Modules Of Finite Projective Dimension Research Articles

Cofiniteness of generalized local cohomology modules over local rings

Let [Formula: see text] be a local commutative Noetherian ring and [Formula: see text] an ideal of [Formula: see text] Assume that [Formula: see text] is a finitely generated [Formula: see text]-module of finite projective dimension [Formula: see text] and [Formula: see text] is a finitely generated [Formula: see text]-module of dimension [Formula: see text] We prove that [Formula: see text] is Artinian. Moreover, if [Formula: see text] is finitely generated, then [Formula: see text] is an [Formula: see text]-cofinite [Formula: see text]-module.

Cohen–Macaulay Noetherian algebras

Let (R,m) be a d-dimensional commutative complete Noetherian local ring and Λ be a Noetherian R-algebra. Motivated by the notion of Cohen–Macaulay Artin algebras of Auslander and Reiten, we say that Λ is Cohen–Macaulay if there is a finitely generated Λ-bimodule ω that is maximal Cohen–Macaulay over R such that the adjoint pair of functors (ω⊗Λ′−,HomΛ′(ω,−)) induces quasi-inverse equivalences between the full subcategories of finitely generated Λ′-modules consisting of modules of finite projective dimension, P∞(Λ′), and the modules of finite injective dimension, I∞(Λ′), whenever Λ′=Λ,Λop. It is proved that such a module ω is unique, up to isomorphism, as a Λ′-module. It is also shown that Λ is a Cohen–Macaulay algebra if and only if there is a semidualizing Λ-bimodule ω of finite injective dimension and P∞(Λ′) and I∞(Λ′) are contained in the Auslander and Bass classes, respectively. We prove that Cohen–Macaulayness behaves well under reduction modulo system of parameters of R. Indeed, it will be observed that if Λ is a Cohen–Macaulay algebra, then for any system of parameters x=x1,…,xd of R, the Artin algebra Λ∕xΛ is Cohen–Macaulay as well. Assume that ω is a semidualizing Λ-bimodule of finite injective dimension that is maximal Cohen–Macaulay as an R-module. It will turn out that Λ being a Cohen–Macaulay algebra is equivalent to saying that the pair (CM(Λ′),I∞(Λ′)) forms a hereditary complete cotorsion theory and the pair (CM(Λ′op),P∞(Λ′)) forms a Tor-torsion theory, where CM(Λ′) is the class of all finitely generated Λ′-modules admitting a right resolution by modules in addω. Finally, it is shown that Cohen–Macaulayness ascends from R to RΓ and RQ, where Γ is a finite group and Q is a finite acyclic quiver.

Finitistic dimension conjectures via Gorenstein projective dimension

It is a well-known result of Auslander and Reiten that contravariant finiteness of the class P∞fin (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, we examine in this work the Gorenstein counterpart of Auslander–Reiten condition, namely contravariant finiteness of the class GP∞fin (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class GP∞fin implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander–Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in the sense that contravariant finiteness of the class GP∞fin implies contravariant finiteness of the class P∞fin over any Artin algebra, and the converse holds for Artin algebras over which the class GP0fin (of finitely generated Gorenstein projective modules) is contravariantly finite.

Quasi-projective dimension

In this paper, we introduce a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. We discuss various properties of quasi-projective dimension. Among other things, we prove the following. (1) Over a quotient of a regular local ring by a regular sequence, every finitely generated module has finite quasi-projective dimension. (2) The Auslander--Buchsbaum formula and the depth formula for modules of finite projective dimension remain valid for modules of finite quasi-projective dimension. (3) Several results on vanishing of Tor and Ext hold for modules of finite quasi-projective dimension.

Support varieties and modules of finite projective dimension for modular Lie superalgebras

We investigate cohomological support varieties for finite-dimensional Lie superalgebras defined over fields of odd characteristic. Verifying a conjecture from our previous work, we show the support variety of a finite-dimensional supermodule can be realized as an explicit subset of the odd nullcone of the underlying Lie superalgebra. We also show the support variety of a finite-dimensional supermodule is zero if and only if the supermodule is of finite projective dimension. As a consequence, we obtain a positive characteristic version of a theorem of B{\o}gvad, showing that if a finite-dimensional Lie superalgebra over a field of odd characteristic is absolutely torsion free, then its enveloping algebra is of finite global dimension.

