The elementary divisors of the incidence matrices between points and linear subspaces of fixed dimension in Pn(Fp) are computed. Introduction Let V be an (n+ 1)-dimensional vector space over Fp. Let Lr denote the set of r-dimensional subspaces of V . Then L1 is the set of points of the projective space P(V ) and Ln the set of hyperplanes. The group G = GL(V ) acts transitively on each of the se...

Beginning with Hilbert’s construction of what is now called the Koszul complex [18], the study of finite free resolutions of modules over commutative rings has always proceeded by a study of certain particular generic resolutions. This has led to information about the structure of all finite free resolutions, as in [5] and [II], and to theorems on the structure and deformation of certain classe...

A split extension of monoids with kernel $k \colon N \to G$, cokernel $e G H$ and splitting $s H G$ is Schreier if there exists a unique set-theoretic map $q N$ such that for all $g \in = kq(g) \cdot se(g)$. extensions have complete characterization been shown to correspond monoid actions $H$ on $N$. If the uniqueness requirement $q$ relaxed, resulting called weakly Schreier. natural example th...

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on -adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitabl...

We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in enumerating matchings of planar graphs, up to matrix operations on their rows and columns. If such a matrix is defined over a principal ideal domain, this is equivalent to considering its Smith normal form or its cokernel. Many variations of the enumeration methods result in equivalent matrices. In particular, Gessel-Viennot ma...

The objective of this paper is to show that point cloud data can under certain circumstances be clustered by strata in a plausible way. For our purposes, we consider a stratified space to be a collection of manifolds of different dimensions which are glued together in a locally trivial manner inside some Euclidean space. To adapt this abstract definition to the world of noise, we first define a...

We describe a “tunneling” proof of the cobordism invariance of the index of Dirac operators. The goal of this note is to present a very short proof of the cobordism invariance of the index. More precisely, if D̂ is a Dirac operator on an odd dimensional manifold M̂ with boundary ∂M̂ = M then we show that the index of its restriction D to M is zero. The novelty of this proof consists in the fact th...

For an ideal I of a preadditive category A, we study when the canonical functor C : A → A/I is local. We prove that there exists a largest full subcategory C of A for which the canonical functor C : C → C/I is local. Under this condition, the functor C turns out to be a weak equivalence between C and C/I. If A is additive (with splitting idempotents), then C is additive (with splitting idempote...

In this article we introduce a heuristic prediction for the distribution of the isomorphism class of GS(p), the Galois group of the maximal pro-p extension of Q unramified outside of S, where S is a ”random” set of primes. That these groups should exhibit any statistical regularity is not at all obvious; our expectations in this direction are guided by the Cohen-Lenstra conjectures, which (amon...

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular we derive that in any local ring R of mixed...