Introduce the filtered derived category. Discuss the algebraic properties of graded and filtered objects such as associated gradeds, coherent cochain complexes, and its connection to complete decreasing filtrations. Construct the Beilinson $t$-structure and give an overview of related constructions.
Introduce the definition of a spectral sequence and the construction of a spectral sequence from filtered objects. Define the décalage functor as the realization of the Whitehead tower of the filtered object and the décalage construction of the spectral sequence. Recall Lurie's definition of a spectral sequence.
Introduce the definition of a spectral sequence and the construction of a spectral sequence from filtered objects. Define the décalage functor as the realization of the Whitehead tower of the filtered object and the décalage construction of the spectral sequence. Recall Lurie's definition of a spectral sequence.
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).
Define even faithfully flat maps and evenly free maps and the connection between even freenees and even faithful flatness with even faithfully flat descent. Briefly introduce the Adams spectral sequence as a black box. Show that the unit map $\mathbb{S}\to\mathrm{MU}$ is evenly free and prove Novikov descent. Recover the even filtration of the sphere spectrum as the d\'{e}calage of the Adams-Novikov filtration.
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