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Results tagged with schemes
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user 2191
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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Arbitrary products of schemes don't exist, do they?
If you want a tensor product satisfying the isomorphism described, you can just define it as the inductive limit of all finite tensor products. For example, if you tensor $k[x_i]$ like this you really …
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150
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An (almost) terminological question: could one shorten the phrase 'the spectrum of the resid...
For a scheme S I want to consider the spectra of the residue fields of points of S. Is there any way to make this phrase shorter? Is there a term for the morphism that connects such a spectrum with S? …
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681
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For a morphism f from a regular scheme, should there exist an open subscheme U of the target...
All schemes are excellent.
If the answer is 'yes', then: could one choose such an $U$ such that the preimage of any regular subscheme of $U$ is regular? Are these conditions on $U$ equivalent? …
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450
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If $f:X\to S$ is a universal homeomorphism, is $f':X\times_S X\to X$ a nil-immersion?
If $f:X\to S$ is a universal homeomorphism, is $f':X\times_S X\to X$ always a nil-immersion? This seems to be easy, yet possibly I miss something. Should I give references to this fact in a paper?
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Model of a scheme regular over the generic point
Let all schemes below be excellent.
Let $X_0$ be a regular (not necessarily smooth, projective) non-empty scheme of finite type over the generic point $\eta$ of a regular connected scheme $S$. …
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If $X,Y$ are regular and of finite type over $S$, can $X\times _S Y$ be embedded into a regu...
Now, I am interested in the following setting: $X,Y$ are regular schemes of finite type over $S$;
$S$ is separated excellent noetherian of finite Krull dimension (and one may assume that $X$ and $Y$ are …
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175
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Regular subscheme of a projective limit of schemes
I am actually only interested in equicharacteristic schemes, and the connecting morphisms are affine and dominant. …
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Is every regular (excellent) scheme separated?
I need schemes that are regular, excellent and separated. Are these three conditions independent? …
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368
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Can any radiciel morphism be presented as the composition of a universal homeomorphism with ...
As in my previous questions, I am interested in excellent schemes of finite Krull dimension. …
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How would you call a subscheme of a smooth $S$-scheme?
In my preprint I propose to call $X/S$ quasi-smooth if $X$ can be embedded into a smooth $X'/S$. Does this sound fine?
Upd. So, smoothly embeddable is better? Is it ok to call a morphism smoothly emb …
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A presentation of a scheme as a limit of smooth ones over finitely generated bases
Which of the following statements are true:
If $S$ is regular, then it can be presented as a projective limit of smooth $\mathbb{Z}$-schemes. … If $S$ is regular, then it can be presented as a projective limit of schemes that are smooth over finite type regular $\mathbb{Z}$-ones. …
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4
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331
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Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular...
Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular? …
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4
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817
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When singular points of a reduced scheme are not dense in it?
A stupid AG question: could singular (Zarisky) points be dense in a reduced (Noetherian) scheme $S$? If yes, which 'standard' restrictions on $S$ could ensure that this does not happen? For example, s …
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Does one need l to be invertible in S in order to consider the l-adic cohomology of S-scheme...
What happens with $\mathbb{Q}_l$-adic cohomology of schemes if $l$ is not invertible in $S$ (but is not equal to the characteristic of $S$)? Will the Tate twist be invertible? …
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More on universal homeomorphisms
Is a universal homeomorphism of connected regular (excellent finite dimensional) schemes an isomorphism if these schemes are not positive characterstic ones? … Suppose that a finite morphism $f:X\to Y$ of connected regular (excellent finite dimensional) schemes is generically purely inseparable. …
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