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I'm trying to learn about Gysin maps on Hodge cohomology, as defined in the Stacks project (https://stacks.math.columbia.edu/tag/0G8A). Namely, if $X \to S$ is a morphism of schemes and $Z \to X$ is …

Suppose that $\mathcal{X}$ is an (algebraic, finite type over an algebraically closed field $k$) moduli stack of some geometric objects (e.g. curves, abelian varieties, etc.), so that for a scheme $S$ …

I guess we can say the following, though I would still be interested if someone else could provide a different perspective and/or corrections. If we have a family of such geometric objects over an int …

If $f: X \to S$ is a smooth, proper morphism of nonsingular complex algebraic varieties with $S$ connected, then the dimensions of the (algebraic) de Rham cohomology groups of the fibers of $f$ are co …

Let $X/\mathbb{C}$ be a scheme over the complex numbers. In "Crystals and the de Rham cohomology of schemes," Grothendieck constructs the infinitesimal ringed site $(X_{\operatorname{inf}}, \mathcal{O …

Let $\mathcal{X}$ be an algebraic stack, and $x \in |\mathcal{X}|$ a point such that the residual gerbe $\mathcal{G}_x$ of $\mathcal{X}$ at $x$ exists. Let $\mathcal{Y}$ be a reduced algebraic stack, …

Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth formu …