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If you want a canonical description of the homotopy type of the general fiber, you can take the homotopy type of the space $X\times_{\mathbb{D}}\mathbb{H}$ where $\mathbb{H}:=\{z\in\mathbb{C}\mid \Im …

No, the definition in Schlichting's first paper are not the "correct" definition of higher Witt groups (in any case they are not the analogue of Balmer's Witt groups), rather they are some shifted hig …

Zariski descent tells us that $$\operatorname{QCoh}(X)=\lim_{U\subseteq X} \operatorname{QCoh}(U)$$ where $U$ ranges through all open affines and the limit is taken in the $(2,1)$-categorical sense. S …

As a first introduction I like these notes by Achim Krause and Thomas Nikolaus. They do require some familiarity with spectra and stable homotopy theory though.

What you are referring to are sometimes called stacks or sheaves of categories. A famously important example is the stack $\mathrm{QCoh}$ sending a scheme $U$ to the category of quasi-coherent sheaves …

The most abstract version of the Tarski-Seidenberg theorem I know of is the following Let $f:A\to B$ be a morphism of finite presentation of commutative rings. Then the induced map $$f^*:\operatornam …

You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least …

You're almost there! The problem is that, as you've surmised, the group $\mathrm{Aut}(x)$ does not capture enough of the geometric structure of $G$. But that's easily solved: For every $x\in X$ we ca …

This question is strongly related to this question. However it seems to me sufficiently distinct to warrant asking it separately. Let $X$ be a quasi-compact, quasi-separated scheme. When is the ∞- …

Let $X$ be a quasi-compact and quasi-separated scheme, and $U\subseteq X$ be a quasi-compact open subscheme. Then we can consider $Rj_*\mathcal{O}_U$ the (derived) pushforward of the structure sheaf o …

As Harry Gindy as said in the comments, there is a refinement of the notion of shape due to Barwick, Glasman and Haine that contains much more information that just the shape. This is not a pro-space, …

Since perfect complexes are dualizable, for every perfect complex $P$ and any complex $Q$ we have $$\mathrm{hom}(P,Q)\cong \mathrm{hom}(P,1)\otimes Q\,.$$ Moreover $\mathrm{hom}(P,1)$ is perfect too ( …

Another reference is Tag 01KO in the Stacks project (note that when $S$ is affine the hypothesis that the two opens map into a common affine is empty). Of course, the condition that the intersection …

Let $\mathrm{Ét}_\mathbb{C}$ be the étale $\infty$-topos of schemes over $\mathbb{C}$, that is the $\infty$-categories of étale sheaves of $\infty$-groupoids over $\mathbb{C}$. This contains necessari …

One way of finding a "fully derived" version of Poincaré duality is Atiyah duality. This says that for any closed manifold $M$ there is an equivalence of spectra (in the sense of algebraic topology) $ …