Polytope Novikov homology

Let M be a closed manifold and $${\mathcal {A}} \subseteq H^1_{\mathrm {dR}}(M)$$ A ⊆ H dR 1 ( M ) a polytope. For each $$a \in {\mathcal {A}}$$ a ∈ A , we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope $${\mathcal {A}}$$ A . The resulting polytope Novikov homology generalizes the ordinary Novikov homology. We prove that any two cohomology classes in a prescribed polytope give rise to chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated with said polytope. As applications, we present a novel approach to the (twisted) Novikov Morse Homology Theorem and prove a new polytope Novikov Principle. The latter generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case.

Hochschild cohomology, finiteness conditions and a generalization of d-Koszul algebras

Given a finite-dimensional algebra [Formula: see text] and [Formula: see text], we construct a new algebra [Formula: see text], called the stretched algebra, and relate the homological properties of [Formula: see text] and [Formula: see text]. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that [Formula: see text] has (Fg) if and only if [Formula: see text] has (Fg). We also consider projective resolutions and apply our results in the case where [Formula: see text] is a [Formula: see text]-Koszul algebra for some [Formula: see text].

Coherency and Constructions for Monoids

A monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck–Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of S to be finitely generated.

Semibricks

In representation theory of finite-dimensional algebras, (semi)bricks are a generalization of (semi)simple modules, and they have long been studied. The aim of this paper is to study semibricks from the point of view of $\tau $-tilting theory. We construct canonical bijections between the set of support $\tau $-tilting modules, the set of semibricks satisfying a certain finiteness condition, and the set of 2-term simple-minded collections. In particular, we unify Koenig–Yang bijections and Ingalls–Thomas bijections generalized by Marks–Št’ovíček, which involve several important notions in the derived categories and the module categories. We also investigate connections between our results and two kinds of reduction theorems of $\tau $-rigid modules by Jasso and Eisele–Janssens–Raedschelders. Moreover, we study semibricks over Nakayama algebras and tilted algebras in detail as examples.

A finiteness condition on the set of overrings of some classes of integral domains

As an extension of the class of Dedekind domains, we have introduced and studied the class of multiplicatively pinched-Dedekind domains (MPD domains) and the class of Globalized multiplicatively pinched-Dedekind domains (GMPD domains) ([T. Dumitrescu and S. U. Rahman, A class of pinched domains, Bull. Math. Soc. Sci. Math. Roumanie 52 (2009) 41–55] and [T. Dumitrescu and S. U. Rahman, A class of pinched domains II, Comm. Algebra 39 (2011) 1394–1403]). The main interest of this paper is to study GMPD domains that have only finitely many overrings.

Polygraphs of finite derivation type

Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition left-FP3. Using this result, he constructed finitely presentable monoids with a decidable word problem, but that cannot be presented by finite convergent rewriting systems. Later, he introduced the condition of finite derivation type, which is a homotopical finiteness property on the presentation complex associated to a monoid presentation. He showed that this condition is an invariant of finite presentations and he gave a constructive way to prove this finiteness property based on the computation of the critical branchings: Being of finite derivation type is a necessary condition for a finitely presented monoid to admit a finite convergent presentation. This survey presents Squier's results in the contemporary language of polygraphs and higher dimensional categories, with new proofs and relations between them.

On the cardinality of β-expansions of some numbers

Let [Formula: see text]. It is well known that every [Formula: see text] has a [Formula: see text]-expansion of the form [Formula: see text] with [Formula: see text], where [Formula: see text] denotes the largest integer not exceeding [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote the sets of all [Formula: see text]-expansions of [Formula: see text] and the set of [Formula: see text]-prefixes of all [Formula: see text]-expansions of [Formula: see text], respectively. We show that [Formula: see text], [Formula: see text] and [Formula: see text] are equivalent under a certain finiteness condition.