Applications and homological properties of local rings with decomposable maximal ideals

We construct a local Cohen–Macaulay ring R with a prime ideal $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC). Given any positive integer n, we also construct a local Cohen–Macaulay ring R with a prime ideal $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R has exactly two non-isomorphic semidualizing modules, but the localization $R_{\mathfrak{p}}$ has $2^n$ non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen–Macaulay fiber products of finite Cohen–Macaulay type.     

On invariant fields of vectors and covectors

Let ${\mathbb{F}}_q$ be the finite field of order q. Let G be one of the three groups $\mathrm{GL}(n,{\mathbb{F}}_q)$, $\mathrm{SL}(n,{\mathbb{F}}_q)$ or $\mathrm{U}(n,{\mathbb{F}}_q)$ and let W be the standard n-dimensional representation of G. For non-negative integers m and d we let $mW\oplus d{W}^{?}$ denote the representation of G given by the direct sum of m vectors and d covectors. We exhibit a minimal set of homogeneous invariant polynomials $\{{?}_1,{?}_2,\dots ,{?}_{(m+d)n}\}?{\mathbb{F}}_q{[mW\oplus d{W}^{?}]}^G$ such that ${\mathbb{F}}_q{(mW\oplus d{W}^{?})}^G={\mathbb{F}}_q({?}_1,{?}_2,\dots ,{?}_{(m+d)n})$ for all cases except when $md=0$ and $G=\mathrm{GL}(n,{\mathbb{F}}_q)$ or $\mathrm{SL}(n,{\mathbb{F}}_q)$.     

The symbolic defect of an ideal

Let I be a homogeneous ideal of $\mathbb{k}[x_0,\dots ,x_n]$. To compare $I^{(m)}$, the m-th symbolic power of I, with $I^m$, the regular m-th power, we introduce the m-th symbolic defect of I, denoted $\mathrm{sdefect}(I,m)$. Precisely, $\mathrm{sdefect}(I,m)$ is the minimal number of generators of the R-module $I^{(m)}/I^m$, or equivalently, the minimal number of generators one must add to $I^m$ to make $I^{(m)}$. In this paper, we take the first step towards understanding the symbolic defect by considering the case that I is either the defining ideal of a star configuration or the ideal associated to a finite set of points in ${\mathbb{P}}^2$. We are specifically interested in identifying ideals I with $\mathrm{sdefect}(I,2)=1$.     

Noetherian-like properties in polynomial and power series rings

There are many Noetherian-like rings. Among them, we are interested in SFT-rings, piecewise Noetherian rings, and rings with Noetherian prime spectrum. Some of them are stable under polynomial extensions but none of them are stable under power series extensions. We give partial answers to some open questions related with stabilities of such rings. In particular, we show that any mixed extensions $R[X_1??[X_n?$ over a zero-dimensional SFT ring R are also SFT-rings, and that if R is an SFT-domain such that $R/P$ is integrally closed for each prime ideal P of R, then $R[X]$ is an SFT-ring. We also give a direct proof that if R is an SFT Prüfer domain, then $R[X_1,?,X_n]$ is an SFT-ring. Finally, we show that the power series extension $R?X?$ over a Prüfer domain R is piecewise Noetherian if and only if R is Noetherian.     

The structure of DGA resolutions of monomial ideals

Let $I?\mathbb{k}[x_1,\dots ,x_n]$ be a squarefree monomial ideal in a polynomial ring. In this paper we study multiplications on the minimal free resolution $\mathbb{F}$ of $\mathbb{k}[x_1,\dots ,x_n]/I$. In particular, we characterize the possible vectors of total Betti numbers for such ideals which admit a differential graded algebra (DGA) structure on $\mathbb{F}$. We also show that under these assumptions the maximal shifts of the graded Betti numbers are subadditive.On the other hand, we present an example of a strongly generic monomial ideal which does not admit a DGA structure on its minimal free resolution. In particular, this demonstrates that the Hull resolution and the Lyubeznik resolution do not admit DGA structures in general.Finally, we show that it is enough to modify the last map of $\mathbb{F}$ to ensure that it admits the structure of a DG algebra.     

Mixed quantum skew Howe duality and link invariants of type A

We define a ribbon category $\mathsf{Sp}(\beta )$, depending on a parameter β, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $\beta =m?n$ the monoidal category of representations of $U_q({\mathfrak{gl}}_{m|n})$ generated by exterior powers of the vector representation and their duals. We identify this category $\mathsf{Sp}(\beta )$ with a direct limit of quotients of a dual idempotented quantum group ${\dot{\mathbf{U}}}_q({\mathfrak{gl}}_{r+s})$, proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category $\mathsf{Sp}(\beta )$ gives a unified natural setting for defining the colored ${\mathfrak{gl}}_{m|n}$ link invariant (for $\beta =m?n$) and the colored HOMFLY-PT polynomial (for β generic).     

Gaps in the number of generators of monomial Togliatti systems

Let $I_{d,n}?k[x_0,?,x_n]$ be a minimal monomial Togliatti system of forms of degree d. In [4], Mezzetti and Miró-Roig proved that the minimal number of generators $\mu (I_{d,n})$ of $I_{d,n}$ lies in the interval $[2n+1,\left(\begin{array}{c}n+d?1\\ n?1\end{array}\right)]$. In this paper, we prove that for $n\ge 4$ and $d\ge 3$, the integer values in $[2n+3,3n?1]$ cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. We classify minimal monomial Togliatti systems $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{d,n})=2n+2$ or 3n (i.e. with the minimal number of generators reaching the border of the non-existence interval). Finally, we prove that for $n=4$, $d\ge 3$ and $\mu \in [9,\left(\begin{array}{c}d+3\\ 3\end{array}\right)]?\{11\}$ there exists a minimal monomial Togliatti system $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{n,d})=\mu$.     

