Mathematics

We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory of the corresponding monoid T(R) . Results of Levy-Wiegand and Levy-Odenthal together with a study of the local case yield an explicit description of T(R) . The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations--a natural class of monoids serving as combinatorial models for the factorization theory of T(R) . As a consequence, the monoid T(R) is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of T(R) and characterize when T(R) is half-factorial. (Factoriality, that is, torsion-free Krull-Remak-Schmidt-Azumaya, is characterized by a theorem of Levy-Odenthal.) The monoids of graph agglomerations introduced here are also of independent interest.

An atomic monoid M is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element x?�M no two distinct factorizations of x have the same length. The notion of length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved that in the setting of integral domains, length-factoriality can be taken as an alternative definition of a unique factorization domain. However, being a length-factorial monoid is in general weaker than being a factorial monoid (i.e., a unique factorization monoid). Here we further investigate length-factoriality. First, we offer two characterizations of a length-factorial monoid M , and we use such characterizations to describe the set of Betti elements and obtain a formula for the catenary degree of M . Then we study the connection between length-factoriality and purely long (resp., purely short) irreducibles, which are irreducible elements that appear in the longer (resp., shorter) part of any unbalanced factorization relation. Finally, we prove that an integral domain cannot contain purely short and a purely long irreducibles simultaneously, and we construct a Dedekind domain containing purely long (resp., purely short) irreducibles but not purely short (resp., purely long) irreducibles.

Levelness and almost Gorensteinness are well-studied properties on graded rings as a generalized notion of Gorensteinness. In the present paper, we study those properties for the edge rings of the complete multipartite graphs, denoted by k[ K r 1 ,?? r n ] with 1??r 1 ?�⋯??r n . We give the complete characterization of which k[ K r 1 ,?? r n ] is level in terms of n and r 1 ,?? r n . Similarly, we also give the complete characterization of which k[ K r 1 ,?? r n ] is almost Gorenstein in terms of n and r 1 ,?? r n .

We show that under some natural conditions, we are able to lift an n -dimensional spectral resolution from one monotone σ -complete unital po-group into another one, when the first one is a σ -homomorphic image of the second one. We note that an n -dimensional spectral resolution is a mapping from R n into a quantum structure which is monotone, left-continuous with non-negative increments and which is going to 0 if one variable goes to −∞ and it goes to 1 if all variables go to +∞ . Applying this result to some important classes of effect algebras including also MV-algebras, we show that there is a one-to-one correspondence between n -dimensional spectral resolutions and n -dimensional observables on these effect algebras which are a kind of σ -homomorphisms from the Borel σ -algebra of R n into the quantum structure. An important used tool are two forms of the Loomis--Sikorski theorem which use two kinds of tribes of fuzzy sets. In addition, we show that we can define three different kinds of n -dimensional joint observables of n one-dimensional observables.

The long standing Lech's conjecture in commutative algebra states that for a flat local extension (R,m)→(S,n) of Noetherian local rings, we have an inequality on the Hilbert--Samuel multiplicities: e(R)≤e(S) . In general the conjecture is wide open as long as dimR>3 , even in equal characteristic. In this paper, we prove Lech's conjecture in all dimensions, provided (R,m) is a standard graded ring over a perfect field (localized at the homogeneous maximal ideal). We introduce the notions of lim Ulrich and weakly lim Ulrich sequence. Roughly speaking these are sequences of finitely generated modules that are not necessarily Cohen--Macaulay, but asymptotically behave like Ulrich modules. We prove that the existence of these sequences imply Lech's conjecture. Though the existence of Ulrich modules is known in very limited cases, we construct weakly lim Ulrich sequences for all standard graded domains over perfect field of positive characteristic.

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo-Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this we show the before-now unknown convergence of Stanley depths of integral closure powers. In addition, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.

The main goal of this paper is to characterize limit key polynomials for a valuation ν on K[x] . We consider the set Ψ α of key polynomials for ν of degree α . We set p be the exponent characteristic of ν . Our first main result (Theorem 1.1) is that if Q α is a limit key polynomial for Ψ α , then the degree of Q α is p r α for some r∈N . Moreover, in Theorem 1.2, we show that there exist Q∈ Ψ α and Q α a limit key polynomial for Ψ α , such that the Q -expansion of Q α only has terms which are powers of p .

In this paper, we study ideals I whose linear strand can be supported on a regular CW complex. We provide a sufficient condition for the linear strand of an arbitrary subideal of I to remain supported on an easily described subcomplex. In particular, we prove that a certain class of rainbow monomial ideals always have linear strand supported on a regular CW complex, including any initial ideal of the ideal of maximal minors of a generic matrix. We also provide a sufficient condition for these ideals to have linear resolution, which is also an equivalence under mild assumptions. We then employ a result of Almousa, Fløystad, and Lohne to apply these results to polarizations of Artinian monomial ideals. We conclude with further questions relating to cellularity of certain classes of squarefree monomial ideals and the relationship between initial ideals of maximal minors and algebra structures on certain resolutions.

A determinantal facet ideal (DFI) is an ideal J ? generated by maximal minors of a generic matrix parametrized by an associated simplicial complex ? . In this paper, we construct an explicit linear strand for the initial ideal with respect to any diagonal term order < of an arbitrary DFI. In particular, we show that if ? has no \emph{1-nonfaces}, then the Betti numbers of the linear strand of J ? and its initial ideal coincide. We apply this result to prove a conjecture of Ene, Herzog, and Hibi on Betti numbers of closed binomial edge ideals in the case that the associated graph has at most 2 maximal cliques. More generally, we show that the linear strand of the initial ideal (with respect to < ) of \emph{any} DFI is supported on a polyhedral cell complex obtained as an induced subcomplex of the \emph{complex of boxes}, introduced by Nagel and Reiner.

We introduce a construction, called linearization, that associates to any monomial ideal I an ideal Lin(I) in a larger polynomial ring. The main feature of this construction is that the new ideal Lin(I) has linear quotients. In particular, since Lin(I) is generated in a single degree, it follows that Lin(I) has a linear resolution. We investigate some properties of this construction, such as its interplay with classical operations on ideals, its Betti numbers, functoriality and combinatorial interpretations. We moreover introduce an auxiliary construction, called equification, that associates to any monomial ideal a new monomial ideal generated in a single degree, in a polynomial ring with one more variable. We study some of the homological and combinatorial properties of the equification, which can be seen as a monomial analogue of the well-known homogenization construction.