Mathematics

For a monomial ideal I , we consider the i th homological shift ideal of I , denoted by HS i (I) , that is, the ideal generated by the i th multigraded shifts of I . Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal I and any monomial prime ideal P , HS i (I(P))⊆ HS i (I)(P) for all i , where I(P) is the monomial localization of I . In particular, we consider the homological shift ideal of some families of monomial ideals with linear quotients. For any c -bounded principal Borel ideal I and for the edge ideal of complement of any path graph, it is proved that HS i (I) has linear quotients for all i . As an example of c -bounded principal Borel ideals, Veronese type ideals are considered and it is shown that the homological shift ideal of these ideals are polymatroidal. This implies that for any polymatroidal ideal which satisfies the strong exchange property, HS j (I) is again a polymatroidal ideal for all j . Moreover, for any edge ideal with linear resolution, the ideal HS j (I) is characterized and it is shown that HS 1 (I) has linear quotients.

New sequences of hyperoperations \cite{BE15,HI26,ACK28,GO47,TAR69} are presented together with their local algebraic properties. The commutative hyperoperations reported by Bennet \cite{BE15} are presented as a sequence of monoids. After identifying the semirings along the sequence, the corresponding fields are constructed via inverse completion.

Given Σ⊂R:=K[ x 1 ,…, x k ] , where K is a field of characteristic 0, any finite collection of linear forms, some possibly proportional, and any 1≤a≤|Σ| , we prove that I a (Σ) , the ideal generated by all a -fold products of Σ , has linear graded free resolution. This allows us to determine a generating set for the defining ideal of the Orlik-Terao algebra of the second order of a line arrangement in P 2 K , and to conclude that for the case k=3 , and Σ defining such a line arrangement, the ideal I |Σ|−2 (Σ) is of fiber type. We also prove several conjectures of symbolic powers for defining ideals of star configurations of any codimension c .

We consider the smallest subring D of R(X) containing every element of the form 1/(1+ x 2 ) , with x∈R(X) . D is a Prüfer domain called the minimal Dress ring of R(X) . In this paper, addressing a general open problem for Prüfer non Bézout domains, we investigate whether 2×2 singular matrices over D can be decomposed as products of idempotent matrices. We show some conditions that guarantee the idempotent factorization in M 2 (D) .

We describe the immediate extensions of a one dimensional valuation ring V which could be embedded in some separation of a ultrapower of V with respect to a certain ultrafilter. For such extensions holds a kind of Artin's approximation.

The goal of the present paper is the study of some algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension d?? . We prove that the inequality d?�reg(?)?�type(?) holds for any (d??) -dimensional Cohen-Macaulay simplicial complex ? satisfying ?=core(?) , where reg(?) (resp. type(?) ) denotes the Castelnuovo-Mumford regularity (resp. Cohen-Macaulay type) of the Stanley-Reisner ring k[?] . Moreover, for any given integers d,r,t satisfying r,t?? and r?�d?�rt , we construct a Cohen-Macaulay simplicial complex ?(G) as an independent complex of a graph G such that dim(?(G))=d?? , reg(?(G))=r and type(?(G))=t .

This is a preliminary draft of Chapter 6 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608:05735. Chapters 4-5 has been posted as arXiv:1707.07190. This installment contains: Chapter 6. Cluster structures in commutative rings

Let S⊂R be an arbitrary subset of a unique factorization domain R and $\K$ be the field of fractions of R . The ring of integer-valued polynomials over S is the set Int(S,R)={f∈K[x]:f(a)∈R ∀ a∈S}. This article is an effort to study the irreducibility of integer-valued polynomials over arbitrary subsets of a unique factorization domain. We give a method to construct special kinds of sequences, which we call d -sequences. We then use these sequences to obtain a criteria for the irreducibility of the polynomials in Int(S,R). In some special cases, we explicitly construct these sequences and use these sequences to check the irreducibility of some polynomials in Int(S,R). At the end, we suggest a generalization of our results to an arbitrary subset of a Dedekind domain.

Let D be an integral domain and L be a field containing D . We study the isolated points of the Zariski space Zar(L|D) , with respect to the constructible topology. In particular, we completely characterize when L (as a point) is isolated and, under the hypothesis that L is the quotient field of D , when a valuation domain of dimension 1 is isolated; as a consequence, we find all isolated points of Zar(D) when D is a Noetherian domain. We also show that if V is a valuation domain and L is transcendental over V then the set of extensions of V to L has no isolated points.

Let R be a standard graded polynomial ring that is finitely generated over a field, and let I be a homogenous prime ideal of R . Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of R/ I t , as t grows arbitrarily large. Such rings are known as thickenings of R/I . We consider R=F[X] where F is a field of characteristic 0, X is a 2×m matrix, and I is the ideal generated by size two minors. We give concrete constructions for the local cohomology modules of thickenings of R/I . Bizarrely, these local cohomology modules can be described using the Taylor series of natural log.