Results 1  10
of
15
Moduli Spaces of Commutative Ring Spectra
, 2003
"... Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as ..."
Abstract

Cited by 19 (0 self)
 Add to MetaCart
(Show Context)
Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as a moduli problem, and give a way  suggested by work of Dwyer, Kan, and Stover  of dissecting the resulting moduli space as a tower with layers governed by appropriate AndreQuillen cohomology groups. A special case is A = E#E itself. The final section applies this to discuss the LubinTate or Morava spectra En .
A universality theorem for voevodsky’s algebraic cobordism spectrum
, 709
"... An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P 1spectra, equipped with the symmetric monoidal structure described in [PPR1]. The algebraic cobordism P 1spectrum MGL is considered as a comm ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P 1spectra, equipped with the symmetric monoidal structure described in [PPR1]. The algebraic cobordism P 1spectrum MGL is considered as a commutative monoid equipped with a canonical orientation th MGL ∈ MGL 2,1 (Th(O(−1))). For a commutative monoid E in the category SH(k) it is proved that assignment ϕ ↦ → ϕ(th MGL) identifies the set of monoid homomorphisms ϕ: MGL → E in the motivic stable homotopy category SH(k) with the set of all orientations of E. The result was stated originally in a slightly different form by G. Vezzosi in [Ve]. 1 Oriented commutative ring spectra We refer to [PPR1, Appendix] for the basic terminology, notation, constructions, definitions, results. For the convenience of the reader we recall the basic definitions. Let S be a Noetherian scheme of finite Krull dimension. One may think of S being the spectrum of
Moduli problems for structured ring spectra
 DANIEL DUGGER AND BROOKE
, 2005
"... In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theore ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theorem on the LubinTate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is nonempty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.
RiemannRoch theorem for oriented cohomology
, 2002
"... Notion of an oriented cohomology pretheory on algebraic varieties is introduced and a RiemannRoch theorem for ring morphisms between oriented pretheories is proved. An explicit formula for the Todd genus related to a ring morphism is given. ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
Notion of an oriented cohomology pretheory on algebraic varieties is introduced and a RiemannRoch theorem for ring morphisms between oriented pretheories is proved. An explicit formula for the Todd genus related to a ring morphism is given.
Realizing Commutative Ring Spectra as E∞ Ring Spectra
, 1999
"... We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E∞ ring spectrum. The obstruction groups are AndréQuillen cohomology groups of an algebra over an E∞ operad. The same cohomology theory is part of a spectral sequence for comput ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E∞ ring spectrum. The obstruction groups are AndréQuillen cohomology groups of an algebra over an E∞ operad. The same cohomology theory is part of a spectral sequence for computing the homotopy type of mapping spaces between E∞ ring spectrum. The obstruction theory arises out of techniques of Dwyer, Kan, and Stover, and the main application here is to prove an analog of a theorem of Haynes Miller and the second author: the LubinTate spectra En are E∞ and the space of E∞ selfmaps has weakly contractible components.
On analytical applications of stable homotopy (the Arnold conjecture, critical points
 Math. Zeitschrift
, 1999
"... Abstract. We prove the Arnold conjecture for closed symplectic manifolds with π2(M) = 0 and cat M = dim M. Furthermore, we prove an analog of the Lusternik– Schnirelmann theorem for functions with “generalized hyperbolicity ” property. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We prove the Arnold conjecture for closed symplectic manifolds with π2(M) = 0 and cat M = dim M. Furthermore, we prove an analog of the Lusternik– Schnirelmann theorem for functions with “generalized hyperbolicity ” property.
On the relation of Voevodsky’s algebraic cobordism to Quillen’s Ktheory
 Invent. Math
"... Quillen’s algebraic Ktheory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P 1spectrum MGL of Voevodsky is considered as a commutative P 1ring spectrum. Setting MGL i = ⊕p−2q=i MGL p,q we regard the bigraded theory MGL p,q as jus ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Quillen’s algebraic Ktheory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P 1spectrum MGL of Voevodsky is considered as a commutative P 1ring spectrum. Setting MGL i = ⊕p−2q=i MGL p,q we regard the bigraded theory MGL p,q as just a graded theory. There is a unique ring morphism φ: MGL 0 (k) → Z which sends the class [X]MGL of a smooth projective kvariety X to the Euler characteristic χ(X,OX) of the structure sheaf OX. Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories
Realizing families of Landweber exact homology theories, from: “New topological contexts for Galois theory and algebraic geometry (BIRS 2008
, 2009
"... I discuss the problem of realizing families of complex orientable homology theories as families of E ∞ring spectra, including a recent result of Jacob Lurie emphasizing the role of pdivisible groups. 1 55N22; 55N34, 14H10 A few years ago, I wrote a paper [10] discussing a realization problem for f ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
I discuss the problem of realizing families of complex orientable homology theories as families of E ∞ring spectra, including a recent result of Jacob Lurie emphasizing the role of pdivisible groups. 1 55N22; 55N34, 14H10 A few years ago, I wrote a paper [10] discussing a realization problem for families of Landweber exact spectra. Since Jacob Lurie [27] now has a major positive result in this direction, it seems worthwhile to revisit these ideas. In brief, the realization problem can be stated as follows. Suppose we are given a flat morphism g: Spec(R) − → Mfg from an affine scheme to the moduli stack of smooth 1dimensional formal groups. Then we get a twoperiodic homology theory E(R, G) with E(R, G)0 ∼ = R and associated formal group G = Spf(E 0 CP ∞) isomorphic to the formal group classified by g. The higher homotopy groups of E(R, G) are zero in odd degrees and E(R, G)2n ∼ = ω ⊗n G where ωG is the module of invariant differentials for G. The module ωG is locally free of rank 1 over R, and free of rank 1 if G has a coordinate. In this case E(R, G) ∗ = R[u ±1] where u ∈ E(R, G)2 is a generator. The fact that g was flat implies E(R, G) is Landweber exact, even if G doesn’t have a coordinate. Now suppose we are given a flat morphism of stacks g: X − → Mfg. 1
Resolutions in Model Categories
, 1997
"... Let C be a closed model category. Following work of Dwyer, Kan, and Stover, we develop a technique for building simplicial resolutions of objects in C which are free, or homotopically free, in a precise sense. Specifically, we specify in advance a set A of small, co brant, coH objects in C and the ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Let C be a closed model category. Following work of Dwyer, Kan, and Stover, we develop a technique for building simplicial resolutions of objects in C which are free, or homotopically free, in a precise sense. Specifically, we specify in advance a set A of small, co brant, coH objects in C and the simplicial resolutions X will have the property that at each level n, the object Xn is homotopy equivalent to a coproduct of objects of A. These Stovertype resolutions are modeled on the free resolutions developed by Quillen, and are meant to be a replacement for projective resolutions in this context. In the end we develop some computational tools for spectra and structured ring spectra.