Questions tagged [homological-algebra]
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
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Comparing the Stacks Project Homotopy limit with limits in the $\infty$-category
In the Stacks project Tag 08TC, there is a definition of a homotopy limit in a derived category, and I expect it to compare with a limit in the $\infty$-categorical enhancement. I guess this is also ...
- 437
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Two spectral sequences arising from a simplicial spectrum
Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization.
Let's assume each $X_n$ is connective.
From this situation, we can form two filtrations on $X$: the ...
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How does the behaviour of a hyperderived functor of many variables change if you use $\prod$-totalisation instead of $\oplus$-totalisation?
$\newcommand{\tot}{\operatorname{Tot}}\newcommand{\A}{\mathscr{A}}\newcommand{\L}{\mathbb{L}}\newcommand{\R}{\mathbb{R}}$Say $T$ is is a functor $\A_1\times\A_2\times\cdots\times\A_n\to\A$ of Abelian ...
- 477
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From exact triangles in the stable category of maximal Cohen--Macaulay modules to short exact sequences
Let $R$ be a local Gorenstein ring. Let $\underline{\text{CM}}(R)$ be the stable category of maximal Cohen--Macaulay modules, it is known to carry a triangulated structure. My question is: If $M\to N\...
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Using the mapping cone to show that a chain map defines a stable equivalence between two symmetric algebras
This question is about an argument in the proof of Theorem 9.8.8 in Linckelmanns Block Theory of Finite Group Algebras. I need to understand the argument in order to do something similar in my ...
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2
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Adjunctions and inverse limits of derived categories
Consider a tower $\dots\to A_{2}\to A_{1}$ of rings. This gives rise to a diagram $\mathbb{N}^{\text{op}}\to\text{Cat}_{\infty}$ of $\infty$-categories (confusing $\mathbb{N}^{\text{op}}$ with its ...
- 437
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Are there known minimal models for the cohomology of semisimple Lie algebras?
My student and I recently found a cute construction of a minimal model for the cohomology of a Lie algebra $\mathfrak{g}$. This is a "minimal model" in the sense that it is a minimal chain-...
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What conditions on an Abelian category allow members of a direct sum to be determined entirely by their components?
EDIT: In comments, with thanks to Maxime Ramzi, this question has a good answer in that what I want to be true is true when $\mathscr{A}$ satisfies axiom $\mathsf{AB}5$, that $\mathscr{A}$ is ...
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Is $\mathrm{Hom}_R(M,R)\neq \mathfrak m \mathrm{Hom}_R(M,R)$ if $M \neq \mathfrak m M$ and $\mathrm{Hom}_R(M,R)\neq 0$?
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring and $M$ be an $R$-module such that $M \neq \mathfrak m M$ and $\mathrm{Hom}_R(M,R)\neq 0$. Then, is it true that $\mathrm{Hom}_R(M,R)\neq \...
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How is the classification of groups extensions by $H^2$ related to Yoneda Ext?
It is well-known that group extensions
$$1\to A \to H \to G \to 1$$
where $A$ is abelian with a $G$-action such that the conjugation action of $G$ on $A$ agree with this fixed action are classified ...
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Why is this map a split monomorphism?
I have a question regarding a lemma in the proof of Hopkins-Neeman Correspondence.
It is the beginning part of Lemma 1.2 in the The Chromatic Tower for D(R)
Let $Y$ be an object of the derived ...
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A complex with homology $=R/p$
Given a Noetherian ring $R$ .
I am looking for a bounded complex $X$ of finitel geenerated projectives over $R$ whose homology is $R/p$. Infact I just need $X$ to have $\operatorname{Supp}(H(X)) = \...
- 151
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Vanishing of $\operatorname{Ext}_R^{1}(M,R)$ when $R$ is a Gorenstein local ring of dimension $1$ and $M$ is not finitely generated
Let $(R,\mathfrak m)$ be a Gorenstein local ring of dimension $1$. Let $M$ be an $R$-module (not finitely generated) such that $M\neq \mathfrak m M$ and there exists a non-zero-divisor $x\in \mathfrak ...
