Let $R$ be a strongly $\mathbb {Z}^2$-graded ring, and let $C$ be a bounded chain complex of finitely generated free $R$-modules. The complex $C$ is $R_{(0,0)}$-finitely dominated, or of type $FP$ over $R_{(0,0)}$, if it is chain homotopy equivalent to a bounded complex of finitely generated projective $R_{(0,0)}$-modules. We show that this happens if and only if $C$ becomes acyclic after taking tensor product with a certain eight rings of formal power series,  the graded analogues of classical Novikov rings. This extends results of Ranicki,  Quinn and the first author on Laurent polynomial rings in one and two indeterminates.

In this work, we explore the dynamical implications of a spectral sequence analysis of a filtered chain complex associated to a non-singular Morse–Smale (NMS) flow $\varphi $ on a closed orientable $3$-manifold $M^3$ with no heteroclinic trajectories connecting saddle periodic orbits. We introduce the novel concepts of cancellations and reductions of pairs of periodic orbits based on Franks’ morsification and Smale’s cancellation theorems. The main goal is to establish an algebraic-dynamical correspondence between the unfolding of this spectral sequence associated to $\varphi $ and a family of flows obtained by cancelling and reducing pairs of periodic orbits of $\varphi $ on $M^3$. This correspondence is achieved through a spectral sequence sweeping algorithm (SSSA), which determines the order in which these cancellations and reductions of periodic orbits occur, producing a family of NMS flows that reaches a core flow when the spectral sequence converges.

We give explicit formulae for differential graded Lie algebra (DGLA) models of $3$ -cells. In particular, for a cube and an $n$ -faceted banana-shaped $3$ -cell with two vertices, $n$ edges each joining those two vertices, and $n$ bi-gon $2$ -cells, we construct a model symmetric under the geometric symmetries of the cell fixing two antipodal vertices. The cube model is to be used in forthcoming work for discrete analogues of differential geometry on cubulated manifolds.

A left R-module M is called two-degree Gorenstein flat if there exists an exact sequence of Gorenstein flat left R-modules ⋅⋅⋅ → G2 → G1 → G0 → G−1 → G−2 → ⋅⋅⋅ such that M ≅ Ker(G0 → G−1) and it remains exact after applying H ⊗R- for any Gorenstein injective right R-module H. In this paper we first give some characterisations of Gorenstein flat objects in the category of complexes of modules and then use them to show that two notions of the two-degree Gorenstein flat and the Gorenstein flat left R-modules coincide when R is right coherent.

In a paper from 2002, Hovey introduced the Gorenstein projective and Gorenstein injective model structures on R-Mod, the category of R-modules, where R is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce: the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable module category of R. If such a ring R has finite global dimension, the graded ring R[x]/(x2) is Gorenstein and the three associated Gorenstein model structures on R[x]/(x2)-Mod, the category of graded R[x]/(x2)-modules, are nothing more than the usual projective, injective and flat model structures on Ch(R), the category of chain complexes of R-modules. Although these correspondences only recover these model structures on Ch(R) when R has finite global dimension, we can set R = ℤ and use general techniques from model category theory to lift the projective model structure from Ch(ℤ) to Ch(R) for an arbitrary ring R. This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when ℤ[x]/(x2) is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring R and the derived category of R and we give some examples of such generalizations.

For a commutative ring $R$ with an ideal $I$, generated by a finite regular sequence, we construct differential graded algebras which provide $R$-free resolutions of $I^s$ and of $R/I^s$ for $s \geq 1$ and which generalise the Koszul resolution. We derive these from a certain multiplicative double complex ${\mathbf K}$. By means of a Cartan–Eilenberg spectral sequence we express ${\rm Tor}_*^R(R/I, R/I^s)$ and ${\rm Tor}_*^R(R/I, I^s)$ in terms of exact sequences and find that they are free as $R/I$-modules. Except for $R/I$, their product structure turns out to be trivial; instead, we consider an exterior product ${\rm Tor}_*^R(R/I, I^s)\,{\otimes_R}\,{\rm Tor}_*^R(R/I, I^t)\,{\to}\,{\rm Tor}_*^R(R/I, I^{s+t})$. This paper is based on ideas by Andrew Baker; it is written in view of applications to algebraic topology.

It is shown that Alperin's weight conjecture is equivalent to the existence of contractible chain complexes whose entries have the right dimension coming from some of the alternating sum formulations. It is conjectured that for the other formulations and for Dade's ordinary conjecture, there also exist such contractible chain complexes.

We settle a conjecture of Goresky, Kottwitz and MacPherson related to Koszul duality, i.e., to the correspondence between differential graded modules over the exterior algebra and those over the symmetric algebra.

Acyclic models is a powerful technique in algebraic topology and homological algebra in which facts about homology theories are verified by first verifying them on "models" (on which the homology theory is trivial) and then showing that there are enough models to present arbitrary objects. One version of the theorem allows one to conclude that two chain complex functors are naturally homotopic and another that two such functors are object-wise homologous. Neither is entirely satisfactory. The purpose of this paper is to provide a uniform account of these two, fixing what is unsatisfactory and also finding intermediate forms of the theorem.

The general problem of what should be a non-abelian cohomology, what is it supposed to do, and what should be the coefficients, form a set of interesting questions which has been around for a long time. In the particular setting of groups, a comprehensible and well motivated cohomology theory has been so far stated in dimensions ≤ 2, the coefficients for being crossed modules. The main effort to define an appropriate for groups has been done by Dedecker [16] and Van Deuren [40]; they studied the obstruction to lifting non-abelian 2-cocycles and concluded with first approach for , which requires “super crossed groups” as coefficients. However, as Dedecker said “some polishing work remains necessary” for his cohomology.

Let C be a cochain complex satisfying the condition that inj.dim. Hq(C) < ∞. Then we prove that C admits an injective approximation.

In the paper "Finite complexes and integral representations" [Illinois Journal of Math, 26, (1982), p 442] an exact sequence relating homotopy types of (G, d)-complexes with objects of integral representation theory together with some known calculations seemed to imply that the group of homotopy types of (G, d)- complexes was always a subquotient of (ℤ|g|)*. This paper gives a new characterization of one of the terms of the above sequence that allows one to conclude that this is not generally true.