# Questions tagged [schemes]

The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.

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### Is there an example of a formally smooth morphism which is not smooth?

A morphism of schemes is formally smooth and locally of finite presentation iff it is smooth. What happens if we drop the finitely presented hypothesis? Of course, locally of finite presentation is ...
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### Ring-theoretic characterization of open affines?

Background Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...
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### Reference for de Rham cohomology in positive characteristic

It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas ...
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### Images and Monomorphisms of Schemes

If $X$ is an object in an arbitrary category, there is a natural definition of a subobject of $X$ as a monomorphism into $X$ (or really an equivalence class of monomorphisms). If $X$ is a scheme, ...
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### About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've asked this question https://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will ...
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### Is there an example of a variety over the complex numbers with no embedding into a smooth variety?

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
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### Do Grothendieck universes matter for an algebraic geometer?

I recently learned that some parts of SGA require axioms beyond ZFC. I am just a simple algebraic geometer so I am trying to understand how can this fact impact my life (you may have engaged in a ...
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### Question about the statement of the going-down theorem of Cohen-Seidenberg in Mumford

In Mumford's red book the statement of the Going-Down Theorem (Chapter II Section 8) is as follows. Let $f: X \to Y$ be a finite morphism. Assume that $Y$ is an irreducible normal scheme. Assume that ...
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### Irreducible of finite Krull dimension implies quasi-compact?

Let $X$ be the underlying space of a scheme. If $X$ is irreducible of finite Krull dimension, is it necessarily quasi-compact? Is it necessarily Noetherian? What if we assume not only that Krull ...
Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms ...