Related papers: What makes a complex a virtual resolution?
We prove that the category of algebras over a cofibrant operad admits a closed model category structure. This leads to the notion of "virtual operad algebra" - the algebra over a cofibrant resolution of the given operad. In particular,…
We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring $R$ that coincides with the $R$-topology defined by Matlis when $R$ is commutative. (2) We…
Building on work of Brandt and Terao in their study of $k$-formality, we introduce a co-chain complex associated to a multi-arrangement and prove that its cohomologies determine freeness of the associated module of multi-derivations. This…
We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals.
A free resolution of free partially commutative monoids is constructed and with its help the homological dimension of these monoids is calculated.
A method is provided for computing an upper bound of the complexity of a module over a local ring, in terms of vanishing of certain cohomology modules. We then specialize to complete intersections, which are precisely the rings over which…
The quantum dimensions of modules for vertex operator algebras are defined and their properties are discussed. The possible values of the quantum dimensions are obtained for rational vertex operator algebras. A criterion for simple currents…
This article describes some complex-analytic aspects of the moduli space of the finite-dimensional complex representations of a finite quiver, which are stable with respect to a fixed rational weight. We construct a natural structure of a…
We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right $R$-modules…
A virtual endomorphism of a group G is a homomorphism f from H into G where H is a subgroup of G of finite index m. The triple (G,H,f) produces a state-closed (or, self-similar) representation t of G on the 1-rooted m-ary tree. This paper…
We define topologically semiperfect (complete, separated, right linear) topological rings and characterize them by equivalent conditions. We show that the endomorphism ring of a module, endowed with the finite topology, is topologically…
We prove that every finitely generated, virtually solvable minimax group can be expressed as a homomorphic image of a virtually torsion-free, virtually solvable minimax group. This result enables us to generalize a theorem of Ch. Pittet and…
Let $(R, \mathfrak{m})$ be a Noetherian local ring. In this paper, we introduce a dual notion for dualizing modules, namely codualizing modules. We study the basic properties of codualizing modules and use them to establish an equivalence…
Let $R$ be a finitely generated positively graded algebra over a Noetherian local ring $B$, and $\mathfrak{m} = [R]_+$ be the graded irrelevant ideal of $R$. We provide a local criterion characterizing the $B$-freeness of all the local…
Sheaf cohomology or, more generally, higher direct images of coherent sheaves along proper morphisms are central to modern algebraic geometry. However, the computation of these objects is a non-trivial and expensive task which easily…
Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
The dual complex associated to a resolution of singularities generalizes the notion of a resolution graph of a surface singularity to any dimension. We show that homotopy type of the dual complex is an invariant of an isolated singularity.
We develop in this paper a stable theory for projective complexes, by which we mean to consider a chain complex of finitely generated projective modules as an object of the factor category of the homotopy category modulo split complexes. As…
We propose a notion of minimal free resolutions for differential modules, and we prove existence and uniqueness results for such resolutions. We also take the first steps toward studying the structure of minimal free resolutions of…