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Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A-\mathrm{Mod}$ and $eAe-\mathrm{Mod}$. This is Example 2.7 in Homological theory of recollements of abelian categories by Chrysostomos Psaroudakis.

There is a "restriction" funtor $e(-):A-\mathrm{Mod}\to eAe-\mathrm{Mod}$ and its right adjoint "direct image" functor $\mathrm{Hom}_{eAe}(eA,-),$ so if we think of $A-\mathrm{Mod}$ as sheaves on something noncommutative, then $eAe-\mathrm{Mod}$ behaves at least somewhat like sheaves on an open chart.

Now let's say we have a collection of orthogonal idempotents $e_1,\ldots,e_m\in A$ with $1=e_1+\ldots + e_m.$Also for $I\subset\{1,\ldots,m\}$ denote $e(I)=\sum_{i\in I} e_i.$ I have in front of me such a situation in which it seems there is a sort of descent for modules over $A$: the category $A-\mathrm{Mod}$ seems to be equivalent to the category of descent data, the objects of which are collections of modules $M_j$ over $e(I_j)Ae(I_j)$ for some collection of indexing sets $I_j$ such that $\bigcup_j I_j=\{1,\ldots,m\}$ with isomorphisms $e(I_i \cap I_j)M_i\simeq e(I_i \cap I_j)M_j$ satisfying the usual cocycle conditions.

For now, in my partial proof of this equivalence I do use the properties of my algebra $A$, but maybe that's not really necessary? Has this sort of descent for module categories been worked out before? I have a gut feeling that this is something known and I just don't know what algebraists would call it.

Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A\text{-}\mathrm{Mod}$ and $eAe\text{-}\mathrm{Mod}$. This is Example 2.7 in Homological theory of recollements of abelian categories by Chrysostomos Psaroudakis.

There is a "restriction" funtor $e(-):A\text{-}\mathrm{Mod}\to eAe\text{-}\mathrm{Mod}$ and its right adjoint "direct image" functor $\mathrm{Hom}_{eAe}(eA,-),$ so if we think of $A\text{-}\mathrm{Mod}$ as sheaves on something noncommutative, then $eAe\text{-}\mathrm{Mod}$ behaves at least somewhat like sheaves on an open chart.

Now let's say we have a collection of orthogonal idempotents $e_1,\ldots,e_m\in A$ with $1=e_1+\ldots + e_m.$ Also for $I\subset\{1,\ldots,m\}$ denote $e(I)=\sum_{i\in I} e_i.$ I have in front of me such a situation in which it seems there is a sort of descent for modules over $A$: the category $A\text{-}\mathrm{Mod}$ seems to be equivalent to the category of descent data, the objects of which are collections of modules $M_j$ over $e(I_j)Ae(I_j)$ for some collection of indexing sets $I_j$ such that $\bigcup_j I_j=\{1,\ldots,m\}$ with isomorphisms $e(I_i \cap I_j)M_i\simeq e(I_i \cap I_j)M_j$ satisfying the usual cocycle conditions.

For now, in my partial proof of this equivalence I do use the properties of my algebra $A$, but maybe that's not really necessary? Has this sort of descent for module categories been worked out before? I have a gut feeling that this is something known and I just don't know what algebraists would call it.

Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A-\mathrm{Mod}$ and $eAe-\mathrm{Mod}$. This is Example 2.7 in Homological theory of recollements of abelian categories by Chrysostomos Psaroudakis.

There is a "restriction" funtor $e(-):A-\mathrm{Mod}\to eAe-\mathrm{Mod}$ and its right adjoint "direct image" functor $\mathrm{Hom}_{eAe}(eA,-),$ so if we think of $A-\mathrm{Mod}$ as sheaves on something noncommutative, then $eAe-\mathrm{Mod}$ behaves at least somewhat like sheaves on an open chart.

Now let's say we have a collection of orthogonal central idempotents $e_1,\ldots,e_m\in A$ with $1=e_1+\ldots + e_m.$ Also for $I\subset\{1,\ldots,m\}$ denote $e(I)=\sum_{i\in I} e_i.$ I have in front of me such a situation in which it seems there is a sort of descent for modules over $A$: the category $A-\mathrm{Mod}$ seems to be equivalent to the category of descent data, the objects of which are collections of modules $M_j$ over $e(I_j)Ae(I_j)$ for some collection of indexing sets $I_j$ such that $\bigcup_j I_j=\{1,\ldots,m\}$ with isomorphisms $e(I_i \cap I_j)M_i\simeq e(I_i \cap I_j)M_j$ satisfying the usual cocycle conditions.

For now, in my partial proof of this equivalence I do use the properties of my algebra $A$, but maybe that's not really necessary? Has this sort of descent for module categories been worked out before? I have a gut feeling that this is something known and I just don't know what algebraists would call it.

Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A-\mathrm{Mod}$ and $eAe-\mathrm{Mod}$. This is Example 2.7 in Homological theory of recollements of abelian categories by Chrysostomos Psaroudakis.

There is a "restriction" funtor $e(-):A-\mathrm{Mod}\to eAe-\mathrm{Mod}$ and its right adjoint "direct image" functor $\mathrm{Hom}_{eAe}(eA,-),$ so if we think of $A-\mathrm{Mod}$ as sheaves on something noncommutative, then $eAe-\mathrm{Mod}$ behaves at least somewhat like sheaves on an open chart.

Now let's say we have a collection of orthogonal idempotents $e_1,\ldots,e_m\in A$ with $1=e_1+\ldots + e_m.$Also for $I\subset\{1,\ldots,m\}$ denote $e(I)=\sum_{i\in I} e_i.$ I have in front of me such a situation in which it seems there is a sort of descent for modules over $A$: the category $A-\mathrm{Mod}$ seems to be equivalent to the category of descent data, the objects of which are collections of modules $M_j$ over $e(I_j)Ae(I_j)$ for some collection of indexing sets $I_j$ such that $\bigcup_j I_j=\{1,\ldots,m\}$ with isomorphisms $e(I_i \cap I_j)M_i\simeq e(I_i \cap I_j)M_j$ satisfying the usual cocycle conditions.

For now, in my partial proof of this equivalence I do use the properties of my algebra $A$, but maybe that's not really necessary? Has this sort of descent for module categories been worked out before? I have a gut feeling that this is something known and I just don't know what algebraists would call it.

Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $\mathrm{Mod}-A$ and $\mathrm{Mod}-eAe$. This is Example 2.7 in Homological theory of recollements of abelian categories by Chrysostomos Psaroudakis.

There is a "restriction" funtor $e(-):\mathrm{Mod}-A\to \mathrm{Mod}-eAe$ and its right adjoint "direct image" functor $\mathrm{Hom}_{eAe}(eA,-),$ so if we think of $\mathrm{Mod}-A$ as sheaves on something noncommutative, then $\mathrm{Mod}-eAe$ behaves at least somewhat like sheaves on an open chart.

Now let's say we have a collection of orthogonal central idempotents $e_1,\ldots,e_m\in A$ with $1=e_1+\ldots + e_m.$ Also for $I\subset\{1,\ldots,m\}$ denote $e(I)=\sum_{i\in I} e_i.$ I have in front of me such a situation in which it seems there is a sort of descent for modules over $A$: the category $\mathrm{Mod}-A$ seems to be equivalent to the category of descent data, the objects of which are collections of modules $M_j$ over $e(I_j)Ae(I_j)$ for some collection of indexing sets $I_j$ such that $\bigcup_j I_j=\{1,\ldots,m\}$ with isomorphisms $e(I_i \cap I_j)M_i\simeq e(I_i \cap I_j)M_j$ satisfying the usual cocycle conditions.

For now, in my partial proof of this equivavence I do use the properties of my algebra $A$, but maybe that's not really necessary? Has this sort of descent for module categories been worked out before? I have a gut feeling that this is something known and I just don't know what algebraists would call it.

Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A-\mathrm{Mod}$ and $eAe-\mathrm{Mod}$. This is Example 2.7 in Homological theory of recollements of abelian categories by Chrysostomos Psaroudakis.

There is a "restriction" funtor $e(-):A-\mathrm{Mod}\to eAe-\mathrm{Mod}$ and its right adjoint "direct image" functor $\mathrm{Hom}_{eAe}(eA,-),$ so if we think of $A-\mathrm{Mod}$ as sheaves on something noncommutative, then $eAe-\mathrm{Mod}$ behaves at least somewhat like sheaves on an open chart.

Now let's say we have a collection of orthogonal central idempotents $e_1,\ldots,e_m\in A$ with $1=e_1+\ldots + e_m.$ Also for $I\subset\{1,\ldots,m\}$ denote $e(I)=\sum_{i\in I} e_i.$ I have in front of me such a situation in which it seems there is a sort of descent for modules over $A$: the category $A-\mathrm{Mod}$ seems to be equivalent to the category of descent data, the objects of which are collections of modules $M_j$ over $e(I_j)Ae(I_j)$ for some collection of indexing sets $I_j$ such that $\bigcup_j I_j=\{1,\ldots,m\}$ with isomorphisms $e(I_i \cap I_j)M_i\simeq e(I_i \cap I_j)M_j$ satisfying the usual cocycle conditions.

For now, in my partial proof of this equivalence I do use the properties of my algebra $A$, but maybe that's not really necessary? Has this sort of descent for module categories been worked out before? I have a gut feeling that this is something known and I just don't know what algebraists would call it.