Commutative Γ-rings do not model all commutative ring spectra

Commutative Γ-rings Do Not Model All Commutative Ring Spectra

We show that the free E∞-algebra on a zero-cell cannot be modeled by a commutative Γ-ring. The proof shows that DyerLashof operations of positive degree must vanish on the zero’th homology of such an object.

-homology of Commutative Rings and of E1 Ring Spectra

Realizing Commutative Ring Spectra as E1 Ring Spectra

We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E 1 ring spectrum. The obstruction groups are Andr e-Quillen cohomology groups of an algebra over an E 1 operad. The same cohomology theory is part of a spectral sequence for computing the homotopy type of mapping spaces between E 1 ring spectrum. The obstruction theory arises ...

ON COMMUTATIVE GELFAND RINGS

A ring is called a Gelfand ring (pm ring ) if each prime ideal is contained in a unique maximal ideal. For a Gelfand ring R with Jacobson radical zero, we show that the following are equivalent: (1) R is Artinian; (2) R is Noetherian; (3) R has a finite Goldie dimension; (4) Every maximal ideal is generated by an idempotent; (5) Max (R) is finite. We also give the following resu1ts:an ideal...

Moduli Spaces of Commutative Ring Spectra

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∗E-comodules, 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 result...