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Let $C$ be an abelian category and $K(C)$ the homotopy category of complexes in $C$. I've seen the following claimed in several sources (without proof): A. The following isomorphisms hold: $$\li …

Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$. One of the possible definitions of an ample line bundle goes as follows: Def 1: A line bundle $\ …

Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned b …

(All functors and categories are derived, all $\mathcal{D}$-modules are holonomic and $f: X \to Y$ is proper for simplicity). We may use the adjunction $\int_f \dashv f^!$ to get some interpretation …

I apologize if this question is slightly vague but I don't know how to ask it non-vaguely. Moreover, my question is about an ideal situation. If there's a close answer which doesn't satisfy all the co …

Let $X$ be a smooth algebraic (/projective if it simplifies things considerably) variety over $\mathbb{C}$ and consider the derived category $\mathcal{C}=D_h^b(\mathcal{D}_X)$ of bounded complexes of …

I'm looking for the simplest possible example (one that's easy to remember) for the situation described in the title. More precisely I'm looking for the following example: A (probably has to be singu …

Let $A$ be non-positively graded commutative DG-algebra almost of finite type over a field $k$ of characteristic $0$. Most of these assumptions (affine, commutative, characteristic, bound) are only to …

I'm trying to build a little dictionary between old Homological algebra for local rings and the slightly more modern approach via derived functors. Let $X = SpecA$ be a spectrum of a local ring $(A, …

This is true. The functor $RHom(\mathcal{O}_X/\mathcal{I}^{n+1},-)$ is in fact a different name for $j^!$ when $j: Z_n \to X$. Where $Z_n$ is the $n$-truncated formal neighborhood of $Z$ in $X$. In th …

Let $X$ be a noetherian scheme $U \subset X$ an open subset with complement $Z = X- U$. Assume $Z$ is cut out by the ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. We have exact sequences: $$0 \to …

Let $(\mathcal{C,W})$ be relative category (equipped with a wide subcategory of weak equivalences satisfying 2 out of 3 property). Consider a profunctor $F: \mathcal{C \times D^{op}} \to \mathsf{Set}$ …