A uniform bound of reducibility index of good parameter ideals for certain class of modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring and M a finitely generated R-module. The invariants $\mathrm{p}(M)$ and $\mathrm{sp}(M)$ of M were introduced in [3] and [17] in order to measure the non-Cohen–Macaulayness and the non-sequential-Cohen–Macaulayness of M, respectively. Let $M=D_0?D_1?\dots ?D_k$ be the filtration of M such that $D_i$ is the largest submodule of M of dimension less than $\dim ?D_{i?1}$ for all $i\le k$ and $\mathrm{p}(D_k)\le 1$. In this paper we prove that if $\mathrm{sp}(M)\le 1$, then there exists a constant c such that ${\mathrm{ir}}_M(\mathfrak{q}M)\le c$ for all good parameter ideals $\mathfrak{q}$ of M with respect to this filtration. Here ${\mathrm{ir}}_M(\mathfrak{q}M)$ is the reducibility index of $\mathfrak{q}$ on M. This is an extension of the main results of [19], [20], [24].     

Cancellation of projective modules over polynomial extensions over a two-dimensional ring

Let R be a commutative Noetherian ring of dimension two with $1/2\in R$ and let $A=R[X_1,?,X_n]$. Let P be a projective A-module of rank 2. In this article, we prove that P is cancellative if ${\wedge }^2(P)\oplus A$ is cancellative.     

Permutations with small maximal k-consecutive sums

Let $n$ and $k$ be positive integers with $n>k$. Given a permutation $({\pi }_1,\dots ,{\pi }_n)$ of integers $1,\dots ,n$, we consider $k$-consecutive sums of $\pi$, i.e., $s_i?{\sum }_{j=0}^{k?1}{\pi }_{i+j}$ for $i=1,\dots ,n$, where we let ${\pi }_{n+j}={\pi }_j$. What we want to do in this paper is to know the exact value of $\mathsf{msum}(n,k)?min\left\{max\{s_i:i=1,\dots ,n\}?\frac{k(n+1)}{2}:\pi \in S_n\right\},$ where $S_n$ denotes the set of all permutations of $1,\dots ,n$. In this paper, we determine the exact values of $\mathsf{msum}(n,k)$ for some particular cases of $n$ and $k$. As a corollary of the results, we obtain $\mathsf{msum}(n,3)$, $\mathsf{msum}(n,4)$ and $\mathsf{msum}(n,6)$ for any $n$.     

Torsion completions are bounded

Given a commutative ring A and a finitely generated ideal I, we prove that $I^{\infty }$-torsion A-modules that are also I-adically complete (or merely derived I-complete) must have bounded $I^{\infty }$-torsion, i.e., they are killed by $I^n$ for some $n\ge 0$.     

Powers of ideals associated to (C4,2K2)-free graphs

Let G be a $(C_4,2K_2)$-free graph with edge ideal $I(G)?\mathbb{k}[x_1,\dots ,x_n]$. We show that $I{(G)}^s$ has linear resolution for every $s\ge 2$. Also, we show that every power of the vertex cover ideal of G has linear quotients. As a result, we describe the Castelnuovo–Mumford regularity of powers of $I{(G)}^{\vee }$ in terms of the maximum degree of G.     

A local–global principle for surjective polynomial maps

Let R be an affine domain of characteristic zero with finite quotients. We prove that a polynomial map over R is surjective if and only if it is surjective over $\stackrel{?}{R_{\mathfrak{m}}}$, the completion of R with respect to $\mathfrak{m}$, for every maximal ideal $\mathfrak{m}?R$. In fact, the completions $\stackrel{?}{R_{\mathfrak{m}}}$ may be replaced by arbitrary subrings containing R. We use this result to yield a characterization of surjective polynomial maps, and remark that there does not exist a similar principle for injective polynomial maps.     

Left dihedral codes over finite chain rings

Let R be a finite commutative chain ring, $D_{2n}$ be the dihedral group of size 2n and $R[D_{2n}]$ be the dihedral group ring. In this paper, we completely characterize left ideals of $R[D_{2n}]$ (called left $D_{2n}$-codes) when $\gcd (char(R),n)=1$. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over $R\times S$, where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where $R={\mathbb{F}}_{p^m}$ is a Galois field, we give a classification for left $D_{2N}$-codes over ${\mathbb{F}}_{p^m}$, for any positive integer N. In both cases we determine dual codes and identify self-dual ones.     

On modal logics arising from scattered locally compact Hausdorff spaces

For a topological space X, let $\mathsf{L}(X)$ be the modal logic of X where □ is interpreted as interior (and hence ◇ as closure) in X. It was shown in [3] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, ${\mathsf{S4}.\mathsf{Grz}}_n$ ($n\ge 1$), and their intersections arise as $\mathsf{L}(X)$ for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to [3, Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either S4.Grz or ${\mathsf{S4}.\mathsf{Grz}}_n$ for some $n\ge 1$. In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space.