The goal of this talk is to define the derived blow-up of a closed immersion of derived schemes, and to mention a few key properties of this construction. After reviewing derived schemes, we will first introduce the infinity category of graded, simplicial rings as a free completion along sifted colimits, and explain how one takes the derived projective spectrum of such a gadget. We then introduce the derived extended Rees algebra via Weil restrictions, which is the key ingredient for derived blow-ups.

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