Layered resolutions of Cohen–Macaulay modules

Let S be a Gorenstein local ring and suppose that M is a finitely generated Cohen–Macaulay S -module of codimension c . Given a regular sequence f_1, \ldots, f_c in the annihilator of M we set R = S/(f_1, \ldots, f_c) and construct layered S -free and R -free resolutions of M . The construction inductively reduces the problem to the case of a Cohen–Macaulay module of codimension c-1 and leads to the inductive construction of a higher matrix factorization for M . In the case where M is a sufficiently high R -syzygy of some module of finite projective dimension over S , the layered resolutions are minimal and coincide with the resolutions defined from higher matrix factorizations we described in [EP]. Our results provide a characterization of all MCM modules over a complete intersection in terms of higher matrix factorizations.

REPETITIVE EQUIVALENCES AND TILTING THEORY

Let$R$be a ring and$T$be a good Wakamatsu-tilting module with$S=\text{End}(T_{R})^{op}$. We prove that$T$induces an equivalence between stable repetitive categories of$R$and$S$(i.e., stable module categories of repetitive algebras$\hat{R}$and${\hat{S}}$). This shows that good Wakamatsu-tilting modules seem to behave in Morita theory of stable repetitive categories as that tilting modules of finite projective dimension behave in Morita theory of derived categories.

Good tilting modules and recollements of derived module categories, II

Homological tilting modules of finite projective dimension are investigated. They generalize both classical and good tilting modules of projective dimension at most one, and produce recollements of derived module categories of rings in which generalized localizations of rings are involved. To decide whether a good tilting module is homological, a sufficient and necessary condition is presented in terms of the internal properties of the given tilting module. Consequently, a class of homological, non-trivial, infinitely generated tilting modules of higher projective dimension is constructed, and the first example of an infinitely generated $n$-tilting module which is not homological for each $n \ge 2$ is exhibited. To deal with both tilting and cotilting modules consistently, the notion of weak tilting modules is introduced. Thus similar results for infinitely generated cotilting modules of finite injective dimension are obtained, though dual technique does not work for infinite-dimensional modules.

On the Loewy length of modules of finite projective dimension

Let (A,m) be a local Gorenstein local ring and let M be an A module of finite length and finite projective dimension. We prove that the Lowey length of M is greater than or equal to order of A. This generalizes a result of Avramov, Buchweitz, Iyengar and Miller (2, 1.1).

On modules of finite projective dimension

AbstractWe 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 ringRof mixed characteristicp> 0, wherepis a nonzero divisor, ifIis an ideal of finite projective dimension overRandp𝜖Iorpis a nonzero divisor onR/I, then every minimal generator ofIis a nonzero divisor. Hence, ifPis a prime ideal of finite projective dimension in a local ringR, then every minimal generator ofPis a nonzero divisor inR.

On modules of finite projective dimension

AbstractWe 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 ringRof mixed characteristicp> 0, wherepis a nonzero divisor, ifIis an ideal of finite projective dimension overRandp𝜖Iorpis a nonzero divisor onR/I, then every minimal generator ofIis a nonzero divisor. Hence, ifPis a prime ideal of finite projective dimension in a local ringR, then every minimal generator ofPis a nonzero divisor inR.

T-Structures and cotilting modules over commutative noetherian rings

For a commutative noetherian ring $$R$$ , we establish a bijection between the resolving subcategories consisting of finitely generated $$R$$ -modules of finite projective dimension and the compactly generated t-structures in the unbounded derived category $$\mathcal {D}(R)$$ that contain $$R[1]$$ in their heart. Under this bijection, the t-structures $$(\mathcal U,\mathcal V)$$ such that the aisle $$\mathcal U$$ consists of objects with homology concentrated in degrees $$<n$$ correspond to the $$n$$ -cotilting classes in $${{\mathrm{Mod}\text {-}R}}$$ . As a consequence of these results, we prove that the little finitistic dimension findim $$R$$ of $$R$$ equals an integer $$n$$ if and only if the direct sum $$\bigoplus _{k=0}^n E_k(R)$$ of the first $$n+1$$ terms in a minimal injective coresolution $$0\rightarrow R\rightarrow E_0(R)\rightarrow E_1(R)\rightarrow \cdots $$ of $$R$$ is an injective cogenerator of $${{\mathrm{Mod}\text {-}R}}$$ .

On the Left Perpendicular Category of the Modules of Finite Projective Dimension

In this article, we characterize several properties of commutative noetherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application, we prove that a local ring is regular if (and only if) there exists a strong test module for projectivity having finite projective dimension. We also obtain corresponding results with respect to a semidualizing module.

A proof of the strong no loop conjecture

The strong no loop conjecture states that a simple module of finite projective dimension over an artin algebra has no non-zero self-extension. The main result of this paper establishes this well known conjecture for finite dimensional algebras over an algebraically closed field.

