Minimal Projective Resolution Research Articles

An elementary exact sequence of modules with an application to tiled orders

Let $m \ge 2$ be an integer. By using $m$ submodules of a given module, we construct a certain exact sequence, which is a well known short exact sequence when $m=2$. As an application, we compute a minimal projective resolution of the Jacobson radical of

MULTIPLICATIVE STRUCTURES FOR KOSZUL ALGEBRAS

Let $\Lambda=kQ/I$ be a Koszul algebra over a field $k$, where $Q$ is a finite quiver. An algorithmic method for finding a minimal projective resolution $\mathbb{F}$ of the graded simple modules over $\Lambda$ is given in Green-Solberg. This resolution is shown to have a "comultiplicative" structure in Green-Hartman-Marcos-Solberg, and this is used to find a minimal projective resolution $\mathbb{P}$ of $\Lambda$ over the enveloping algebra $\Lambda^e$. Using these results we show that the multiplication in the Hochschild cohomology ring of $\L$ relative to the resolution $\mathbb{P}$ is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of $\L$ with respect to a canonical basis over $k$ associated to the resolution $\mathbb{F}$. The natural map from the Hochschild cohomology to the Koszul dual of $\Lambda$ is shown to be surjective onto the graded centre of the Koszul dual.

Hochschild cohomology for self-injective algebras of tree class D n . I

The minimal projective bimodule resolution is constructed for algebras in a family of self-injective algebras of finite representation type with tree class Dn. Using this resolution, we calculate the dimensions of the Hochschild cohomology groups for the algebras under consideration. The described resolution is periodic, and thus the Hochschild cohomology of these algebras is periodic as well. Bibliography: 11 titles.

An algorithmic approach to resolutions

We provide an algorithmic method for constructing projective resolutions of modules over quotients of path algebras. This algorithm is modified to construct minimal projective resolutions of linear modules over Koszul algebras.

Constructing Minimal Projective Resolutions

Let Λ be a monomial algebra. For any Λ-module M, we describe graphically its minimal projective resolution as a weighted graph Δ(M). If M is finitely generated we give two algorithms for computing Δ(M). We also give a short computation-like argument to reprove a famous Syzygy Theorem.

Hochschild cohomology of Möbius algebras

The additive structure of the Hochschild cohomology ring for a Mobius algebra is described. We use the structure of the bimodule minimal projective resolution of the algebra, constructed in a previous paper. Bibliography: 4 titles.

Filtered spectra arising from permutative categories

Given a special Γ-category 𝒞 satisfying some mild hypotheses, we construct a sequence of spectra interpolating between the spectrum associated to 𝒞 and the Eilenberg-Mac Lane spectrum Hℤ. Examples of categories to which our construction applies are: the category of finite sets, the category of finite-dimensional vector spaces, and the category of finitely-generated free modules over a reasonable ring. In the case of finite sets, our construction recovers the filtration of Hℤ by symmetric powers of the sphere spectrum. In the case of finite-dimensional complex vector spaces, we obtain an apparently new sequence of spectra, {Am}, that interpolate between bu and . We think of Am as a “bu-analogue” of Spm(S) and describe far-reaching formal similarities between the two sequences of spectra. For instance, in both cases the mth subquotient is contractible unless m is a power of a prime, and in vk-periodic homotopy the filtration has only k + 2 non-trivial terms. There is an intriguing relationship between the bu-analogues of symmetric powers and Weiss's orthogonal calculus, parallel to the not yet completely understood relationship between the symmetric powers of spheres and the Goodwillie calculus of homotopy functors. We conjecture that the sequence {Am}, when rewritten in a suitable chain complex form, gives rise to a minimal projective resolution of the connected cover of bu. This conjecture is the bu-analogue of a theorem of Kuhn and Priddy about the symmetric power filtration. The calculus of functors provides substantial supporting evidence for the conjecture.

On minimal projective resolution

The aim of this paper is to prove that for any unbounded chain complex Y in the category Ch(R) of chain complexes whose homology modules Hi(Y ) are of finite type has a minimal projective resolution where R is a commutative hereditary local ring. Mathematics Subject Classification: 55U15

Projective resolutions and Yoneda algebras for algebras of dihedral type: The family $$D(3\mathcal{Q})$$

This paper provides a method for the computation of Yoneda algebras for algebras of dihedral type. The Yoneda algebras for one infinite family of algebras of dihedral type (the family \(D(3\mathcal{Q})\)) in K. Erdmann’s notation) are computed. The minimal projective resolutions of simple modules were calculated by an original computer program implemented by one of the authors in the C++ language. The algorithm of the program is based on a diagrammatic method presented in this paper and inspired by that of D. Benson and J. Carlson.

On projective resolutions of simple modules over the Borel subalgebra [formula omitted] of the Schur algebra [formula omitted] for [formula omitted

We construct projective resolutions for simple modules over the Borel subalgebras S + ( 2 , r ) and S + ( 3 , r ) of the Schur algebras S ( 2 , r ) and S ( 3 , r ) over a field of positive characteristic. In the case S + ( 2 , r ) we get a minimal projective resolution. We find also projective resolutions of minimal possible length for Weyl modules over the Schur algebra S ( 2 , r ) , corresponding to the regular weights.

