Mark Hovey

Spectra and symmetric spectra in general model categories

Abstract We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as the A 1 -homotopy theory of Morel and Voevodsky (preprint, 1998) and Voevodsky (Proceedings of the International Congress of Mathematicians, Vol. I, Berlin, Doc. Math. Extra Vol. I, 1998, pp. 579–604 (electronic)). One is based on the standard notion of spectra originated by Vogt (Boardmans Stable Homotopy Category, Lecture Notes Series, Vol. 21, Matematisk Institut Aarhus Universitet, Aarhus, 1970). Its input is a well-behaved model category D and an endofunctor T, generalizing the suspension. Its output is a model category Sp N ( D ,T) on which T is a Quillen equivalence. The second construction is based on symmetric spectra (Hovey et al., J. Amer. Math. Soc. 13(1) (2000) 149–208) and applies to model categories C with a compatible monoidal structure. In this case, the functor T must be given by tensoring with a cofibrant object K. The output is again a model category Sp Σ ( C ,K) where tensoring with K is a Quillen equivalence, but now Sp Σ ( C ,K) is again a monoidal model category. We study general properties of these stabilizations; most importantly, we give a sufficient condition for these two stabilizations to be equivalent that applies both in the known case of topological spaces and in the case of A 1 -homotopy theory.

MODEL CATEGORY STRUCTURES ON CHAIN COMPLEXES OF SHEAVES

In this paper, we try to determine when the derived category of an abelian category is the homotopy category of a model structure on the category of chain complexes. We prove that this is always the case when the abelian category is a Grothendieck category, as has also been done by Morel. But this model structure is not very useful for dening derived tensor products. We therefore consider another method for constructing a model structure, and apply it to the category of sheaves on a well-behaved ringed space. The resulting flat model structure is compatible with the tensor product and all homomorphisms of ringed spaces.

A-cordial graphs

Abstract We introduce A -cordial graphs, for an abelian group A . (If A = Z k we call them k -cordial graphs.) These generalize harmonious, elegant, and cordial graphs. They also provide a graph-theoretic realization of the function υ γ studied by Graham and Sloane (1980). We show that trees are 3, 4 and 5-cordial and provide a finite (though long) test that, if passed, guarantees that all trees are A -cordial. We conjecture that trees are k -cordial for all k . We provide a partial classification of which cycles and complete graphs are k -cordial, and we show that for k even and >4, most graphs are not k -cordial.

Invertible Spectra in the E(n)-Local Stable Homotopy Category

Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such invertible objects. The Picard group need not be a set in general, but if it is then it is an abelian group canonically associated with C. There are many examples of symmetric monoidal categories in stable homotopy theory. In particular, one could take the whole stable homotopy category S. In this case, it was proved by Hopkins that the Picard group is just Z, where a representative for n can be taken to be simply the n-sphere S [HMS94, Str92]. It is more interesting to consider Picard groups of the E-local category, for various spectra E (all of which will be p-local for some fixed prime p in this paper). Here the smash product of two E-local spectra need not be E-local, so one must relocalize the result by applying the Bousfield localization functor LE . The most well-known case is E = K(n), the nth Morava K-theory, considered in [HMS94]. In this paper we study the case E = E(n), where E(n) is the Johnson-Wilson spectrum. In this case the E-localization functor is universally denoted Ln, and we denote the category of E-local spectra by L. Our main theorem is the following result.

Classifying subcategories of modules

In this paper, we classify certain subcategories of modules over a ring R. A wide subcategory of R-modules is an Abelian subcategory of R-Mod that is closed under extensions. We give a complete classification of wide subcategories of finitely presented modules when R is a quotient of a regular commutative coherent ring by a finitely generated ideal. This includes all finitely presented algebras over a principal ideal domain, as well as polynomial rings on infinitely many variable over a PID. The classification is in terms of subsets of Spec R, and depends heavily on Thomasons classification of thick subcategories of small objects in the derived category. We also classify all wide subcategories closed under arbitrary coproducts for any Noetherian commutative ring R. These correspond to arbitrary subsets of Spec R, and this classification depends on Neemans classification of localizing subcategories of the derived category.

Morita theory for Hopf algebroids and presheaves of groupoids

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology T on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasi-coherent sheaves. The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules. We apply this to formal group laws, where we get considerable generalizations of the Miller-Ravenel and Hovey-Sadofsky change of rings theorems in algebraic topology.

GORENSTEIN MODEL STRUCTURES AND GENERALIZED DERIVED CATEGORIES

In a paper from 2002, Hovey introduced the Gorenstein projective and Gorenstein injective model structures on R -Mod, the category of R -modules, where R is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce: the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable module category of R . If such a ring R has finite global dimension, the graded ring R [ x ]/( x 2 ) is Gorenstein and the three associated Gorenstein model structures on R [ x ]/( x 2 )-Mod, the category of graded R [ x ]/( x 2 )-modules, are nothing more than the usual projective, injective and flat model structures on Ch( R ), the category of chain complexes of R -modules. Although these correspondences only recover these model structures on Ch( R ) when R has finite global dimension, we can set R = ℤ and use general techniques from model category theory to lift the projective model structure from Ch(ℤ) to Ch( R ) for an arbitrary ring R . This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when ℤ[ x ]/( x 2 ) is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring R and the derived category of R and we give some examples of such generalizations.

THE 7-CONNECTED COBORDISM RING AT p = 3

In this paper, we study the cobordism spectrum MOh8i at the prime 3. This spectrum is important because it is conjectured to play the role for elliptic cohomology that Spin cobordism plays for real K-theory. We show that the torsion is all killed by 3, and that the Adams-Novikov spectral sequence collapses after only 2 dierentials. Many of our methods apply more generally.

Cohomological Bousfield classes

In this paper, we begin the study of Bousfield classes for cohomology theories defined on spectra. Our main result is that a map f:X → Y induces an isomorphism on E(n)-cohomology if and only if it induces an isomorphism on E(n)-homology. We also prove this for variants of E(n) such as elliptic cohomology and real K-theory. We also show that there is a nontrivial map from a spectrum Z to the K(n)-local sphere if and only if K(n)∗(Z) ≠ 0.

The ghost dimension of a ring

We introduce the concept of the ghost dimension gh.dim. R of a ring R. This is the longest nontrivial chain of maps in the derived category emanating from a perfect complex such that each map is zero on homology. We show that w.dim. R < gh.dim. R with equality if R is coherent or w.dim. R = 1.