebraic characterizations of these rings, one of which is the algebraic analogue of the Urysohn's lemma. Finally we look at a special class of those graded rings called pm$^+$ graded rings which can be viewed as graded ring with a Gelfand strong property.
We discuss some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central roles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions …
Let R be a commutative ring with unity, M be an unitary R-module and Γ be a simple graph. This research article is an interplay of combinatorial and algebraic properties of M . We show a combinatorial object completely determines an algebraic object and characterise all finite abelian groups. We discuss the correspondence between essential ideals of R, submodules of M and vertices of graphs arising from M . We examine various types of equivalence relations on objects of M. We study essential ideals corresponding to elements of an object over hereditary and regular rings. Further, we study isom…
The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with m…
This paper is a review of concepts from graded commutative algebra with specific attention given to length and multiplicity. The author's motivation for this paper comes from the study of equivariant cohomology in algebraic topology where the graded commutative algebra of the module is intimately connected to topological properties of the space (as shown by Quillen). Results (and their proofs) which are often left as exercises, or considered 'folklore' in the commutative algebra community are included in this paper, as are references to relevant applications in topology. As such, this paper is…
These notes illustrates the power of formulating ideas of commutative algebra in a homotopy invariant form. They can then be applied to derived categories of rings or ring spectra. These ideas are powerful in classical algebra, in representation theory of groups, in classical algebraic topology and elsewhere. The notes grew out of a series of lectures given during the `Interactions between Representation Theory, Algebraic Topology and Commutative Algebra' (IRTATCA) at the CRM (Barcelona) in Spring 2015.
Reflexive functors of modules naturally appear in Algebraic Geometry, mainly in the theory of linear representations of group schemes, and in "duality theories". In this paper we study and determine reflexive functors and we give many properties of reflexive functors.
Given a finite set of points $Γ$ in $\mathbb P^{k-1}$ not all contained in a hyperplane, the "fitting problem" asks what is the maximum number $hyp(Γ)$ of these points that can fit in some hyperplane and what is (are) the equation(s) of such hyperplane(s). If $Γ$ has the property that any $k-1$ of its points span a hyperplane, then $hyp(Γ)=nil(I)+k-2$, where $nil(I)$ is the index of nilpotency of an ideal constructed from the homogeneous coordinates of the points of $Γ$. Note that in $\mathbb P^2$ any two points span a line, and we find that the maximum number of collinear points of any given …
Motivated by the Poisson Dixmier-Moeglin equivalence problem, a systematic study of commutative unitary rings equipped with a {\em biderivation}, namely a binary operation that is a derivation in each argument, is here begun, with an eye toward the geometry of the corresponding {\em $B$-varieties}. Foundational results about extending biderivations to localisations, algebraic extensions and transcendental extensions are established. Resolving a deficiency in Poisson algebraic geometry, a theory of base extension is achieved, and it is shown that dominant $B$-morphisms admit generic $B$-fibres.…
Fix a prime number $p$. We report on some recent developments in algebraic geometry (broadly construed) over $p$-adically complete commutative rings. These developments include foundational advances within the subject as well as external applications.
CAKR: Commutative algebra k-mer representations for genomics
King Fahd University of Petroleum and Minerals（阿卜杜勒阿齐兹国王石油大学）；Michigan State University（密歇根州立大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；University of Georgia（佐治亚大学）；Georgia Institute of Technology（佐治亚理工学院）
We prove that two finite-dimensional commutative algebras over an algebraically closed field are isomorphic if and only if they give rise to isomorphic representations of the category of finite sets and surjective maps.
