Free Summands Research Articles

The definable content of homological invariants I: Ext$\mathrm{Ext}$ and lim1$\mathrm{lim}^1$

AbstractThis is the first installment in a series of papers illustrating how classical invariants of homological algebra and algebraic topology may be enriched with additional descriptive set theoretic information. To effect this enrichment, we show that many of these invariants may be naturally regarded as functors to the category, introduced herein, of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification. In the present work we focus on the first derived functors of and . The resulting definable for pairs of countable abelian groups and definable for towers of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable is a fully faithful contravariant functor from the category of finite‐rank torsion‐free abelian groups with no free summands; this contrasts with the fact that there are uncountably many non‐isomorphic such groups with isomorphic classical invariants . To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover; within this framework we prove several rigidity results for non‐Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of ‐adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem of classifying all group extensions of by up to base‐free isomorphism, when for prime numbers and .

Vasconcelos’ conjecture on the conormal module

For any ideal I of finite projective dimension in a commutative noetherian local ring R, we prove that if the conormal module $$I/I^2$$ has finite projective dimension over R/I, then I must be generated by a regular sequence. This resolves a conjecture of Vasconcelos. We prove a similar result for the first Koszul homology module of I. When R is a localisation of a polynomial ring over a field K of characteristic zero, Vasconcelos conjectured that R/I is a reduced complete intersection if the module $$\Omega _{(R/I)/K}$$ of Kähler differentials has finite projective dimension; we prove this contingent on the Eisenbud-Mazur conjecture. The arguments exploit the structure of the homotopy Lie algebra associated to I in an essential way. By work of Avramov and Halperin, if every degree 2 element of the homotopy Lie algebra is radical, then I is generated by a regular sequence. Iyengar has shown that free summands of $$I/I^2$$ give rise to central elements of the homotopy Lie algebra, and we establish an analogous criterion for constructing radical elements, from which we deduce our main result.

Various Row Invariants on Cohen-Macaulay Rings

AbstractWe define a numerical invariant
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ˆ -th syzygy module (with or without free summands). We show that
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‚ Þ šsa
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‚Ý ޚsa
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‚ . Key words : Row Invariants, Cohen-Macaulay Local Ring, Maximal Cohen-Macaulay Module, Free Summands 1. Introduction 1 Throughout this paper, we assume that ޚi
‹s is a Noetherian local ring, and all modules are unitary. It is proved [1] that there are certain restrictions on the entries of the maps in the minimal free resolutions of finitely generated modules of infinite projective dimension over Noetherian local rings. This fact provides not only a new way to understand some previously known results in commutative ring theory (see for instance Corollary 2.8 [1] , or Proposition 2.2 [1] ), but also new interesting invariants of local rings. These invariants have turned out to be quite useful; for example, the Auslander index of šcan be described as a column invariant when š is Gorenstein

The monomial conjecture and order ideals

In this article first we prove that a special case of the order ideal conjecture, originating from the work of Evans and Griffith in equicharacteristic, implies the monomial conjecture due to M. Hochster. We derive a necessary and sufficient condition for the validity of this special case in terms of certain syzygies of canonical modules of normal domains possessing free summands. We also prove some special cases of this observation.

The Hilbert Scheme of Buchsbaum space curves

We consider the Hilbert scheme H(d,g) of space curves C with homogeneous ideal I(C):=H * 0 (ℐ C ) and Rao module M:=H * 1 (ℐ C ). By taking suitable generizations (deformations to a more general curve) C ′ of C, we simplify the minimal free resolution of I(C) by e.g making consecutive free summands (ghost-terms) disappear in a free resolution of I(C ′ ). Using this for Buchsbaum curves of diameter one (M v ≠0 for only one v), we establish a one-to-one correspondence between the set 𝒮 of irreducible components of H(d,g) that contain (C) and a set of minimal 5-tuples that specializes in an explicit manner to a 5-tuple of certain graded Betti numbers of C related to ghost-terms. Moreover we almost completely (resp. completely) determine the graded Betti numbers of all generizations of C (resp. all generic curves of 𝒮), and we give a specific description of the singular locus of the Hilbert scheme of curves of diameter at most one. We also prove some semi-continuity results for the graded Betti numbers of any space curve under some assumptions.

Tame parts of free summands in coproducts of Priestley spaces

It is well known that a sum (coproduct) of a family { X i : i ∈ I } of Priestley spaces is a compactification of their disjoint union, and that this compactification in turn can be organized into a union of pairwise disjoint order independent closed subspaces X u , indexed by the ultrafilters u on the index set I. The nature of those subspaces X u indexed by the free ultrafilters u is not yet fully understood. In this article we study a certain dense subset X u ∂ ⊆ X u satisfying exactly those sentences in the first-order theory of partial orders which are satisfied by almost all of the X i 's. As an application we present a complete analysis of the coproduct of an increasing family of finite chains, in a sense the first non-trivial case which is not a Čech–Stone compactification of the disjoint union ⋃ I X i . In this case, all the X u 's with u free turn out to be isomorphic under the Continuum Hypothesis.

