We show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds in the derived category of a ring R if and only if R is von Neumann regular. This extends results of the second author. We also characterize rings for which the original form (the faithful version) of the generating hypothesis holds in the derived category of R. These must be close to von Neumann regular in a precise sense, and, given any of a number of finiteness hypotheses, must be von Neumann regular. However, we construct an example of such a ring that is not von Neumann regular, and therefore does not satisfy the strong form of the generating hypothesis.
We construct a "diagonal" cofibrantly generated model structre on the category of simplicial objects in the category of topological categories sCat_{Top}, which is the category of diagrams [Δ^{op}, Cat_{Top}]. Moreover, we prove that the diagonal model structures is left proper and cellular. We also prove that the category of \infty-groupoids (the full subcategory of topological categories) has a cofibrantly generated model structure and is Quillen equivalent to the model category of simplicial sets, which proves the Grothendieck's homotopy hypothesis.
We consider the derived category of a specific non-Noetherian ring Λ, and show that there are objects in D(Λ) that are not Bousfield equivalent to any module. This answers a question posed by Dwyer and Palmieri.
In this paper, we prove a version of Freyd's generating hypothesis for triangulated categories: if D is a cocomplete triangulated category and S is an object in D whose endomorphism ring is graded commutative and concentrated in degree zero, then S generates (in the sense of Freyd) the thick subcategory determined by S if and only if the endomorphism ring of S is von Neumann regular. As a corollary, we obtain that the generating hypothesis is true in the derived category of a commutative ring R if and only if R is von Neumann regular. We also investigate alternative formulations of the generat…
Freyd's generating hypothesis, interpreted in the stable module category of a finite p-group G, is the statement that a map between finite-dimensional kG-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd's generating hypothesis holds for a non-trivial finite p-group G if and only if G is either C_2 or C_3. We also give various conditions which are equivalent to the generating hypothesis.
Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C|=BC. We study the ring homomorphism HH*(kC) --> H*(|C|,k) and prove it is split surjective. This generalizes the well-known results for groups and posets. Based on this result, we construct a seven-dimensional category algebra whose Hochschild cohomology ring modulo nilpotents is not finitely generated, against a conjecture of Snashall and Solberg.
The invertibility hypothesis for a monoidal model category S asks that localizing an S-enriched category with respect to an equivalence results in an weakly equivalent enriched category. This is the most technical among the axioms for S to be an excellent model category in the sense of Lurie, who showed that the category of S-enriched categories then has a model structure with characterizable fibrant objects. We use a universal property of cubical sets, as a monoidal model category, to show that the invertibility hypothesis is consequence of the other axioms.
We show that if $2k\geq n$, then a k-connected weak n-category $A$ can be ``delooped'' to a k+1-connected weak n+1-category $Y$ with $Hom_Y(y,y)\cong A$. This is the essential part of the ``stabilization hypothesis'' of Baez and Dolan q-alg/9503002, math/9802029.
We study the category of discrete modules over the ring of degree zero stable operations in p-local complex K-theory. We show that the p-local K-homology of any space or spectrum is such a module, and that this category is isomorphic to a category defined by Bousfield and used in his work on the K-local stable homotopy category (Amer. J. Math., 1985). We also provide an alternative characterisation of discrete modules as locally finitely generated modules.
Let M be a bicomplete, closed symmetric monoidal category. Let P be an operad in M, i.e., a monoid in the category of symmetric sequences of objects in M, with its composition monoidal structure. Let R be a P-co-ring, i.e., a comonoid in the category of P-bimodules. The co-ring R induces a natural ``fattening'' of the category of P-(co)algebras, expanding the morphism sets while leaving the objects fixed. Co-rings over operads are thus ``relative operads,'' parametrizing morphisms as operads parametrize (co)algebras. Let A denote the associative operad in the category of chain complexes. We de…
We state the generating hypothesis in the homotopy category of G-spectra for a compact Lie group G, and prove that if G is finite, then the generating hypothesis implies the strong generating hypothesis, just as in the non-equivariant case. We also give an explicit counterexample to the generating hypothesis in the category of rational S^1-equivariant spectra.
We classify bounded t-structures on the category of perfect complexes over a commutative, Noetherian ring of finite Krull dimension, extending a result of Alonso Tarrio, Jeremias Lopez and Saorin which covers the regular case. In particular, we show that there are no bounded t-structures in the singular case, verifying the affine version of a conjecture of Antieau, Gepner and Heller, and also that there are no non-trivial t-structures at all in the singular, irreducible case.
In this short paper we outline (mostly without proofs) our new approach to the derived category of sheaves of commutative DG rings. The proofs will appear in a subsequent paper. Among other things, we explain how to form the derived intersection of two closed subschemes inside a given algebraic scheme X, without recourse to simplicial or higher homotopical methods, and without any global assumptions on X.
We construct a number of different examples of non-Noetherian graded rings that are injective as modules over themselves (or have some related but weaker properties). We discuss how these are related to the theory of triangulated categories, and to Freyd's Generating Hypothesis in stable homotopy theory.
In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category …
We introduce and study a notion of cylinder coherator similar to the notion of Grothendieck coherator which define more flexible notion of weak infinity groupoids. We show that each such cylinder coherator produces a combinatorial semi-model category of weak infinity groupoids, whose objects are all fibrant and which is in a precise sense "freely generated by an object". We show that all those semi model categories are Quillen equivalent together and Quillen to the model category of spaces. A general procedure is given to produce such coherator, and several explicit examples are presented: one…
Given a monoidal $\infty$-category $C$ equipped with a monoidal recollement, we give a simple criterion for an object in $C$ to be dualizable in terms of the dualizability of each of its factors and a projection formula relating them. Predicated on this, we then characterize dualizability in any monoidally stratified $\infty$-category in terms of stratumwise dualizability and a projection formula for the links. Using our criterion, we prove a 1-dimensional bordism hypothesis for symmetric monoidal recollements. Namely, we provide an algebraic enhancement of the 1-dimensional framed bordism $\i…
We show that the cellular objects in the module category over a motivic E infinity ring spectrum E can be described as the module category over a graded topological spectrum if E is strongly periodizable in our language. A similar statement is proven for triangulated categories of motives. Since MGL is strongly periodizable we obtain topological incarnations of motivic Landweber spectra. Under some categorical assumptions the unit object of the model category for triangulated motives is as well strongly periodizable giving motivic cochains whose module category models integral triangulated cat…
It is well known that very special $Γ$-spaces and grouplike $\E_\infty$ spaces both model connective spectra. Both these models have equivariant analogues. Shimakawa defined the category of equivariant $Γ$-spaces and showed that special equivariant $Γ$-spaces determine positive equivariant spectra. Costenoble and Waner showed that grouplike equivariant $\E_\infty$-spaces determine connective equivariant spectra. We show that with suitable model category structures the category of equivariant $Γ$-spaces is Quillen equivalent to the category of equivariant $\E_\infty$ spaces. We define the units…
We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category R we construct a natural additive group completion R' that retains the multiplicative structure, hence has become a ring category. If we start with a commutative rig category R (also known as a symmetric bimonoidal category), the additive group completion R' will be a commutative ring category. In an accompanying paper we show how this can be used to prove the conjecture from [BDR] that the algebraic K-theory of the connective top…
Every homology or cohomology theory on a category of E-infinity ring spectra is Topological Andre-Quillen homology or cohomology with appropriate coefficients. Analogous results hold for the category of A-infinity ring spectra and for categories of algebras over many other operads.
In the context of a well generated tensor triangulated category, Section 3 investigates the relationship between the Bousfield lattice of a quotient and quotients of the Bousfield lattice. In Section 4 we develop a general framework to study the Bousfield lattice of the derived category of a commutative or graded-commutative ring, using derived functors induced by extension of scalars. Section 5 applies this work to extend results of Dwyer and Palmieri [DP08] to new non-Noetherian rings.
In this paper, we construct a version of Auslander-Reiten sequences for the $K(n)$-local stable homotopy category. In particular, the role of the Auslander-Reiten translation is played by the local Brown-Comenetz duality functor. As an application, we produce counterexamples to the $K(n)$-local generating hypothesis for all heights $n>0$ and all primes. Furthermore, our methods apply to other triangulated categories, as for example the derived category of quasi-coherent sheaves on a smooth projective scheme.
A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a `r…
We propose a new notion of `n-category with duals', which we call a Whitney n-category. There are two motivations. The first is that Baez and Dolan's Tangle Hypothesis is (almost) tautological when interpreted as a statement about Whitney categories. The second is that we can functorially construct `fundamental Whitney n-categories' from each smooth stratified space X. These are obtained by considering the homotopy theory of smooth maps into X which are transversal to all strata. This makes concrete another idea of Baez and Dolan's which is that a suitable version of homotopy theory for strati…
We classify complactly generated t-structures on the derived category of modules over a commutative Noetherian ring R in terms of decreasing filtrations by supports on Spec(R). A decreasing filtration by supports ϕ: Z -> Spec(R) satisfies the weak Cousin condition if for any integer i \in Z, the set ϕ(i) contains all the inmediate generalizations of each point in ϕ(i+1). Every t-structure on D^b_fg(R) (equivalently, on D^-_fg(R)) is induced by complactly generated t-structures on D(R) whose associated filtrations by supports satisfy the weak Cousin condition. If the ring R has dualizing comple…
Pursuing ideas of Jeff Smith, we develop a homotopy theory of ideals of monoids in a symmetric monoidal model category. This includes Smith ideals of structured ring spectra and of differential graded algebras. Such Smith ideals are NOT subobjects, and as a result the theory seems to require us to consider all Smith ideals of all monoids simultaneously, rather then restricting to the Smith ideals of one particular monoid. However, we can take a quotient by a Smith ideal and get a monoid homomorphism. In the stable case, we show that this construction is part of a Quillen equivalence between a …
A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that - similar to spaces - the derived category obtained by inverting the quasi-isomorphisms is naturally left triangulated. Also, homology theories on rings are studied. These must be homotopy invariant in the algebraic sense, meet the Mayer-Vietoris property and plus some minor natural axioms. To …
We give a new construction of the algebraic $K$-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and output. This requires us to define multiplicative structure on the category of small permutative categories. The framework we use is the concept of multicategory, a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction. Our method ends up in Smith's category of symmetric…
Let $\hat\Z_p$ be the ring of $p$-adic integers. We prove in the present paper that the category of polynomial functors from finitely generated free abelian groups to $\hat \Z_p$-modules of degree at most $p$ is equivalent to the category of finitely generated modules over a particularly well understood ring, called Green order. That this is the case was conjectured by Yuri Drozd.
