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Here is a phrasing of some Cartan Theorem B statements:

Consider the following conditions:

1. $X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible subset of $\mathbb{R}^n$}.
2. $\mathcal{F}$ is a {coherent, quasi-coherent, coherent, locally constant} sheaf (of $\mathcal{O}_X$-modules) on $X$.
3. The sheaf cohomology $\operatorname{H}^p(X,\mathcal{F})$ is zero for $p\geqslant1$.

Then (1) + (2) $\implies$ (3).

Here this statement applies to four geometric contexts: {complex-analytic, complex-algebraic, real-analytic, smooth}. In all of these, a "sharpness" statement also holds:

Given arbitrary $X$, if every sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) that satisfies (2) also satisfies (3), then $X$ satisfies (1).

(For the two complex contexts, this is "classical"; for the real-analytic case, I believe this should be covered in Cartan's paper "Variétés analytiques réelles et variétés analytiques complexes" but I should check this; for the smooth case, this is quasi-folklore).

Question. For the four contexts above, is the remaining implication known to be true? i.e. is the statement

Let $X$ satisfy (1); if an arbitrary sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) satisfies (3) then it satisfies (2).

true?

I assume this should be false, at least in the smooth case, given the existence of e.g. injective sheaves that are not locally constant, and I think in the complex-algebraic case it's also false (thanks to skyscraper sheaves) but I'm not so sure about the other cases.

Ideally an answer would consist of four "yes/no" answers with some references, but I'd be happier with some partial answers than with no answers at all :-)

Here is a phrasing of some Cartan Theorem B statements:

Consider the following conditions:

1. $X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible subset of $\mathbb{R}^n$}.
2. $\mathcal{F}$ is a {coherent, quasi-coherent, coherent, locally constant} sheaf (of $\mathcal{O}_X$-modules) on $X$.
3. The sheaf cohomology $\operatorname{H}^p(X,\mathcal{F})$ is zero for $p\geqslant1$.

Then (1) + (2) $\implies$ (3).

Here this statement applies to four geometric contexts: {complex-analytic, complex-algebraic, real-analytic, smooth}. In all of these, a "sharpness" statement also holds:

Given arbitrary $X$, if every sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) that satisfies (2) also satisfies (3), then $X$ satisfies (1).

(For the two complex contexts, this is "classical"; for the real-analytic case, I believe this should be covered in Cartan's paper "Variétés analytiques réelles et variétés analytiques complexes" but I should check this; for the smooth case, this is quasi-folklore.)

Question. For the four contexts above, is the remaining implication known to be true? i.e. is the statement

Let $X$ satisfy (1); if an arbitrary sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) satisfies (3) then it satisfies (2).

true?

I assume this should be false, at least in the smooth case, given the existence of e.g. injective sheaves that are not locally constant, and I think in the complex-algebraic case it's also false (thanks to skyscraper sheaves) but I'm not so sure about the other cases.

Ideally an answer would consist of four "yes/no" answers with some references, but I'd be happier with some partial answers than with no answers at all :-)

Here is a phrasing of some Cartan Theorem B statements:

Consider the following conditions:

1. $X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible subset of $\mathbb{R}^n$}.
2. $\mathcal{F}$ is a {coherent, quasi-coherent, coherent, locally constant} sheaf (of $\mathcal{O}_X$-modules) on $X$.
3. The sheaf cohomology $\operatorname{H}^p(X,\mathcal{F})$ is zero for $p\geqslant1$.

Then (1) + (2) $\implies$ (3).

Here this statement applies to four geometric contexts: {complex-analytic, complex-algebraic, real-analytic, smooth}. In all of these, a "sharpness" statement also holds:

Given arbitrary $X$, if every sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) that satisfies (2) also satisfies (3), then $X$ satisfies (1).

(For the two complex contexts, this is "classical"; for the real-analytic case, I believe this should be covered in Cartan's paper "Variétés analytiques réelles et variétés analytiques complexes" but I should check this; for the smooth case, this is quasi-folklore).

Question. For the four contexts above, is the remaining implication known to be true? i.e. is the statement

Let $X$ satisfy (1); if an arbitrary sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) satisfies (3) then it satisfies (2).

true?

I assume this should be false, at least in the smooth case, given the existence of e.g. injective sheaves that are not locally constant, and I think in the complex-algebraic case it's also false (by skyscraper sheaves, but the top two answers to this question seem to contradict each other? or at least imply that if we take $X$ to be locally of finite type then skyscraper sheaves do not give a counterexample) but I'm not so sure about the other cases.

