Talk 4: Evenness I.

Give some basic intuition for $\mathbb{E}_{\infty}$ rings as spectra where $\pi_{0}(R)$ is a commutative ring and $\pi_{n}(R)$ a $\pi_{0}(R)$-module for each $n$ with graded-commutative multiplication. Introduce the notion of evenness of $\mathbb{E}_{\infty}$-rings and discuss how they arise naturally in algebraic topology (eg. $\mathrm{MU}$). State and prove results relating to the commutative algebra of even rings. Conclude by constructing the faithfully flat topology (ie. ignore the $p$-complete and $S^{1}$ settings).