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In this question, it is asked why we like to consider $\mathsf{D}_\textrm{qc}(X)$ rather than $\mathsf{D}(\mathsf{QCoh}(X)).$ Professor Cisinski answers rather convincingly that the $\infty$-categoric …

This question is a follow-up to a previous question I asked. If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h …

In the following, an algebraic stack means a stack over the big fppf site of a scheme, admitting a smooth, representable, surjective morphism from a scheme (no separation hypotheses), and let $\mathsf …

As Harrison notes in the comments, we may define $\mathcal{D}_{\mathsf{B}}(\mathsf{A})$ as the full subcategory of $\mathcal{D}(\mathsf{A})$ consisting of objects $X$ such that $\pi_0(X[n])\in\mathsf{ …

Conventions: In the below, unless otherwise stated, terms regarding derived algebraic geometry will follow the conventions of Yaylali. an algebraic stack will be a stack $\mathscr{S}$ over a base sch …

I learned the following partial answer from Peter Haine (any errors are of course my own). In the following I will ignore any set-theoretic issues which may arise. Let $\mathsf{A}$ be an abelian cate …

Ogus states that he draws a log scheme $(X,\mathcal{M})$ by first drawing $X$ and then adding a picture of $\operatorname{Spec}\mathcal{M}_x$ at each $x\in X.$ (He says this on page 21.) In this case, …

Thanks to David Benjamin Lim's comment above, I have found an answer in the main case of interest to me, detailed below. However, I will leave this question open, since I am also interested in the ans …