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For questions about ring spectra (in homotopy theory).
7
votes
Morphisms of $\mathbb E_l$-rings between $\mathbb E_k$-rings for $l<k$
Let $R$ be an $E_\infty$-ring spectrum and write $R\{t\}$ (resp. $R[t]$) for the free $E_\infty$-$R$-algebra (resp. free $E_1$-$R$-algebra) on one generator $t$ (in degree zero). This notation is com …
7
votes
Accepted
When do the polynomial algebra and free algebra coincide in brave new algebra?
Any morphism of $R$-algebras $\varphi : R\{t\} \to R[t]$ is determined up to homotopy by an element of $\pi_0(R[t]) \approx \pi_0(R)[t]$. If $\varphi$ is an equivalence, then this element must be the …
23
votes
Accepted
Motivation and potential applications of spectral algebraic geometry
This is not really an answer to your question, just an attempt to address your question from the comments.
There are various flavours of homotopical or higher algebraic geometry that are commonly con …