相关论文: Relative Gr\"obner bases of modules and applicatio…
In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely…
Multipersistence homology modules were introduced by G.Carlsson and A.Zomorodian which gave, together with G.Singh, an algorithm to compute their Groebner bases. Although their algorithm has polynomial complexity when the chain modules are…
We investigate homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. A central theme is the study of criteria for freeness and reflexivity derived from the…
We develop a Gr\"obner basis theory for a class of algebras that generalizes both PBW-algebras and rings of differential algebras on smooth varieties. Emphasis lies on methods to compute filtrations and graded structures defined by weight…
We study existence and computability of finite bases for ideals of polynomials over infinitely many variables. In our setting, variables come from a countable logical structure A, and embeddings from A to A act on polynomials by renaming…
In the present paper we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding…
Using recent work by Erman-Sam-Snowden, we show that finitely generated ideals in the ring of bounded-degree formal power series in infinitely many variables have finitely generated Gr\"obner bases relative to the graded reverse…
In this paper we give versions of Hilbert's syzygy theorem for finitely generated modules over polynomial rings over direct product of principal ideal rings.
Persistence modules serve as the algebraic foundation for topological data analysis, typically studied as representations of posets over a field. This article extends the structural and decomposition theory of persistence modules to the…
This is an English translation of the author's Ph.D. thesis, accumulating his results on a construction of Cohen-Macaulay modules over a polynomial ring that appeared in the study of Cauchy-Fueter equations. This construction is generalized…
Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n(R) \subset GL_n(R)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U_n(R)$ vary with…
Boij-S\"oderberg theory characterizes syzygies of graded modules and sheaves on projective space. This paper continues earlier work with S. Sam, extending the theory to the setting of $GL_k$-equivariant modules and sheaves on Grassmannians.…
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…
In this paper we will define analogs of Gr\"obner bases for $R$-subalgebras and their ideals in a polynomial ring $R[x_1,\ldots,x_n]$ where $R$ is a noetherian integral domain with multiplicative identity and in which we can determine ideal…
It is known that the Frobenius algebra of the injective hull of the residue field of a complete Stanley--Reisner ring (i.e. a formal power series ring modulo a squarefree monomial ideal) can be only principally generated or infinitely…
The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary…
A flat-injective presentation of a multiparameter persistence module $M$ characterizes $M$ as the image of a morphism from a flat to an injective persistence module. Like flat or injective presentations, flat-injective presentations can be…
In this note we study the logarithmic derivation module of a non-free arrangement. We prove a generalized addition theorem for all arrangements. This addition theorem allows us to find various relationships between non-free arrangements,…
Fix a pair of positive integers d and n. We create a ring R and a complex G of R-modules with the following universal property. Let P be a polynomial ring in d variables over a field and let I be a grade d Gorenstein ideal in P which is…
We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner…