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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.
6
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305
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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? …
- 16.9k
6
votes
1
answer
814
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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. …
- 16.9k
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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 …
- 16.9k
5
votes
1
answer
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? …
- 16.9k
5
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1
answer
1k
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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 $S$. …
- 16.9k
5
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0
answers
734
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Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situati...
This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases?
Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to define the p …
- 16.9k
5
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Spectrum of the Grothendieck ring of varieties
This ring is very important for motivic integration; so it might be useful for you to read surveys on this subject.
Yet I would say that this ring is too large and complicated. A reasonable factor-rin …
- 16.9k
4
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1
answer
228
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Which valuations of a field yield codimension $1$ subschemes of their 'models'
For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). …
- 16.9k
4
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2
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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? …
- 16.9k
4
votes
1
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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 …
- 16.9k
4
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1
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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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4
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525
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Which schemes can be presented as limits of smooth varieties?
In this text I only treat schemes that are excellent separated of finite Krull dimension. So, I have the following questions. … Is there an interesting subclass in the class of all limit schemes of this sort? I don't want to restrict myself to affine schemes. …
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Are "strongly finite dimensional" homotopy invariant sheaves with transfers (locally) constant?
Moreover, I am actually interested in the extension of $S$ to pro-smooth (say, affine) $k$-schemes; and my finite dimensionality assumption corresponds to the finite dimensionality of $S(\operatorname{ …
- 16.9k
3
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0
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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. …
- 16.9k
2
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When inverse image is conservative; a reference or a generalization?
I am interested in the following question: for $f$ being a morphism of schemes, which conditions ensure that $Rf^*_{et}$ is conservative? …
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