Comments (8)
Hm, I haven't watched the talk yet, but I believe that for a map
In classical algebraic geometry,
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You can always write $\ast blah \ast$
) instead of $blah$ ($*blah*$
) in order to not get the Markdown parser confused...
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We can also try to find out if we have a synthetic
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Where is the notation
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(No idea why the latex is not working...)
Interesting. This is the definition of
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Hm, I haven't watched the talk yet, but I believe that for a map f:X→Y of schemes and a bundle E=(Ey)y:Y of modules on E, the bundle (Ef(x))x:X on X should definitely be called f∗E and not f−1E, as it is already a module over the structure ring of X (namely the constant ring bundle (R)x:X).
In classical algebraic geometry, f−1E is a sheaf of modules over f−1OY, and only after tensoring ⋅⊗f−1OYOX do we obtain f∗E. But we don't need this additional tensoring. In some sense it's already built in.
The classical counterpart of bundles $E = (E_y){y : Y}$ is the espace étale of a sheaf. The topological pullback of the espace étale $E$ along a continuous map $f\colon X \to Y$ is fibrewise given by $(E{f(x)})_{x:X}$. Moreover, the constant ring bundle over
In other words, the synthetic pullback of bundles behaves as the topological pullback of (the espace étale of) sheaves. The latter is denoted by
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Verbatim because the LaTeX formatter is broken:
> Hm, I haven't watched the talk yet, but I believe that for a map f:X→Y of schemes and a bundle E=(Ey)y:Y of modules on E, the bundle (Ef(x))x:X on X should definitely be called f∗E and not f−1E, as it is already a module over the structure ring of X (namely the constant ring bundle (R)x:X).
>
> In classical algebraic geometry, f−1E is a sheaf of modules over f−1OY, and only after tensoring ⋅⊗f−1OYOX do we obtain f∗E. But we don't need this additional tensoring. In some sense it's already built in.
The classical counterpart of bundles $E = (E_y)_{y : Y}$ is the _espace étale_ of a sheaf. The topological pullback of the espace étale $E$ along a continuous map $f\colon X \to Y$ is fibrewise given by $(E_{f(x)})_{x:X}$. Moreover, the constant ring bundle over $X$ is just the topological pullback of the constant ring bundle of the bundle over $Y$.
In other words, the synthetic pullback of bundles behaves as the topological pullback of (the espace étale of) sheaves. The latter is denoted by $f^{-1}$ in the context of algebraic geometry and is exact. As we discussed earlier, we should not fall into the trap of viewing quasicoherent sheaves as the classical counterpart of our synthetic bundles.
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BTW, where are the videos? I presume it is meant this years? Not on the same YT channel that I can see...
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Related Issues (20)
- Should (quasi-) projective schemes be closed under dependent sums? HOT 2
- Replace "finite type" with "of finite presentation" HOT 11
- Synthetic Proj construction? HOT 10
- Are Z-torsors on A1 trivial? HOT 2
- Is B(formally etale GL1) an A1-local type? HOT 1
- Is there a flat-modality in A1-homotopy theory? HOT 4
- What is the correct condition corresponding externally to "X is a scheme over a field"? HOT 13
- Remove cache warning HOT 2
- CITATION.cff HOT 6
- Basic opens of the affine parts... HOT 2
- What are external sets, internally? HOT 3
- Better definition for reduced schemes HOT 1
- Document discussion on open problems in SAG
- Discussion on renamings HOT 16
- Dimension of flat morphisms/schemes
- Use of cohesive and/or fractured structures in synthetic algebraic geometry HOT 14
- Do basic open propositions have choice? HOT 1
- The name 'weakly quasi-coherent' HOT 1
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