Modules Of Finite Length Research Articles

Boij-Söderberg conjectures for differential modules

Boij-Söderberg theory gives a combinatorial description of the set of Betti tables belonging to finite length modules over the polynomial ring S=k[x1,…,xn]. We posit that a similar combinatorial description can be given for analogous numerical invariants of graded differential S-modules, which are natural generalizations of chain complexes. We prove several results that lend evidence in support of this conjecture, including a categorical pairing between the derived categories of graded differential S-modules and coherent sheaves on Pn−1 and a proof of the conjecture in the case where S=k[t].

Koszul modules with vanishing resonance in algebraic geometry

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K⊆⋀2V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\subseteq \\bigwedge ^2 V$$\\end{document}, where V is a vector space. Previously Koszul modules of finite length have been used to give a proof of Green’s Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion.

On quasi-morphic modules

An R-module M is called quasi-morphic if for any f∈EndR(M), there exist g,h∈EndR(M) such that Imf=Kerg and Kerf=Imh. In addition, MR is said to be morphic whenever g=h in the above definition. The main objective of this paper is investigating quasi-morphic property for several classes of modules. First we obtain general properties of quasi-morphic modules via exact sequence approach. Moreover, we investigate conditions under which a finite length quasi-morphic module is morphic. As a result, we show that for uniserial finite length modules, the notions of morphic and quasi-morphic coincide. Over a principal ideal domain R, direct sums of cyclic modules which are (quasi-)morphic are characterized. Among applications of our results, nonsingular extending (quasi-)morphic modules are characterized completely. We also prove that over a commutative Noetherian domain R which is not a field, quasi-morphic nonsingular extending modules are precisely direct sums of copies of Q (the quotient field of R). (Quasi-)Morphic singular extending abelian groups are also characterized.

Multiplicities and Betti numbers in local algebra via lim Ulrich points

This work concerns finite free complexes with finite length homology over a commutative noetherian local ring $R$. The focus is on complexes that have length $\mathrm{dim}\, R$, which is the smallest possible value, and in particular on free resolutions of modules of finite length and finite projective dimension. Lower bounds are obtained on the Euler characteristic of such short complexes when $R$ is a strict complete intersection, and also on the Dutta multiplicity, when $R$ is the localization at its maximal ideal of a standard graded algebra over a field of positive prime characteristic. The key idea in the proof is the construction of a suitable Ulrich module, or, in the latter case, a sequence of modules that have the Ulrich property asymptotically, and with good convergence properties in the rational Grothendieck group of $R$. Such a sequence is obtained by constructing an appropriate sequence of sheaves on the associated projective variety.

Homological duality for covering groups of reductive $p$-adic groups

In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre duality with respect to the Bernstein center and provide a proof of the folklore result that on admissible modules this functor is nothing but the contragredient duality. We single out a necessary and sufficient condition for when these three dualities agree on finite length modules in a given block. In particular, we show this is the case for all cuspidal blocks as well as, due to a result of Roche, on all blocks with trivial stabilizer in the relative Weyl group.

A Composite Order Generalization of Modular Moonshine

We introduce a generalization of Brauer character to allow arbitrary finite length modules over discrete valuation rings. We show that the generalized super Brauer character of Tate cohomology is a linear combination of trace functions. Using this result, we find a counterexample to a conjecture of Borcherds about vanishing of Tate cohomology for Fricke elements of the Monster.

Tensor Structure on the Kazhdan–Lusztig Category for Affine 𝔤𝔩(1|1)

AbstractWe show that the Kazhdan–Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for the affine Lie superalgebra $\widehat{\mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure and that its full subcategory $\mathcal{O}_k^{fin}$ of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple $\widehat{\mathfrak{gl}(1|1)}$-module in $KL_k$ has a projective cover in ${\mathcal{O}}_k^{fin}$, and we determine all fusion rules involving simple and projective objects in ${\mathcal{O}}_k^{fin}$. Then using Knizhnik–Zamolodchikov equations, we prove that $KL_k$ and $\mathcal{O}_k^{fin}$ are rigid. As an application of the tensor supercategory structure on $\mathcal{O}_k^{fin}$, we study certain module categories for the affine Lie superalgebra $\widehat{\mathfrak{sl}(2|1)}$ at levels $1$ and $-\frac{1}{2}$. In particular, we obtain a tensor category of $\widehat{\mathfrak{sl}(2|1)}$-modules at level $-\frac{1}{2}$ that includes relaxed highest-weight modules and their images under spectral flow.