Hochschild (Co)homology of Exterior Algebras

The minimal projective bimodule resolutions of the exterior algebras are explicitly constructed. They are applied to calculate the Hochschild (co)homology of the exterior algebras. Thus the cyclic homology of the exterior algebras can be calculated in case the underlying field is of characteristic zero. Moreover, the Hochschild cohomology rings of the exterior algebras are determined by generators and relations.

Projective Resolutions and Yoneda Algebras for Algebras of Dihedral Type

This paper provides a method for the computation of Yoneda algebras for algebras of dihedral type. The Yoneda algebras for one infinite family of algebras of dihedral type (the family \(D(3\mathcal R)\) in K. Erdmann’s notation) are computed. The minimal projective resolutions of simple modules were calculated by an original computer program implemented by one of the authors in C++ language. The algorithm of the program is based on a diagrammatic method presented in this paper and inspired by that of D. Benson and J. Carlson.

Bimodule resolutions of Möbius algebras

The minimal projective bimodule resolution of Möbius algebras that form a class of self-injective algebras of finite representation type is constructed. Bibliography: 7 titles.

Bimodule resolution of the Liu-Schulz algebras

The minimal projective bimodule resolution of symmetric 8-dimensional algebras, which are discovered by Liu and Schulz, is constructed. Bibliography: 4 titles.

Direct sums of exact covers of complexes

A ring $R$ is left noetherian if and only if the direct sum of injective envelopes of any family of left $R$-modules is the injective envelope of the direct sum of the given family of modules (or equivalently, if and only if the direct sum of any family of injective left $R$-modules is also injective). This result of Bass ([2]) led to a series of similar closure questions concerning classes of modules and classes of envelopes and covers (Chase in [4] considers the question of the closure of the class of flat modules with respect to products). Motivated by Bass' result we consider the question of direct sums of exact covers of complexes. From the close connection between minimal injective resolutions of modules and exact covers of complexes it seemed reasonable to conjecture that we get this closure over left noetherian rings. In this paper we show that this is not the case and that under various additional hypotheses on the ring that in fact the ring must have finite left global dimension for this to happen. Our results raise what we consider an interesting question about characterizing the local rings of finite global dimension in terms of a certain property of minimal projective resolutions of finitely generated modules over the local ring. We also consider the closely related question of when the direct sum of DG-injective complexes is DG-injective.

Finite-dimensional algebras with smallest resolutions of simple modules

Let $X$ be a finitely generated left module over a left artinian ring $R$, and let $p(X)=\{l_i\}$ be the infinite sequence of nonnegative integers where $l_i$ is the length of the $i$-th term of the minimal projective resolution of $X$. We introduce a preorder relation $\le$ on the set $\{p(X)\}$ and characterize the elementary finite-dimensional algebras $\Lambda$ with the following property. Let $S$ be a simple $\Lambda$-module, and let $T$ be a finitely generated module over an arbitrary left artinian ring $R$. If the projective dimension of $S$ does not exceed the projective dimension of $T$, then $p(S)\le p(T)$. We characterize the indicated algebras by quivers with relations.

Complexity and periodicity

Let $M$ be a finitely generated module over an Artin algebra. By considering the lengths of the modules in the minimal projective resolution of $M$, we obtain the Betti sequence of $M$. This sequence must be bounded if $M$ is eventually periodic, but the

Castelnuovo–Mumford regularity for complexes and weakly Koszul modules

Let A be a noetherian AS-regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and gr A the category of finitely generated graded left A -modules. Following Jørgensen, we define the Castelnuovo–Mumford regularity reg ( M • ) of a complex M • ∈ D b ( gr A ) in terms of the local cohomologies or the minimal projective resolution of M • . Let A ! be the quadratic dual ring of A . For the Koszul duality functor G : D b ( gr A ) → D b ( gr A ! ) , we have reg ( M • ) = max { i ∣ H i ( G ( M • ) ) ≠ 0 } . Using these concepts, we interpret results of Martinez-Villa and Zacharia concerning weakly Koszul modules (also called componentwise linear modules) over A ! . As an application, refining a result of Herzog and Römer, we show that if J is a monomial ideal of an exterior algebra E = ⋀ 〈 y 1 , … , y d 〉 , d ≥ 3 , then the ( d − 2 ) nd syzygy of E / J is weakly Koszul.

Resolutions over Koszul algebras

In this paper we show that if $\Lambda=\amalg_{i\geq 0}\Lambda_i$ is a Koszul algebra with $\Lambda_0$ isomorphic to a product of copies of a field, then the minimal projective resolution of $\Lambda_0$ as a right $\Lambda$-module provides all the information necessary to construct both a minimal projective resolution of $\Lambda_0$ as a left $\Lambda$-module and a minimal projective resolution of $\Lambda$ as a right module over the enveloping algebra of $\Lambda$. The main tool for this is showing that there is a comultiplicative structure on a minimal projective resolution of $\Lambda_0$ as a right $\Lambda$-module.

τ-Categories I: Ladders

In this series of papers, we introduce τ-categories, which are additive categories with some kind of Auslander–Reiten sequences. We apply them to study the category of lattices over orders. In this first paper, we study minimal projective resolutions in functor categories over τ-categories. Then we give a structure theorem of completely graded τ-categories using mesh categories.