Strongly tilting truncated path algebras

For any truncated path algebra Λ, we give a structural description of the modules in the categories \({\mathcal{P}^{<\infty}(\Lambda\text{-}{\rm mod})}\) and \({\mathcal{P}^{<\infty}(\Lambda\text{-}{\rm mod})}\) , consisting of the finitely generated (resp. arbitrary) Λ-modules of finite projective dimension. We deduce that these categories are contravariantly finite in Λ−mod and Λ-Mod, respectively, and determine the corresponding minimal \({\mathcal{P}^{<\infty}}\) -approximation of an arbitrary Λ-module from a projective presentation. In particular, we explicitly construct—based on the Gabriel quiver Q and the Loewy length of Λ—the basic strong tilting module ΛT (in the sense of Auslander and Reiten) which is coupled with \({\mathcal{P}^{<\infty}(\Lambda\text{-}{\rm mod})}\) in the contravariantly finite case. A main topic is the study of the homological properties of the corresponding tilted algebra \({\tilde{\Lambda} = {\rm End}_\Lambda(T)^{\rm op}}\) , such as its finitistic dimensions and the structure of its modules of finite projective dimension. In particular, we characterize, in terms of a straightforward condition on Q, the situation where the tilting module \({T_{\tilde{\Lambda}}}\) is strong over \({\tilde{\Lambda}}\) as well. In this Λ-\({\tilde{\Lambda}}\)-symmetric situation, we obtain sharp results on the submodule lattices of the objects in \({\mathcal{P}^{<\infty}({\rm Mod}\text{-}\tilde{\Lambda})}\) , among them a certain heredity property; it entails that any module in \({\mathcal{P}^{<\infty}({\rm Mod}\text{-}\tilde{\Lambda})}\) is an extension of a projective module by a module all of whose simple composition factors belong to \({\mathcal{P}^{<\infty}({\rm Mod}\text{-}\tilde{\Lambda})}\) .

Homological aspects of semidualizing modules

We investigate the notion of the $C$-projective dimension of a module, where $C$ is a semidualizing module. When $C=R$, this recovers the standard projective dimension. We show that three natural definitions of finite $C$-projective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results that demonstrate the deep connections between modules of finite projective dimension and modules of finite $C$-projective dimension. In parallel, we develop the dual theory for injective dimension and $C$-injective dimension.

Homology of perfect complexes

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed fiber of some flat local homomorphism. Other applications include, as special cases, uniform proofs of known results on free actions of elementary abelian groups and of tori on finite CW complexes. The arguments use numerical invariants of objects in general triangulated categories, introduced here and called levels. They allow one to track, through changes of triangulated categories, homological invariants like projective dimension, as well as structural invariants like Loewy length. An intermediate result sharpens, with a new proof, the New Intersection Theorem for commutative algebras over fields. Under additional hypotheses on the ring R stronger estimates are proved for Loewy lengths of modules of finite projective dimension.

Characterization of modules of finite projective dimension via Frobenius functors

Let M be a finitely generated module over a local ring R of characteristic p > 0. If depth(R) = s, then the property that M has finite projective dimension can be characterized by the vanishing of the functor \({{\rm Ext}^i_R(M, ^{f^n}R)}\) for s + 1 consecutive values i > 0 and for infinitely many n. In addition, if R is a d-dimensional complete intersection, then M has finite projective dimension can be characterized by the vanishing of the functor \({{\rm Ext}^i_R(M, ^{f^n}R)}\) for some i ≥ d and some n > 0.

Self-Orthogonal Modules of Finite Projective Dimension

Let R be a ring and R ω a self-orthogonal module. We introduce the notion of the right orthogonal dimension (relative to R ω) of modules. We give a criterion for computing this relative right orthogonal dimension of modules. For a left coherent and semilocal ring R and a finitely presented self-orthogonal module R ω, we show that the projective dimension of R ω and the right orthogonal dimension (relative to R ω) of R/J are identical, where J is the Jacobson radical of R. As a consequence, we get that R ω has finite projective dimension if and only if every left (finitely presented) R-module has finite right orthogonal dimension (relative to R ω). If ω is a tilting module, we then prove that a left R-module has finite right orthogonal dimension (relative to R ω) if and only if it has a special ω⊥-preenvelope.

An Euler characteristic for modules of finite G-dimension

We extend Auslander and Buchsbaum's Euler characteristic from the category of finitely generated modules of finite projective dimension to the category of modules of finite G-dimension using Avramov and Martsinkovsky's notion of relative Betti numbers. We prove analogues of some properties of the classical invariant and provide examples showing that other properties do not translate to the new context. One unexpected property is in the characterization of the extremal behavior of this invariant: the vanishing of the Euler characteristic of a module $M$ of finite G-dimension implies the finiteness of the projective dimension of $M$. We include two applications of the Euler characteristic as well as several explicit calculations.