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I had a number of questions regarding Gabber's rigidity, I'm not sure whether I am understanding it correctly, so please let me know if I'm making any mistakes.

Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), Gabber's rigidity in algebraic $K$-theory for henselian pairs implies that $K_*(A/I, \mathbb{Z}/l\mathbb{Z})\simeq K_*(\hat{A}, \mathbb{Z}/l\mathbb{Z})$. Here algebraic $K$-theory of the completion is taken in the sense that $\hat{A}$ is just a Noetherian ring and we can construct $BGL(\hat{A})^+$ just as we can do for any ring i.e. the $K$-theory of finitely generated projective modules on $\hat{A}$. There is another way to define $K$-theory and that is to take the $K$-theory of formal vector bundles on $\hat{A}$, which means a system of vector bundles on all finite thickenings of $I$ with compatibility conditions under pullbacks. I don't think these two coincide in general. (Correct me if I am wrong)

Q1) Here is my first question: Given the affine formal scheme $\text{Spf}(\hat{A})$ we can associate to it the scheme $\text{Spec}(\hat{A})$. Does this generalize and give a well-defined assignement in the non-affine case? For the formal completion $X_Z$ ($X$ along a closed subscheme $Z$) can we assign a scheme that on affines behaves just like what I described? If so does this functor have name? (I am going to denote this hypothetical functor by $\mathcal{F}$)

Q2) My original question is about whether there is a global version of this rigidity in the setting of formal completions or not? The following is my attempt on this question. (I am not sure whether I am making any mistakes or not)

Let's assume we have Noetherian schemes $X$ and a closed subscheme $Z$ and we are looking at the formal completion $X_Z$. Well as we are dealing with formal schemes it makes sense to consider formal vector bundles and compare its $K$-theory with the $K$-theory of $Z$ with coefficient in $\mathbb{Z}/l\mathbb{Z}$. But this comparison possibly fails even in the affine case according to the observation above. So I think the natural generalization would be instead of considering $X_Z$ as a formal scheme we can consider it as a scheme just as in the affine case (This means if Q1 has a positive answer we consider $\mathcal{F}(X_Z)$). Since completion with respect to an ideal preserves being Noetherian and having a finite Krull dimension and because of Zariski descent of connective algebraic $K$-theory this implies that $K_*(Z,\mathbb{Z}/l\mathbb{Z})\simeq K_*(\mathcal{F}(X_Z),\mathbb{Z}/l\mathbb{Z})$ where $\mathcal{F}(X_Z)$ is the associated scheme to $X_Z$ according to Q1.

Few points about Q1: The relation between $\text{Spf}(\hat{A})$ and $\text{Spec}(\hat{A})$ is that the former is the colimit of the schemes $\text{Spec}(A/I^n)$ in the category of ringed spaces and $\text{Spec}(\hat{A})$ is the colimit of the same schemes in the category of schemes. For a proof of the latter see here. So question Q1 can be restated in the following way: Does the colimit of the schemes $X_Z^{(n)}$ ($n$-th thickening of $Z$) exist in the category of schemes? Stated in this way it seems obvious (maybe it is not) because glueing schemes is a colimit and colimits should commute. So we can take the colimit on affine charts and then glue together and this should give the functor $\mathcal{F}$.

Regarding the question when the $K$-theory of formal scheme $X_Z$ should match with its $K$-theory as a scheme i.e. $\mathcal{F}(X_Z)$, I think if the pair $(X,Z)$ satisfy the property that every formal vector bundle on $X_Z$ can be extended to a neighborhood of $Z$ they should be compatible.

I had a question regarding Gabber's rigidity.

Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), Gabber's rigidity in algebraic $K$-theory for henselian pairs implies that $K_*(A/I, \mathbb{Z}/l\mathbb{Z})\simeq K_*(\hat{A}, \mathbb{Z}/l\mathbb{Z})$. Here algebraic $K$-theory of the completion is taken in the sense that $\hat{A}$ is just a Noetherian ring and we can construct $BGL(\hat{A})^+$ just as we can do for any ring i.e. the $K$-theory of finitely generated projective modules on $\hat{A}$. There is another way to define $K$-theory and that is to take the $K$-theory of formal vector bundles on $\hat{A}$, which means a system of vector bundles on all finite thickenings of $I$ with compatibility conditions under pullbacks. These two probably are the same. (There is an equivalence between the formal vector bundles and $\hat{A}$ bundles given by taking limit and restriction to infinitesimal neighborhoods.)

