Investigate the construction of polynomial ring spectra and a potential monoid ring functor for E₁ ring spectra | Step-by-Step Solution

🎓 Understanding Monoid Rings and Polynomial Ring Spectra

What We're Solving: We're exploring how to construct polynomial-like objects in the world of ring spectra, specifically looking at how to build a "monoid ring" functor for E₁ ring spectra and understand the difference between commuting and non-commuting polynomial constructions.

The Approach: This is a research-oriented problem that requires synthesizing advanced concepts from homotopy theory and algebraic topology. The goal is to guide you through the conceptual framework and research approach to tackle this independently!

Step-by-Step Research Strategy:

Step 1: Establish Your Foundations

• Review what E₁ ring spectra are (associative but not necessarily commutative ring spectra)
• Understand classical monoid rings R[M] for a ring R and monoid M
• Study how these constructions generalize to the spectral setting

Step 2: Investigate Existing Literature

• Look into work by Elmendorf-Kriz-Mandell-May on ring spectra
• Research Basterra and Mandell's work on commutative ring spectra
• Examine how the "free E₁-algebra" functor works

Step 3: Develop Your Construction Framework Consider these key questions:

• How do we define R[M] when R is an E₁ ring spectrum?
• What categorical properties should this functor satisfy?
• How does the monoidal structure of spectra interact with monoid actions?

Step 4: Compare Commutative vs Non-Commutative Cases Analyze:

• For E∞ ring spectra: polynomial rings R[x₁,...,xₙ] where variables commute
• For E₁ ring spectra: "polynomial" rings R⟨x₁,...,xₙ⟩ where variables may not commute
• How these relate to free algebra constructions

The Framework You Should Develop:

Instead of a single answer, you should produce:

• 1. Definition Section: Precise definition of your monoid ring functor
• 2. Construction Details: How to build R[M] using categorical tools
• 3. Properties Analysis: What universal properties your construction satisfies
• 4. Examples Section: Concrete computations with simple monoids
• 5. Comparison Study: How commutative vs non-commutative cases differ

Memory Tip: Think of this as "building algebraic structures in the land of spectra" - you're taking familiar algebraic constructions and carefully translating them to work in the more flexible world of ring spectra, where you have to be very careful about what "commutative" means!

Encouragement: This is graduate-level research mathematics! Start with the classical case, understand it deeply, then gradually work toward the spectral generalization. Each small step builds your understanding! 🌟