Free submodules for the central representation in the cohomology of Lie algebras

If Z Z is the centre of the Lie algebra L L , its cohomology H ∗ ( L ) H^*(L) is naturally a module over the exterior algebra Λ Z \Lambda Z . Under suitable hypotheses on L L , motivated by recent work by Pouseele and Tirao, we find free summands in H ∗ ( L ) H^*(L) for this module structure, thus establishing the Toral Rank Conjecture for a new class of Lie algebras.

Free Direct Summands of Maximal Rank and Rigidity in Projective Dimension Two

ABSTRACT Let M be a finitely generated module of rank r and finite projective dimension over a commutative Noetherian local ring S. We show that M has a free direct summand of rank r if and only if its Buchsbaum–Eisenbud invariant J(M) is a principal ideal. As an application, we consider the problem of finding rigid triples. Recall that the module M is said to be rigid if for each i ≥ 0 and each finitely generated S-module N the condition implies for every j ≥ i. A triple of positive integers b = (b 0, b 1, b 2) is said to be rigid if for every commutative Noetherian local ring S one has that every S-module with minimal free resolution of the form 0 → S b 2 → S b 1 → S b 0 → 0 is rigid. We use our criterion for free direct summands of maximal rank to derive a necessary and sufficient condition for a triple b to be rigid in terms of the rigidity of the universal modules of type b over complete regular local rings. In particular, we exhibit the rigidity of any triple of the form (k, m + 1,m), thus providing a new class of rigid modules.

Column Invariants Over Cohen–Macaulay Local Rings

We define some numerical invariants over Cohen–Macaulay local rings. These invariants are related to columns of the presenting matrices of maximal Cohen–Macaulay modules and syzygy modules without free summands. We study the relationship between these invariants, and the invariant col(A). #Communicated by I. Swanson.

Splitting off free summands of torsion-free modules over complete DVRs

If R is a complete discrete valuation ring and M is a reduced, torsion-free R-module of rank \kappa, where \aleph_0 \leq \kappa < 2^ (\aleph_0), we show that M \prop\oplus_(\aleph_0) R \oplus C for some R-module C. As a consequence, it must be the case that M \prop M \oplus (\oplus{_\alpha}R), where \alpha \leq \aleph_0, and {\rm (End)_R}M/\rm (Fin)M has rank at least 2^ (\aleph_0), where Fin M denotes the set of endomorphisms of M with finite rank image.

Free summands of conormal modules and central elements in homotopy Lie algebras of local rings

If (Q, n) -* (R, m) is a surjective local homomorphism with kernel I, such that I C n2 and the conormal module I/12 has a free summand of rank n, then the degree 2 central subspace of the homotopy Lie algebra of R has dimension greater than or equal to n. This is a corollary of the Main Theorem of this note. The techniques involved provide new proofs of some well known results concerning the conormal module. Let R be a noetherian local ring with maximal ideal m and residue field k. Ring theoretic properties of R are reflected on the algebra structures carried by TorR (k, k) and EXtR (k, k) . Recall that the former has the rh-product of Cartan and Eilenberg, and the latter the Yoneda multiplication. The k-algebra structure of TorR (k, k) is rather simple: It is free in the appropriate category, and so determined by the dimension of the k-vector space TornR (k, k), for all integers i > 0, that is to say, the Betti numbers of k over R. The multiplicative structure of EXtR (k, k) is quite another matter. The simplicity of the product on TorR (k, k) arises from the fact that it is endowed with additional structures: It is a commutative algebra, in the graded sense, with a family of divided powers, and has a diagonal map that is compatible with the divided powers algebra structure on it. In other words, the k-algebra TorR (k, k) is a commutative Hopf algebra with divided powers, and so is free as a divided powers algebra. In contrast, the multiplication on EXtR(k, k) is decidedly non-commutative, unless R happens to be a complete intersection of a special kind. The graded kdual of the product on TorR (k, k) turns EXtR (k, k) = TorR (k, k) * into a Hopf algebra. Attention has focussed on a certain subspace of primitives of this Hopf algebra, denoted ir(R), which is a Lie algebra with the bracket operation defined by the commutator in EXtR (k, k) . This object is called the homotopy Lie algebra of R. The importance of this Lie algebra is attested to by its defining property: Its universal enveloping algebra is EXtR (k, k) . In this note, we are concerned with the centre of the homotopy Lie algebra of R, denoted ((R). This subspace, besides being a measure of the non-commutativity of ir(R), and hence of ExtR (k, k), plays an important part in the change of rings Received by the editors April 7, 1999 and, in revised form, May 12, 1999. 1991 Mathematics Subject Classification. Primary 13C15, 13D03, 13D07, 18G15.