Q) My question is about whether there is a global version of this rigidity in the setting of formal completions or not?

Let's assume we have Noetherian schemes $X$ and a closed subscheme $Z$ and we are looking at the formal completion $X_Z$. Since we are dealing with formal schemes it makes sense to consider formal vector bundles and compare its $K$-theory with the $K$-theory of $Z$ with coefficient in $\mathbb{Z}/l\mathbb{Z}$. One question is whether there is a Zariski descent for formal schemes with finite coefficients? if so that would imply $K_*(X_Z,\mathbb{Z}/l)\simeq K_*(Z, \mathbb{Z}/l) $. Or there could be possibly a different approach. This could also be wrong in the global case, which I'd like to know some examples.

I had a number of questions regarding Gabber's rigidity, I'm not sure whether I am understanding it correctly, so please let me know if I'm making any mistakes.

Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), Gabber's rigidity in algebraic $K$-theory for henselian pairs implies that $K_*(A/I, \mathbb{Z}/l\mathbb{Z})\simeq K_*(\hat{A}, \mathbb{Z}/l\mathbb{Z})$. Here algebraic $K$-theory of the completion is taken in the sense that $\hat{A}$ is just a Noetherian ring and we can construct $BGL(\hat{A})^+$ just as we can do for any ring i.e. the $K$-theory of finitely generated projective modules on $\hat{A}$. There is another way to define $K$-theory and that is to take the $K$-theory of formal vector bundles on $\hat{A}$, which means a system of vector bundles on all finite thickenings of $I$ with compatibility conditions under pullbacks. I don't think these two coincide in general. (Correct me if I am wrong)

Q1) Here is my first question: Given the affine formal scheme $\text{Spf}(\hat{A})$ we can associate to it the scheme $\text{Spec}(\hat{A})$. Does this generalize and give a well-defined assignement in the non-affine case? For the formal completion $X_Z$ ($X$ along a closed subscheme $Z$) can we assign a scheme that on affines behaves just like what I described? If so does this functor have name? (I am going to denote this hypothetical functor by $\mathcal{F}$)

Q2) My original question is about whether there is a global version of this rigidity in the setting of formal completions or not? The following is my attempt on this question. (I am not sure whether I am making any mistakes or not)

Let's assume we have Noetherian schemes $X$ and a closed subscheme $Z$ and we are looking at the formal completion $X_Z$. Well as we are dealing with formal schemes it makes sense to consider formal vector bundles and compare its $K$-theory with the $K$-theory of $Z$ with coefficient in $\mathbb{Z}/l\mathbb{Z}$. But this comparison possibly fails even in the affine case according to the observation above. So I think the natural generalization would be instead of considering $X_Z$ as a formal scheme we can consider it as a scheme just as in the affine case (This means if Q1 has a positive answer we consider $\mathcal{F}(X_Z)$). Since completion with respect to an ideal preserves being Noetherian and having a finite Krull dimension and because of Zariski descent of connective algebraic $K$-theory this implies that $K_*(Z,\mathbb{Z}/l\mathbb{Z})\simeq K_*(\mathcal{F}(X_Z),\mathbb{Z}/l\mathbb{Z})$ where $\mathcal{F}(X_Z)$ is the associated scheme to $X_Z$ according to Q1.

Regarding the question when the $K$-theory of formal scheme $X_Z$ should match with its $K$-theory as a scheme i.e. $\mathcal{F}(X_Z)$, I think if the pair $(X,Z)$ satisfy the property that every formal vector bundle on $X_Z$ can be extended to a neighborhood of $Z$ they should be compatible.

I had a number of questions regarding Gabber's rigidity, I'm not sure whether I am understanding it correctly, so please let me know if I'm making any mistakes.

Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), Gabber's rigidity in algebraic $K$-theory for henselian pairs implies that $K_*(A/I, \mathbb{Z}/l\mathbb{Z})\simeq K_*(\hat{A}, \mathbb{Z}/l\mathbb{Z})$. Here algebraic $K$-theory of the completion is taken in the sense that $\hat{A}$ is just a Noetherian ring and we can construct $BGL(\hat{A})^+$ just as we can do for any ring i.e. the $K$-theory of finitely generated projective modules on $\hat{A}$. There is another way to define $K$-theory and that is to take the $K$-theory of formal vector bundles on $\hat{A}$, which means a system of vector bundles on all finite thickenings of $I$ with compatibility conditions under pullbacks. I don't think these two coincide in general. (Correct me if I am wrong)

Q1) Here is my first question: Given the affine formal scheme $\text{Spf}(\hat{A})$ we can associate to it the scheme $\text{Spec}(\hat{A})$. Does this generalize and give a well-defined assignement in the non-affine case? For the formal completion $X_Z$ ($X$ along a closed subscheme $Z$) can we assign a scheme that on affines behaves just like what I described? If so does this functor have name? (I am going to denote this hypothetical functor by $\mathcal{F}$)

Q2) My original question is about whether there is a global version of this rigidity in the setting of formal completions or not? The following is my attempt on this question. (I am not sure whether I am making any mistakes or not)

Let's assume we have Noetherian schemes $X$ and a closed subscheme $Z$ and we are looking at the formal completion $X_Z$. Well as we are dealing with formal schemes it makes sense to consider formal vector bundles and compare its $K$-theory with the $K$-theory of $Z$ with coefficient in $\mathbb{Z}/l\mathbb{Z}$. But this comparison possibly fails even in the affine case according to the observation above. So I think the natural generalization would be instead of considering $X_Z$ as a formal scheme we can consider it as a scheme just as in the affine case (This means if Q1 has a positive answer we consider $\mathcal{F}(X_Z)$). Since completion with respect to an ideal preserves being Noetherian and having a finite Krull dimension and because of Zariski descent of connective algebraic $K$-theory this implies that $K_*(Z,\mathbb{Z}/l\mathbb{Z})\simeq K_*(\mathcal{F}(X_Z),\mathbb{Z}/l\mathbb{Z})$ where $\mathcal{F}(X_Z)$ is the associated scheme to $X_Z$ according to Q1.

Few points about Q1: The relation between $\text{Spf}(\hat{A})$ and $\text{Spec}(\hat{A})$ is that the former is the colimit of the schemes $\text{Spec}(A/I^n)$ in the category of ringed spaces and $\text{Spec}(\hat{A})$ is the colimit of the same schemes in the category of schemes. For a proof of the latter see here. So question Q1 can be restated in the following way: Does the colimit of the schemes $X_Z^{(n)}$ ($n$-th thickening of $Z$) exist in the category of schemes? Stated in this way it seems obvious (maybe it is not) because glueing schemes is a colimit and colimits should commute. So we can take the colimit on affine charts and then glue together and this should give the functor $\mathcal{F}$.

Regarding the question when the $K$-theory of formal scheme $X_Z$ should match with its $K$-theory as a scheme i.e. $\mathcal{F}(X_Z)$, I think if the pair $(X,Z)$ satisfy the property that every formal vector bundle on $X_Z$ can be extended to a neighborhood of $Z$ they should be compatible.

I had a number of questions regarding Gabber's rigidity, I'm not sure whether I am understanding it correctly, so please let me know if I'm making any mistakes.

Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), Gabber's rigidity in algebraic $K$-theory for henselian pairs implies that $K_*(A/I, \mathbb{Z}/l\mathbb{Z})\simeq K_*(\hat{A}, \mathbb{Z}/l\mathbb{Z})$. Here algebraic $K$-theory of the completion is taken in the sense that $\hat{A}$ is just a Noetherian ring and we can construct $BGL(\hat{A})^+$ just as we can do for any ring i.e. the $K$-theory of finitely generated projective modules on $\hat{A}$. There is another way to define $K$-theory and that is to take the $K$-theory of formal vector bundles on $\hat{A}$, which means a system of vector bundles on all finite thickenings of $I$ with compatibility conditions under pullbacks. I don't think these two coincide in general. (Correct me if I am wrong)

My original question is about whether there is a global version of this rigidity in the setting of formal completions or not? The following is my attempt on this question. (I am not sure whether I am making any mistakes or not)

