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A partial converse ghost lemma for the derived category of a commutative Noetherian ring
Abstract
In this article a condition is given to detect the containment among thick subcategories of the bounded derived category of a commutative noetherian ring. More precisely, for a commutative noetherian ring R R and complexes of R R -modules with finitely generated homology M M and N N , we show N N is in the thick subcategory generated by M M if and only if the ghost index of N p N_\mathfrak {p} with respect to M p M_\mathfrak {p} is finite for each prime p \mathfrak {p} of R R . To do so, we establish a “converse coghost lemma” for the bounded derived category of a non-negatively graded DG algebra with noetherian homology.
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We study homological properties of a locally complete intersection ring by importing facts from homological algebra over exterior algebras. One application is showing that the thick subcategories of the bounded derived category of a locally complete intersection ring are self-dual under Grothendieck duality. This was proved by Stevenson when the ring is a quotient of a regular ring modulo a regular sequence; we offer two independent proofs in the more general setting. Second, we use these techniques to supply new proofs that complete intersections possess symmetry of complexity.
Conditions on the Koszul complex of a noetherian local ring R guarantee
that is non-zero for infinitely many i, when M
and N are finitely generated R-modules of infinite projective dimension.
These conditions are obtained from results concerning Tor of differential
graded modules over certain trivial extensions of commutative differential
graded algebras.
Local cohomology functors are constructed for the category of cohomological
functors on an essentially small triangulated category T equipped with an
action of a commutative noetherian ring. This is used to establish a
local-global principle and to develop a notion of stratification, for T and the
cohomological functors on it, analogous to such concepts for compactly
generated triangulated categories.
The paper explores dualizing differential graded (DG) modules over DG algebras. The focus is on DG algebras that are commutative local, and finite. One of the main results established is that, for this class of DG algebras, a finite DG module is dualizing precisely when its Bass number is 1. As a corollary, one obtains that the Avramov–Foxby notion of Gorenstein DG algebras coincides with that due to Frankild and Jørgensen. One other key result is that, under suitable hypotheses, any two dualizing DG modules are quasiisomorphic up to a suspension. In addition, it is established that a number of naturally occurring DG algebras possess dualizing DG modules.
We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal of morphisms with certain properties and that if has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik–Schnirelmann category of a space, and we calculate this stable analogue for low-dimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case ofA∞modules over anA∞ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps fromXtoYcan always be described as agroup. The last two sections focus on algebraic examples. In the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of Kelly on the vanishing of composites of such maps. We also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non-trivially.
We show, for a wide class of abelian categories relevant in representation
theory and algebraic geometry, that the bounded derived categories have no
non-trivial strongly finitely generated thick subcategories containing all
perfect complexes. In order to do so we prove a strong converse of the Ghost
Lemma for bounded derived categories.
A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a substitute for the length of a free complex--and on the rank of a differential module in terms of invariants of its homology. These results specialize to basic theorems in commutative algebra and algebraic topology. One instance is a common generalization of the equicharacteristic case of the New Intersection Theorem of Hochster, Peskine, P. Roberts, and Szpiro, concerning complexes over noetherian commutative rings, and of a theorem of G. Carlsson on differential graded modules over graded polynomial rings. Comment: 27 pages. Minor changes; mainly stylistic. To appear in Inventiones Mathematicae
The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp’s conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.
This 1987 volume presents a collection of papers given at the 1985 Durham Symposium on homotopy theory. They survey recent developments in the subject including localisation and periodicity, computational complexity, and the algebraic K-theory of spaces.
Sullivan Model and Rationalization of a Non-Simply Connected Space Homotopy Lie Algebra of a Space and Fundamental Group of the Rationalization, Model of a Fibration Holonomy Operation in a Fibration Malcev Completion of a Group and Examples Lusternik-Schnirelmann Category Depth of a Sullivan Lie Algebra Growth of Rational Homotopy Groups Structure of Rational Homotopy Lie Algebras Weighted Lie Algebras
We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a 'Morita theorem` (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories. Summary We give an account of the contents of this paper for the special case of DG algebras. Let k be a commutative ring and A a DG (k-)algebra, i.e. a Z-graded k-algebra A = a p2Z A p endowed with a differential d of degree 1 such that d(ab) = (da)b + (\Gamma1) p a(db) for all a 2 A p , b 2 A. A DG (right) A-module is a Z-graded A-module M = ` p2Z M p endowed with a differential d of degree 1 such that d(ma) = (dm)a + (\Gamma1) p m(da) for all m 2 M p , a 2 A. A morphism of DG A-modules is a homogeneous morphism of degree 0 of the underlying graded A-modules commuting with the differentials. The DG A-modules form an abelian category CA. A morphism f : M ! N of CA is null-homotopic if f = dr + rd for some homogeneous morphism r : M ! N of degree-1 of the underlying graded A-modules.
The purpose of this paper is to prove Theorems 1 and 2 below by exhibiting them as special cases of a more general result, Theorem 3, which admits of a simple proof.(Received January 14 1965)
This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matemàtica, Institut d’Estudis Catalans, July 15–26, 1996.
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.
We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension.
We obtain, via the formalism of tensor actions, a complete classification of
the localizing subcategories of the stable derived category of any affine
scheme with hypersurface singularities and of any local complete intersection
over a field; in particular this classifies the thick subcategories of the
singularity categories of such rings. The analogous result is also proved for
certain locally complete intersection schemes. It is also shown that from each
of these classifications one can deduce the (relative) telescope conjecture.
We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.
We define a dimension for a triangulated category. We prove a representabilityTheorem for a certain class of functors on finite dimensional triangulatedcategories. We study the dimension of the boundedderived category of an algebra or a scheme and we show in particularthat the bounded derived category of coherent sheaves over avariety has a finite dimension. For a self-injective algebra, a lowerbound for Auslander's representation dimension is given by the dimensionof the stable category. We use this to compute the representationdimension of exterior algebras. This provides the first known examplesof representation dimension >3. We deduce that theLoewy length of the group algebra over F_2 of a finite group is strictly bounded below by2-rank of the group (a conjecture of Benson).
Some ghost lemmas, lecture notes for the conference “the representation dimension of artin algebras
- Apostolos Beligiannis