Some Restrictions on the Maps in Minimal Resolutions

We show that there are certain restrictions on the entries of the maps in the minimal free resolutions of finitely generated modules of infinite projective dimension over Noetherian local rings. These restrictions provide a way to slightly improve Herzog's characterization of modules of finite projective and injective dimensions in characteristicp>0. We also discuss other homological situations related to these restrictions, including previously known results on the existence of free summands in syzygy modules.

Antiisomorphisms of endomorphism rings of modules close to free ones which are induced by Morita antiequivalences

The paper deals with antiisomorphisms of endomorphism rings of modules close to free ones (those modules are strong generators, i.e., they contain free cyclic direct summands). A criterion for such antiisomorphisms to be induced by a Morita antiequivalence is obtained. Specifically, any antiisomorphism of endomorphism rings of progenerators (e.g., of matrix rings) is induced by a Morita antiequivalence. Bibliography: 9 titles.

Vanishing of free summands in cohen-macaulay approximations

Vanishing of free summands in cohen-macaulay approximations

Free summands in maximal Cohen-Macaulay approximations and Eisenbud operators over hypersurface rings

The Eisenbud operator of a module over a complete hypersurface ring completely determines the delta invariants of this module.

A glance back at the Quillen-Suslin theorem and the recent status o f the Bass-Quillen conjecture

In his famous FAC-article J.P. Serre asked whether all finitely generated projective modules over K[X1,...,Xn] (K a field) are free. This question soon became known in the mathematical society as “Serre’s Conjecture”. This conjecture was proved independently in 1976 by A. A. Suslin and D. Quillen. Important further developments followed from the ideas in Quillen’s proof. In this article we discuss the geometric motivation of the original problem as well as certain aspects of Quillen’s proof. Then we discuss further developments, in particular a proof that finitely generated projective modules over R[X1,...,Xn], R a Bezout domain, are free. We also give attention to new results on the existence of free summands in projective modules. In the case of polynomial rings, this work strengthens Serre’s famous free summand theorem: if R is a Noetherian ring and P is a finitely generated projective R-module such that rank P > dim R, then there exists a free rank one summand in P. Finally we discuss the present status of the most important open problem in this field, namely the Bass-Quillen Conjecture: if R is a regular local ring, is every finitely generated projective R[X1,...,Xn] module free?

BIVARIANT LONG EXACT SEQUENCES I

Abstract Given a pair of short exact sequences in an abelian category A, extending results due to Pressman [7]. We construct nine bivariant long exact sequences that involve the groups extn(δ,β), extn(δ,α) and extn(γ,α) that interlock with the univariant hom-ext sequences determined by (i). Two applications of these sequences in the category of abelian groups are given. Firstly we describe an isomorphism that yields a “constructive” way to compute the adjusted cotorsion part of a reduced cotorsion group and secondly, given a torsion-free, reduced group G with no free summands, we indicate how to determine the p-ranks of the divisible group Ext(G,Z).

Modules over rings with a power basis

A homological category is constructed and also a functor into it from the category of finitely generated modules over a ring with a power basis such that from an object that corresponds to a module the latter is uniquely determined to within free direct summands, and from a morphism corresponding to a module homomorphism this homomorphism can be uniquely reconstructed to within norms.

Free Summands of Stably Free Modules

AbstractIn this note we show that the generic orthogonal stably free modules of type (2, 7) and (3, 8) have one free summand. This completes the work of other authors on free summands of orthogonal stably free modules.

Projective summands in generators

An R-module M is a generator (of the category of modules) provided every module is a homomorphic image of a suitable direct sum of copies of M. Equivalently, some M(k) has R as a summand. Except in the last section, all rings are assumed to be commutative, Noetherian domains, and modules are usually finitely generated. In this context generators are exactly those modules that have non-zero free summands locally. Of course, generators can fail to have free summands (e.g., over Dedekind domains), and we ask whether they necessarily have non-zero projective summands. The answer is “yes” for rings of dimension 1, as we point out in § 3, and for the polynomial ring in one variable over a Dedekind domain. In § 1 we show that for 2-dimensional rings the answer is intimately connected with the structure of projective modules. Our main result in the positive direction, Theorem 1.3, grew out of the attempt, in conversations with T. Stafford, to understand the case R = k[x, y]. In § 2 we give examples of rings having generators with no projective summands. The last section contains miscellaneous observations, some of them on rings without chain conditions.