The projective dimension of three cubics is at most 5

Let R be a polynomial ring over a field and I an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension $\mathrm{pd}(R/I)$ of $R/I$ is at most 36, although the example with largest projective dimension he constructed has $\mathrm{pd}(R/I)=5$. Based on computational evidence, it had been conjectured that $\mathrm{pd}(R/I)\le 5$. In the present paper we prove this conjectured sharp bound.     

Endomorphisms and cores of quadratic forms graphs in odd characteristic

A graph G is called a pseudo-core if every endomorphism of G is either an automorphism or a colouring. A graph G is a core if every endomorphism of G is an automorphism. Let ${\mathbb{F}}_q$ be the finite field with q elements where q is a power of an odd prime number. The quadratic forms graph, denoted by $\mathrm{Quad}(n,q)$ where $n\ge 2$, has all quadratic forms on ${\mathbb{F}}_q^n$ as vertices and two vertices f and g are adjacent whenever $\mathrm{rk}(f-g)=1$ or 2. We prove that every $\mathrm{Quad}(n,q)$ is a pseudo-core. Further, when n is even, $\mathrm{Quad}(n,q)$ is a core. When n is odd, $\mathrm{Quad}(n,q)$ is not a core. On the other hand, we completely determine the independence number of $\mathrm{Quad}(n,q)$.     

The reconstruction conjecture for finite simple graphs and associated directed graphs

In this paper, we study the Reconstruction Conjecture for finite simple graphs. Let Γ and ${\mathrm{\Gamma }}^{\prime }$ be finite simple graphs with at least three vertices such that there exists a bijective map $f:V(\mathrm{\Gamma })\to V({\mathrm{\Gamma }}^{\prime })$ and for any $v\in V(\mathrm{\Gamma })$, there exists an isomorphism ${?}_v:\mathrm{\Gamma }?v\to {\mathrm{\Gamma }}^{\prime }?f(v)$. Then we define the associated directed graph $\stackrel{?}{\mathrm{\Gamma }}=\stackrel{?}{\mathrm{\Gamma }}(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime },f,{\{{?}_v\}}_{v\in V(\mathrm{\Gamma })})$ with two kinds of arrows from the graphs Γ and ${\mathrm{\Gamma }}^{\prime }$, the bijective map f and the isomorphisms ${\{{?}_v\}}_{v\in V(\mathrm{\Gamma })}$. By investigating the associated directed graph $\stackrel{?}{\mathrm{\Gamma }}$, we study when are the two graphs Γ and ${\mathrm{\Gamma }}^{\prime }$ isomorphic.     

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.     

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 $\mathrm{sdefect}(I,2)=1$.     

GE2-rings and a graph of unimodular rows

For a commutative ring A we consider a related graph, $\mathrm{\Gamma }(A)$, whose vertices are the unimodular rows of length 2 up to multiplication by units. We prove that $\mathrm{\Gamma }(A)$ is path-connected if and only if A is a ${\mathrm{GE}}_2$-ring, in the terminology of P. M. Cohn. Furthermore, if $Y(A)$ denotes the clique complex of $\mathrm{\Gamma }(A)$, we prove that $Y(A)$ is simply connected if and only if A is universal for ${\mathrm{GE}}_2$. More precisely, our main theorem is that for any commutative ring A the fundamental group of $Y(A)$ is isomorphic to the group $K_2(2,A)$ modulo the subgroup generated by symbols.     

Framed motives of smooth affine pairs

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular, the motivically fibrant Ω-resolution in positive degrees of the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }{X}_{+}$, where $X_{+}=X??$, for a smooth scheme $X\in {\mathrm{Sm}}_k$ over an infinite perfect field k, is computed.The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $({\mathbb{A}}^l\times X)/(({\mathbb{A}}^l?0)\times X)$, $X\in {\mathrm{Sm}}_k$, is one of ingredients in the theory. In the article we extend this result to the case of a pair $(X,U)$ given by a smooth affine variety X over k and an open subscheme $U?X$.The result gives an explicit motivically fibrant Ω-resolution in positive degrees for the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }(X_{+}/U_{+})$ of the quotient-sheaf $X_{+}/U_{+}$.     

Time-dependent global attractor for extensible Berger equation

In this paper, we investigate the asymptotic behavior of the nonautonomous Berger equation

$\epsilon (t)u_{tt}+{\mathrm{\Delta }}^{2}u?(Q+\underset{\mathrm{\Omega }}{\int }|\mathrm{?}u{|}^{2}dx)\mathrm{\Delta }u+g(u_t)+\phi (u)=f,t>\tau ,$

on a bounded smooth domain $\mathrm{\Omega }?{\mathbb{R}}^N$ with hinged boundary condition, where $\epsilon (t)$ is a decreasing function vanishing at infinity. Under suitable assumptions, we establish an invariant time-dependent global attractor within the theory of process on time-dependent space.     

A generalization of lifting non-proper tropical intersections

Let X and $X^{\prime }$ be closed subschemes of an algebraic torus T over a non-archimedean field. We prove the rational equivalence as tropical cycles in the sense of [11, §2] between the tropicalization of the intersection product $X?X^{\prime }$ and the stable intersection $\mathrm{trop}(X)?\mathrm{trop}(X^{\prime })$, when restricted to (the inverse image under the tropicalization map of) a connected component C of $\mathrm{trop}(X)\cap \mathrm{trop}(X^{\prime })$. This requires possibly passing to a (partial) compactification of T with respect to a suitable fan. We define the compactified stable intersection in a toric tropical variety, and check that this definition is compatible with the intersection product in [11, §2]. As a result we get a numerical equivalence between $\stackrel{￣}{X}?{\stackrel{￣}{X}}^{\prime }{|}_{\stackrel{￣}{C}}$ and $\stackrel{￣}{\mathrm{trop}(X)?\mathrm{trop}(X^{\prime }){|}_C}$ via the compactified stable intersection, where the closures are taken inside the compactifications of T and ${\mathbb{R}}^n$. In particular, when X and $X^{\prime }$ have complementary codimensions, this equivalence generalizes [15, Theorem 6.4], in the sense that $X\cap X^{\prime }$ is allowed to be of positive dimension. Moreover, if $\stackrel{￣}{X}\cap {\stackrel{￣}{X}}^{\prime }$ has finitely many points which tropicalize to $\stackrel{￣}{C}$, we prove a similar equation as in [15, Theorem 6.4] when the ambient space is a reduced subscheme of T (instead of T itself).     

Existence of solutions of a non-linear eigenvalue problem with a variable weight

We study the non-linear minimization problem on $H_0^1(\mathrm{\Omega })?L^q$ with $q=\frac{2n}{n?2}$, $\alpha >0$ and $n\ge 4$:

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }a(x,u)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2$

where $a(x,s)$ presents a global minimum α at $(x_0,0)$ with $x_0\in \mathrm{\Omega }$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behavior of $a(x,s)$ with respect to s. The model case is

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }(\alpha +|x{|}^{\beta }|u{|}^k)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2.$

In a previous paper dedicated to the same problem with $\lambda =0$, we showed that minimizers exist only in the range $\beta <kn/q$, which corresponds to a dominant non-linear term. On the contrary, the linear influence for $\beta \ge kn/q$ prevented their existence. The goal of this present paper is to show that for $0<\lambda \le \alpha {\lambda }_1(\mathrm{\Omega })$, $0\le k\le q?2$ and $\beta >kn/q+2$, minimizers do exist.