Now we have Noetherian schemes $X$ and a closed subscheme $Z$ and we are looking at the formal completion $X_Z$. Well as we are dealing with formal schemes it makes sense to consider formal vector bundles and compare its $K$-theory with the $K$-theory of $Z$ with coefficient in $\mathbb{Z}/l\mathbb{Z}$. But this comparison possibly fails even in the affine case according to the observation above. So I think the natural generalization would be instead of considering $X_Z$ as a formal scheme we can consider it as a scheme just as in the affine case. This means if on an affine chart $X_Z$ is the formal scheme $\hat{A}$, we consider it just as a scheme i.e. $\text{Spec}(\hat{A})$ (Please let me know if this construction is flawed because I've never since a formal scheme to be considered as a scheme but I do not see any issues on this case). Since completion with respect to an ideal preserves being Noetherian and having a finite Krull dimension and because of Zariski descent of connective algebraic $K$-theory this implies that $K_*(Z,\mathbb{Z}/l\mathbb{Z})\simeq K_*(X_Z,\mathbb{Z}/l\mathbb{Z})$ where $X_Z$ is regarded as a scheme rather than a formal scheme.

For the question when the $K$-theory of formal scheme $X_Z$ should match with its $K$-theory as a scheme, I think if the pair $(X,Z)$ satisfy the property that every formal vector bundle on $X_Z$ can be extended to a neighborhood of $Z$ they should be compatible.

I had a number of questions regarding Gabber's rigidity, I'm not sure whether I am understanding it correctly, so please let me know if I'm making any mistakes.

Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), Gabber's rigidity in algebraic $K$-theory for henselian pairs implies that $K_*(A/I, \mathbb{Z}/l\mathbb{Z})\simeq K_*(\hat{A}, \mathbb{Z}/l\mathbb{Z})$. Here algebraic $K$-theory of the completion is taken in the sense that $\hat{A}$ is just a Noetherian ring and we can construct $BGL(\hat{A})^+$ just as we can do for any ring i.e. the $K$-theory of finitely generated projective modules on $\hat{A}$. There is another way to define $K$-theory and that is to take the $K$-theory of formal vector bundles on $\hat{A}$, which means a system of vector bundles on all finite thickenings of $I$ with compatibility conditions under pullbacks. I don't think these two coincide in general. (Correct me if I am wrong)

Q1) Here is my first question: Given the affine formal scheme $\text{Spf}(\hat{A})$ we can associate to it the scheme $\text{Spec}(\hat{A})$. Does this generalize and give a well-defined assignement in the non-affine case? For the formal completion $X_Z$ ($X$ along a closed subscheme $Z$) can we assign a scheme that on affines behaves just like what I described? If so does this functor have name? (I am going to denote this hypothetical functor by $\mathcal{F}$)

Q2) My original question is about whether there is a global version of this rigidity in the setting of formal completions or not? The following is my attempt on this question. (I am not sure whether I am making any mistakes or not)

Let's assume we have Noetherian schemes $X$ and a closed subscheme $Z$ and we are looking at the formal completion $X_Z$. Well as we are dealing with formal schemes it makes sense to consider formal vector bundles and compare its $K$-theory with the $K$-theory of $Z$ with coefficient in $\mathbb{Z}/l\mathbb{Z}$. But this comparison possibly fails even in the affine case according to the observation above. So I think the natural generalization would be instead of considering $X_Z$ as a formal scheme we can consider it as a scheme just as in the affine case (This means if Q1 has a positive answer we consider $\mathcal{F}(X_Z)$). Since completion with respect to an ideal preserves being Noetherian and having a finite Krull dimension and because of Zariski descent of connective algebraic $K$-theory this implies that $K_*(Z,\mathbb{Z}/l\mathbb{Z})\simeq K_*(\mathcal{F}(X_Z),\mathbb{Z}/l\mathbb{Z})$ where $\mathcal{F}(X_Z)$ is the associated scheme to $X_Z$ according to Q1.

Regarding the question when the $K$-theory of formal scheme $X_Z$ should match with its $K$-theory as a scheme i.e. $\mathcal{F}(X_Z)$, I think if the pair $(X,Z)$ satisfy the property that every formal vector bundle on $X_Z$ can be extended to a neighborhood of $Z$ they should be compatible.