An approach to identify the normal subgroups determined by ideals in free group rings with the help of the derived functors of non-additive functors is explored. A similar approach, i.e., via derived functors, for computing limits of functors from the category of free presentations to the category of abelian groups, arising from commutator structure of free groups, is also discussed.
In this paper we study representation theory of the category FI$^m$ introduced by Gadish which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FI$^m$ in a natural way. In particular, we prove the representation stability property of finitely generated FI$^m$-modules over fields of characteristic 0.
We develop homotopical algebraic geometry (see math.AG/0207028) in the special context where the base symmetric monoidal model category is the category S of spectra, i.e. what might be called, after Waldhausen, ``brave new algebraic geometry''. We discuss various model topologies on the model category of commutative algebras in S, the associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry), and finally show how to define global moduli spaces of associative ring spectra structures and a moduli space related to topo…
It is well known that the zeroth stable homotopy group of a genuine equivariant commutative ring spectrum has multiplicative transfers (norms), making it into a Tambara functor. We prove here that all Tambara functors can be obtained in this way. In fact, we prove that the homotopy category of Eilenberg MacLane commutative ring spectra is equivalent to the category of Tambara functors. Several algebraic results on Tambara functors are derived as corollaries. Finally, we rule out the existence of a lax symmetric monoidal construction for Eilenberg MacLane spectra when the group is nontrivial.
Freyd's generating hypothesis for the stable module category of a non-trivial finite group G is the statement that a map between finitely generated kG-modules that belongs to the thick subcategory generated by k factors through a projective if the induced map on Tate cohomology is trivial. In this paper we show that Freyd's generating hypothesis fails for kG when the Sylow p-subgroup of G has order at least 4 using almost split sequences. By combining this with our earlier work, we obtain a complete answer to Freyd's generating hypothesis for the stable module category of a finite group. We al…
This very speculative sketch suggests that a theory of fundamental groupoids for tensor triangulated categories could be used to describe the ring of integers as the singular fiber in a family of ring-spectra parametrized by a structure space for the stable homotopy category, and that Bousfield localization might be part of a theory of `nearby' cycles for stacks or orbifolds.
We prove two results from Morita theory of stable model categories. Both can be regarded as topological versions of recent algebraic theorems. One is on recollements of triangulated categories, which have been studied in the algebraic case by Jørgensen. We give a criterion which answers the following question: When is there a recollement for the derived category of a given symmetric ring spectrum in terms of two other symmetric ring spectra? The other result is on well generated triangulated categories in the sense of Neeman. Porta characterizes the algebraic well generated categories as local…
This is a survey paper, based on lectures given at the Workshop on "Structured ring spectra and their applications" which took place January 21-25, 2002, at the University of Glasgow. The term `Morita theory' is usually used for results concerning equivalences of various kinds of module categories. We focus on the covariant form of Morita theory, so the basic question is: When do two `rings' have `equivalent' module categories ? We discuss this question in different contexts and illustrate it by examples: (Classical) When are the module categories of two rings equivalent as categories ? (Deriv…
The dihedral homology functor $HD:A_\infty^{\rm inv}(K)\to GrM(K)$ from the category $A_\infty^{\rm inv}(K)$ of involutive $A_\infty$-algebras over any commutative unital ring $K$ to the category $GrM(K)$ of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy equivalences of involutive $A_\infty$-algebras into isomorphisms of graded modules.
We give, for every finite group G, a combinatorial description of the ring of G-Witt vectors on a polynomial algebra over the integers. Using this description we show that the functor, which takes a ring with trivial action of G to its ring of Witt vectors, coincides with the left adjoint of the algebraic functor from the category of G-Tambara functors to the category of rings with an action of G.
