Questions tagged [abelian-categories]
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On a simple alternative correction to Roos' theorem on $\varprojlim^1$
Here is a discussion about an incorrect theorem of Roos, later corrected, some counterexamples and so on. Reading over this, I was a bit shocked because it contradicted something from Weibel's ...
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contravariant finiteness and limit closure: is there dual to a result of Crawley-Boevey?
Let $\mathcal A$ be a locally finitely presented category. Theorem 4.2 of https://doi.org/10.1080/00927879408824927 says that given a full additive subcategory $\mathcal D$ of finitely presented ...
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Thick subcategory containment in bounded derived category vs. singularity category
Let $R$ be a commutative Noetherian ring, and $D^b(\operatorname{mod } R)$ the bounded derived category of the abelian category of finitely generated $R$-modules. Let me abbreviate this as $D^b(R)$. ...
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Endomorphisms of simple dualizable objects in a linear abelian monoidal categories
In When is an object in a linear or abelian category simple? Or: How should I define fusion categories? endomorphisms of simple objects in a k-linear abelian category are discussed. In the answer it ...
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Automorphism groups for simple objects in abelian linear categories
Let $\mathcal{A}$ be an abelian category that is also $k$-linear, where $k$ is some algebraically closed field. Let $X$ be a simple object in $\mathcal{A}$. What can we say about $\mathrm{Aut}(X)$? I ...
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Is taking Freyd envelopes adjoint to taking stable module categories?
Let $T$ be an (idempotent-complete) triangulated category. Then the Freyd envelope $mod(T)$ is an abelian category, the universal recipient of a homological functor $T \to mod(T)$. The Freyd envelope ...
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Reference request regarding faithfully exact functors between abelian categories
I am looking for a reference for the following result (or any subresult) in any book or notes:
Lemma. Let $F:\mathcal{A}\to\mathcal{B}$ be an exact functor between abelian categories. The following ...
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Dual objects in an abelian monoidal category
Let $(\mathcal{M},\otimes)$ be a monoidal category that is also abelian, and assume that $\otimes$ is bilinear with respect to the biproduct. Let $X$ and $Y$ be two objects that admit (left) duals ${}^...
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Is there a way to “derive” a (non-exact) functor which preserves images?
Let $F : \mathcal A \to \mathcal B$ be an additive functor between abelian categories. If $F$ is left exact, then under certain conditions $F$ admits right derived functors which “measure” its failure ...
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Kernels and cokernels in a quotient of an abelian category
I am trying to understand the construction of the quotient of an abelian category called the Serre quotient or Gabriel quotient. From the description here: https://en.wikipedia.org/wiki/...
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Why are Gabriel categories closed under localization?
Let $\mathcal K$ be a Grothendieck category. Recall the Gabriel filtration $0 \subseteq \mathcal K_1 \subseteq \cdots \mathcal K$ of localizing subcategories, where $\mathcal K_{\alpha+1}$ is ...
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Example of a Grothendieck category which is not Gabriel?
Following Gabriel, for $\mathcal C$ a Grothendieck category, set $\mathcal T(\mathcal C)$ to be the localizing subcategory generated by the objects of finite length, and $\mathcal C' = \mathcal C / \...
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Left duals and right duals are also isomorphic in a semisimple category
In the n-Lab page
https://ncatlab.org/nlab/show/rigid+monoidal+category
it is written that
Left duals and right duals are also isomorphic in a semisimple
category.
For a left dual semisimplicity ...
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Comparing stabilization of stable category modulo injectives and a Verdier localization
Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...
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Is uniform dimension monotonic in quotients when there is a unique indecomposable injective?
The notion of uniform or Goldie dimension is something I’ve only seen discussed for categories of modules, but I believe the theory works the same way in any Grothendieck category $\mathcal C$. Recall ...
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