The isomorphism problem for uniserial modules over an arbitrary ring

First, we give a partial solution to the isomorphism problem for uniserial modules of finite length with the help of the morphisms between these modules. Later, under suitable assumptions on the lattice of the submodules, we give a method to partially solve the isomorphism problem for uniserial modules over an arbitrary ring. Particular attention is given to the natural class of uniserial modules defined over algebras given by quivers.

Representation-theoretic properties of balanced big Cohen–Macaulay modules

Let $$(R, {\mathfrak {m}}, k)$$ be a complete Cohen–Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen–Macaulay module, called $$\underline{\mathsf {h}}$$ -length. Among other results, it is proved that, for a given balanced big Cohen–Macaulay R-module M with an $${\mathfrak {m}}$$ -primary cohomological annihilator, if there is a bound on the $$\underline{\mathsf {h}}$$ -length of all modules appearing in $$\mathsf {CM}$$ -support of M, then it is fully decomposable, i.e. it is a direct sum of finitely generated modules. While the first Brauer–Thrall conjecture fails in general by a counterexample of Dieterich dealing with multiplicities to measure the size of maximal Cohen–Macaulay modules, our formalism establishes the validity of the conjecture for complete Cohen–Macaulay local rings. In addition, the pure-semisimplicity of a subcategory of balanced big Cohen–Macaulay modules is settled. Namely, it is shown that R is of finite $$\mathsf {CM}$$ -type if and only if R is an isolated singularity and the category of all fully decomposable balanced big Cohen–Macaulay modules is closed under kernels of epimorphisms. Finally, we examine the mentioned results in the context of Cohen–Macaulay artin algebras admitting a dualizing bimodule $$\omega $$ , as defined by Auslander and Reiten. It will turn out that, $$\omega $$ -Gorenstein projective modules with bounded $$\mathsf {CM}$$ -support are fully decomposable. In particular, a Cohen–Macaulay algebra $$\Lambda $$ is of finite $$\mathsf {CM}$$ -type if and only if every $$\omega $$ -Gorenstein projective module is of finite $$\mathsf {CM}$$ -type, which generalizes a result of Chen for Gorenstein algebras. Our main tool in the proof of results is Gabriel–Roiter (co)measure, an invariant assigned to modules of finite length, and defined by Gabriel and Ringel. This, in fact, provides an application of the Gabriel–Roiter (co)measure in the category of maximal Cohen–Macaulay modules.

On a question of Buchweitz about ranks of syzygies of modules of finite length

Let R be a local ring of dimension d. Buchweitz asks if the rank of the d-th syzygy of a module of finite lengh is greater than or equal to the rank of the d-th syzygy of the residue field, unless the module has finite projective dimension. Assuming that R is Gorenstein, we prove that if the question is affrmative, then R is a hypersurface. If moreover R has dimension two, then we show that the converse also holds true.

Noncommutative resolutions using syzygies

Given a noether algebra with a noncommutative resolution, a general construction of new noncommutative resolutions is given. As an application, it is proved that any finite length module over a regular local or polynomial ring gives rise, via suitable syzygies, to a noncommutative resolution.

Total Betti numbers of modules of finite projective dimension

The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that if M is a non-zero module of finite length and finite projective dimension over a local ring R of dimension d, then the i-th Betti number of M is at least d choose i. This conjecture implies that the sum of all the Betti numbers of such a module must be at least 2^d. We prove the latter holds in a large number of cases.

Frobenius Betti numbers and modules of finite projective dimension

Let $(R,\mathfrak{m} ,K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, $\beta ^F_i(M,R)$, defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\mathfrak{m} ,R)=\beta ^F_0(K,R)$). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of $\beta ^F_i(M,R)$ implies that $M$ has finite projective dimension. In particular, we give a complete characterization of the vanishing of $\beta ^F_i(M,R)$ for one-dimensional rings. As a consequence of our methods we give conditions for the non-existence of syzygies of finite length.

