Since its original publication in 1916 under the title ”The algebraic theory of modular systems”, the book by F.S. Macaulay has attracted a lot of scientists with a view towards pure matematics (D. Eisenbud,...) or applications to control theory (U. Oberst,...). However, a carefull examination of the quotations clearly shows that people had only a look at the first three chapters respectively d...

Let [Formula: see text] be a simple undirected graph. The family of all matchings forms simplicial complex called the matching text]. Here, we give classification graphs with Gorenstein complex. Also study when is Cohen–Macaulay and, in certain classes graphs, fully characterize those which have In particular, graph girth at least five or complete Cohen–Macaulay.

Suppose that f : P → Q is a poset map whose fibers f−1(Q≤q) are sufficiently well connected. Our main result is a formula expressing the homotopy type of P in terms of Q and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences or special cases. Homology, Cohen-Macaulay, and equivariant versions are given, and some application...

Let D be a weighted oriented graph, whose underlying graph is G , and let I (D) its edge ideal. If has no 3-, 5-, or 7-cycles, K?nig, we characterize when unmixed. 3- 5-cycles, Cohen–Macaulay. We prove that unmixed if only Cohen–Macaulay girth greater than 7 K?nig 4-cycles.

We examine some recent work of Phillip Griffith on étale covers and fibered products from the point of view of tight closure theory. While it is known that cyclic covers of Gorenstein rings with rational singularities are Cohen-Macaulay, we show this is not true in general in the absence of the Gorenstein hypothesis. Specifically, we show that the canonical cover of a Q-Gorenstein ring with rat...

We study in detail locally Cohen-Macaulay curves in P 3 which are contained in a double plane 2H , thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes Hd,g(2H) of locally Cohen-Macaulay curves in 2H of degree d and arithmetic genus g, and we show that Hd,g(2H) is connected. We also discuss the Rao module of ...

Abstract We study local models that describe the singularities of Shimura varieties non-PEL type for orthogonal groups at primes where level subgroup is given by stabilizer a single lattice. In particular, we use Pappas–Zhu construction and give explicit equations an open subset around “worst” point These display affine chart model as hypersurface in determinantal scheme. Using this, prove spec...

This paper describes a trial of Macaulay and Crerar’s method of mapping a workplace soundscape [1] to assess its fitness as a basis for an extended soundscape mapping method. Twelve participants took part within 14 separate environments, which included academic, commercial and domestic locations. Results were visualized and subsequently collapsed to produce typical responses to typical environm...

We show that the linked Grassmannian scheme, which arises in a functorial compactification of spaces of limit linear series, and in local models of certain Shimura varieties, is Cohen-Macaulay, reduced, and flat. We give an application to spaces of limit linear series.