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Published: 2018-02-19T20:30:00-05:00
We prove upper and lower bounds for leading coefficient of Kolchin dimension polynomial of systems of partial linear differential equations in codimension two.
We consider artinian algebras $A=\mathbb{C}[x_0,\ldots,x_m]/I$, with $I$ generated by a regular sequence of homogeneous forms of the same degree $d\geq 2$. We show that the multiplication by a general linear form from $A_{d-1}$ to $A_d$ is injective. We prove that the Weak Lefschetz Property holds for artinian complete intersection algebras as above, with $d=2$ and $m\leq 4$. Apparently, this was previously known only for $m\leq 3$. Although we are proposing only very limited progress towards the WLP conjecture for complete intersections, we hope that the methods of the present article can illustrate some geometrical aspects of the general problem.
In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to vertex weighted rooted forests and oriented cycles. As some consequences, we give some exact formulas for the depth of these ideals.
We introduce and investigate the category of factorization of a multiplicative, commutative, cancellative, pre-ordered monoid $A$, which we denote $\mathcal{F}(A)$. We pay particular attention to the divisibility pre-order and to the monoid $A=D\setminus\{0\}$ where $D$ is an integral domain. The objects of $\mathcal{F}(A)$ are factorizations of elements of $A$, and the morphisms in $\mathcal{F}(A)$ encode combinatorial similarities and differences between the factorizations.
Among other results, we show that $\mathcal{F}(A)$ is a category with weak equivalences and compute the associated homotopy category. Also, we use this construction to characterize various factorization properties of integral domains: atomicity, unique factorization, and so on.
Let R be a local Noetherian commutative ring. We prove that R is an Artinian Gorenstein ring if and only if every ideal in R is a trace ideal. We discuss when the trace ideal of a module coincides with its double annihilator.
We describe the typical homological properties of monomial ideals defined by random generating sets. We show that, under mild assumptions, random monomial ideals (RMI's) will almost always have resolutions of maximal length; that is, the projective dimension will almost always be $n$, where $n$ is the number of variables in the polynomial ring. We give a rigorous proof that Cohen-Macaulayness is a "rare" property. We characterize when an RMI is generic/strongly generic, and when it "is Scarf"---in other words, when the algebraic Scarf complex of $M\subset S=k[x_1,\ldots,x_n]$ gives a minimal free resolution of $S/M$. As a result we see that, outside of a very specific ratio of model parameters, RMI's are Scarf only when they are generic. We end with a discussion of the average magnitude of Betti numbers.
We bring additional support to the conjecture saying that a rational cuspidal plane curve is either free or nearly free. This conjecture was confirmed for curves of even degree, and in this note we prove it for many odd degrees. In particular, we show that this conjecture holds for the curves of degree at most 34.
We prove that the polynomial invariants of a permutation group are Cohen-Macaulay for any choice of coefficient field if and only if the group is generated by transpositions, double transpositions, and 3-cycles. This unites and generalizes several previously known results. The "if" direction of the argument uses Stanley-Reisner theory and a recent result of Christian Lange in orbifold theory. The "only-if" direction uses a local-global result based on a theorem of Raynaud to reduce the problem to an analysis of inertia groups, and a combinatorial argument to identify inertia groups that obstruct Cohen-Macaulayness.
For an $F$-finite scheme $X$ separated over a perfect field $k$ of characteristic $p>0$ which admits an embedding into a smooth $k$-scheme, we establish an equivalence between the bounded derived categories of Cartier crystals on $X$ and constructible $\mathbb{Z}/p\mathbb{Z}$-sheaves on the \'{e}tale site $X_{\text{\'{e}t}}$. The key intermediate step is to extend the category of locally finitely generated unit $\mathcal{O}_{F,X}$-modules for smooth schemes introduced by Emerton and Kisin to embeddable schemes. On the one hand, this category is equivalent to Cartier crystals. On the other hand, by using Emerton-Kisin's Riemann-Hilbert correspondence, we show that it is equivalent to Gabber's category of perverse sheaves in $D_c^b(X_{\text{\'{e}t}},\mathbb{Z}/p\mathbb{Z})$. Furthermore, we define intermediate extensions for Cartier crystals and show that our equivalence between Cartier crystals and perverse constructible \'{e}tale sheaves commutes with the intermediate extension functor.
We study modules for the divided power algebra $D$ in a single variable over a commutative noetherian ring $k$. Our first result states that $D$ is a coherent ring. In fact, we show that there is a theory of Gr\"obner bases for finitely generated ideals, and so computations with finitely presented $D$-modules are in principle algorithmic. We go on to determine much about the structure of finitely presented $D$-modules, such as: existence of certain nice resolutions, computation of the Grothendieck group, results about injective dimension, and how they interact with torsion modules. Our results apply not just to the classical divided power algebra, but to its $q$-variant as well, and even to a much broader class of algebras we introduce called "generalized divided power algebras." On the other hand, we show that the divided power algebra in two variables over $\mathbf{Z}_p$ is not coherent.
The set of periodic distributions, with usual addition and convolution, forms a ring, which is isomorphic, via taking a Fourier series expansion, to the ring ${\mathcal{S}}'({\mathbb{Z}}^d)$ of sequences of at most polynomial growth with termwise operations. In this article, we establish several algebraic properties of these rings.
We study some properties of a family of rings $R(I)_{a,b}$ that are obtained as quotients of the Rees algebra associated with a ring $R$ and an ideal $I$. In particular, we give a complete description of the spectrum of every member of the family and describe the localizations at a prime ideal. Consequently, we are able to characterize the Cohen-Macaulay and Gorenstein properties, generalizing known results stated in the local case. Moreover, we study when $R(I)_{a,b}$ is an integral domain, reduced, quasi-Gorenstein, or satisfies Serre's conditions.
Four sets of necessary and sufficient conditions are obtained for the first-order rigidity of a crystallographic bond-node framework C in R^d. In particular, an extremal rank characterisation is obtained which incorporates a multi-variable matrix-valued transfer function Psi_\C(z) defined on the product space C^d_* = (C \{0})^d. The first-order flex space of a crystal framework is shown to be finite-dimensional if and only if its geometric spectrum, associated with Psi_\C(z), is a finite set in C^d_*. More generally the first-order flex space of a crystal framework is shown to be the closed linear span of a set of vector-valued polynomially weighted geometric multi-sequences whose geometric multi-factors in C^d_* lie in a finite set. Paradoxically, first-order rigid crystal frameworks may possess nontrivial continuous motions which (necessarily) are non-differentiable. The examples given are associated with aperiodic displacive phase transitions between periodic states.