A canonical torsion theory for pro-p Iwahori–Hecke modules

Let F be a locally compact nonarchimedean field with residue characteristic p and G the group of F-rational points of a connected split reductive group over F. We define a torsion pair in the category Mod(H) of modules over the pro-p-Iwahori Hecke k-algebra H of G, where k is an arbitrary field. We prove that, under a certain hypothesis, the torsionfree class embeds fully faithfully into the category ModI(G) of smooth k-representations of G generated by their pro-p-Iwahori fixed vectors.If the characteristic of k is different from p then this hypothesis is always satisfied and the torsionfree class is the whole category Mod(H).If k contains the residue field of F then we study the case G=SL2(F). We show that our hypothesis is satisfied, and we describe explicitly the torsionfree and the torsion classes. If F≠Qp and p≠2, then an H-module is in the torsion class if and only if it is a union of supersingular finite length submodules; it lies in the torsionfree class if and only if it does not contain any nonzero supersingular finite length module. If F=Qp, the torsionfree class is the whole category Mod(H), and we give a new proof of the fact that Mod(H) is equivalent to ModI(G). These results are based on the computation of the H-module structure of certain natural cohomology spaces for the pro-p-Iwahori subgroup I of G.

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).

Artinian dimension and isoradical of modules

This paper is a continuation of our previous article [9]. We introduce an “artinian dimension” of modules, which allows us to study isoartinian modules, and an “isoradical” of a module, which is the analogue of the (Jacobson) radical of a module. We study the modules generated by isosimple submodules, and modules of finite I-length, which are analogous to modules of finite composition length.

Uniform almost relative injective modules

The concept of a module M being almost N-injective where N is some module, was introduced by Baba (1989). For a given module M, the class of modules N, for which M is almost N-injective, is not closed under direct sums. Baba gave a necessary and sufficient condition under which a uniform finite length module U is almost V-injective, where V is a finite direct sum of uniform, finite length modules, in terms of extending properties of simple submodules of V. Let U be a uniform module and V be a finite direct sum of indecomposable modules. Recently, Singh (2016), has determined some conditions under which U is almost V injective, which generalize Baba's result. In the present paper some more results in this direction are proved. A module M is said to be completely almost self-injective, if for any two subfactors A, B of M, A is almost B-injective. A necessary and sufficient condition for a module M to be completely almost self-injective is given. Using this, it is proved that a Von Neumann ring R is completely almost right self-injective if and only if Rsoc(RR) is semi-simple and every minimal right ideal of R is injective.

Equivariant $L^2$-Euler characteristics of $G\textrm{-}CW$-complexes

We show that if $X$ is a cocompact $G\textrm{-}CW$-complex such that each isotropy subgroup $G_\sigma$ is $L^{(2)}$-good over an arbitrary commutative ring $k$, then $X$ satisfies some fixed-point formula which is an $L^{(2)}$-analogue of Brown’s formula in 1982. Using this result we present a fixed point formula for a cocompact proper $G\textrm{-}CW$-complex which relates the equivariant $L^{(2)}$-Euler characteristic of a fixed point $CW$-complex $X^s$ and the Euler characteristic of $X/G$. As corollaries, we prove Atiyah’s theorem in 1976, Akita’s formula in 1999 and a result of Chatterji–Mislin in 2009. We also show that if X is a free $G\textrm{-}CW$-complex such that $C_{*} (X)$ is chain homotopy equivalent to a chain complex of finitely generated projective $Z \pi_1 (X)$-modules of finite length and $X$ satisfies some fixed-point formula over $\mathbb{Q}$ or $\mathbb{C}$ which is an $L^{(2)}$-analogue of Brown’s formula, then $\chi (X/G) = \chi^{(2)} (X)$. As an application, we prove that the weak Bass conjecture holds for any finitely presented group $G$ satisfying the following condition: for any finitely dominated $CW$-complex $Y$ with $\pi_1 (Y)=G, \widetilde{Y}$ satisfies some fixed-point formula over $\mathbb{Q}$ or $\mathbb{C}$ which is an $L^{(2)}$-analogue of Brown’s formula.

The cone of Betti tables over three non-collinear points in the plane

We describe the cone of Betti tables of all finitely generated graded modules over the homogeneous coordinate ring of three non-collinear points in the projective plane. We also describe the cone of Betti tables of all finite length modules.

Finite homological dimension and a derived equivalence

For a Cohen-Macaulay ring R R , we exhibit the equivalence of the bounded derived categories of certain resolving subcategories, which, amongst other results, yields an equivalence of the bounded derived category of finite length and finite projective dimension modules with the bounded derived category of projective modules with finite length homologies. This yields isomorphisms of K K -theory and Witt groups (amongst other invariants) and improves on terms of associated spectral sequences and Gersten complexes.