Ideally an answer would consist of four "yes/no" answers with some references, but I'd be happier with some partial answers than with no answers at all :-)

Here is a phrasing of some Cartan Theorem B statements:

Consider the following conditions:

1. $X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible subset of $\mathbb{R}^n$}.
2. $\mathcal{F}$ is a {coherent, quasi-coherent, coherent, locally constant} sheaf (of $\mathcal{O}_X$-modules) on $X$.
3. The sheaf cohomology $\operatorname{H}^p(X,\mathcal{F})$ is zero for $p\geqslant1$.

Then (1) + (2) $\implies$ (3).

Here this statement applies to four geometric contexts: {complex-analytic, complex-algebraic, real-analytic, smooth}. In all of these, a "sharpness" statement also holds:

Given arbitrary $X$, if every sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) that satisfies (2) also satisfies (3), then $X$ satisfies (1).

(For the two complex contexts, this is "classical"; for the real-analytic case, I believe this should be covered in Cartan's paper "Variétés analytiques réelles et variétés analytiques complexes" but I should check this; for the smooth case, this is quasi-folklore).

Question. For the four contexts above, is the remaining implication known to be true? i.e. is the statement

Let $X$ satisfy (1); if an arbitrary sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) satisfies (3) then it satisfies (2).

true?

I assume this should be false, at least in the smooth case, given the existence of e.g. injective sheaves that are not locally constant, and I think in the complex-algebraic case it's also false (thanks to skyscraper sheaves) but I'm not so sure about the other cases.

Ideally an answer would consist of four "yes/no" answers with some references, but I'd be happier with some partial answers than with no answers at all :-)

Here is a phrasing of some Cartan Theorem B statements:

Consider the following conditions:

1. $X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible subset of $\mathbb{R}^n$}.
2. $\mathcal{F}$ is a {coherent, quasi-coherent, coherent, locally constant} sheaf (of $\mathcal{O}_X$-modules) on $X$.
3. The sheaf cohomology $\operatorname{H}^p(X,\mathcal{F})$ is zero for $p\geqslant1$.

Then (1) + (2) $\implies$ (3).

Here this statement applies to four geometric contexts: {complex-analytic, complex-algebraic, real-analytic, smooth}. In all of these, a "sharpness" statement also holds:

Given arbitrary $X$, if every sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) that satisfies (2) also satisfies (3), then $X$ satisfies (1).

(For the two complex contexts, this is "classical"; for the real-analytic case, I believe this should be covered in Cartan's paper "Variétés analytiques réelles et variétés analytiques complexes" but I should check this; for the smooth case, this is quasi-folklore).

Question. For the four contexts above, is the remaining implication known to be true? i.e. is the statement

Let $X$ satisfy (1); if an arbitrary sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) satisfies (3) then it satisfies (2).

true?

I assume this should be false, at least in the smooth case, given the existence of e.g. injective sheaves that are not locally constant, and I think in the complex-algebraic case it's also false (by skyscraper sheaves, but the top two answers to this question seem to contradict each other? or at least imply that if we take $X$ to be locally of finite type then skyscraper sheaves do not give a counterexample) but I'm not so sure about the other cases.

Ideally an answer would consist of four "yes/no" answers with some references, but I'd be happy with partial answers than with no answers at all!

Here is a phrasing of some Cartan Theorem B statements:

Consider the following conditions:

1. $X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible subset of $\mathbb{R}^n$}.
2. $\mathcal{F}$ is a {coherent, quasi-coherent, coherent, locally constant} sheaf (of $\mathcal{O}_X$-modules) on $X$.
3. The sheaf cohomology $\operatorname{H}^p(X,\mathcal{F})$ is zero for $p\geqslant1$.

Then (1) + (2) $\implies$ (3).

Here this statement applies to four geometric contexts: {complex-analytic, complex-algebraic, real-analytic, smooth}. In all of these, a "sharpness" statement also holds:

Given arbitrary $X$, if every sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) that satisfies (2) also satisfies (3), then $X$ satisfies (1).

(For the two complex contexts, this is "classical"; for the real-analytic case, I believe this should be covered in Cartan's paper "Variétés analytiques réelles et variétés analytiques complexes" but I should check this; for the smooth case, this is quasi-folklore).

Question. For the four contexts above, is the remaining implication known to be true? i.e. is the statement

Let $X$ satisfy (1); if an arbitrary sheaf $\mathcal{F}$ (of $\mathcal{O}_X$-modules) satisfies (3) then it satisfies (2).

true?

I assume this should be false, at least in the smooth case, given the existence of e.g. injective sheaves that are not locally constant, and I think in the complex-algebraic case it's also false (by skyscraper sheaves, but the top two answers to this question seem to contradict each other? or at least imply that if we take $X$ to be locally of finite type then skyscraper sheaves do not give a counterexample) but I'm not so sure about the other cases.

Ideally an answer would consist of four "yes/no" answers with some references, but I'd be happier with some partial answers than with no answers at all :-)