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The Krull dimension of the tensor product of rings
The Krull dimension of a ring $R$ is defined as the length of the longest chain of prime ideals in it. Let $R_i$, for $i\in\mathbb{N}$ denote a sequence of commutative Noetherian rings of Krull ...
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When is a functor of chain complexes triangulated?
Let $\textsf{A}, \textsf{B}$ be abelian categories.
Let $F: \operatorname{Ch}(\textsf{A}) \to \operatorname{Ch}(\textsf{B})$ be an additive functor of chain complexes. If $F$ preserves chain ...
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Homotopy theory of differential objects
In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
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Pontryagin product on the homology of cyclic groups
Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the ...
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Confusion about signs in the definition of an $A_\infty$-algebra
We are trying to understand the definition of $A_\infty$-algebras. But we are puzzled by what appear to be two different sign conventions (and we cannot figure out how these two are equivalent)
We see ...
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$\mathbb{Z}[T]$-Solidification in light condensed setting
In the lectures to "Analytic Stacks" Scholze and Clausen introduced a new concept of "light" condensed mathematics. In Lecture 7 Clausen introduces the derived $T$-solidification ...
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On infinity-morphisms between algebras over algebraic operads
I posted this question in the "Mathematics" stack exchange, but it hasn't got much attention... I hope it will get more here.
Let $P$ be a Koszul operad.
In the book of Loday-Vallette "...
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When do faithfully semiinjective complexes exist?
Question:
For which (perhaps noncommutative but always unital and associative) rings $R$ do faithfully semiinjective complexes of right or left $R$-modules exist?
Hopefully the answer is: "for ...
- 477
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Lengths and additive invariants which preserve positivity
The length of a module is well-known to be an additive invariant of finite-length modules. That is, if $R$ is a ring and $Art(R)$ its category of finite-length modules, then $length : Ob (Art(R)) \to \...
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Why should we study the total complex?
Recall that for every double complex $C_{\bullet,\bullet}$, there is a canonical construction called the total complex $\operatorname{Tot}(C_{\bullet,\bullet})$ associated to it. This complex can be ...
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What happens if I take a doubly-free simplicial abelian group?
Suppose that I have a simplicial set $X_\bullet$. I can take the free abelian group generated by $X_\bullet$, $\mathbb{Z}X_\bullet$. But then I can forget that this has an abelian group structure, ...
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Hattori-Stallings trace
Let $R$ be a (possibly non-commutative) unital ring and $M$ be a left $R$-module. If $M$ is finitely generated and projective, the natural map $$\iota:\mathrm{Hom}_R(M,R)\otimes_R M\to \mathrm{Hom}_R(...
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Rings where all indecomposable modules are projective or injective
Let $A$ be a semi-perfect noetherian ring.
Is there a nice classification of such $A$ such that every indecomposable finitely generated $A$-module is projective or injective?
Im also interested in ...
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Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample
Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
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Representation theory of $\mathrm{GL}_n(\mathbb{Z})$
I want to understand the (complex) representation theory of $\mathrm{GL}_n(\mathbb{Z})$, the general linear group of the integers. I have gone through several representation theory texts but all of ...
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Projective objects in chain complexes of an abelian category: Further question
Yes, I see there are other Q&A's on this, for instance here: Projective objects in the category of chain complexes
I am wondering why a level-wise projective chain complex $P$ which is split ...
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Group homology for a metacyclic group
Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
We work with the first homology group
$$ H_1(G,M).$$
For any ...
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Geometric meaning of Beilinson's unipotent Jordan block construction of nearby cycles
In his paper how to glue perverse sheaves, Beilinson used unipotent Jordan blocks to construct the famous (unipotent) nearby cycle functor, vanishing cycle functor and the maximal extension functor. ...
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Vershik's conjecture about generic quadratic algebras
Is it still unknown whether very general (lying in a countable intersection of some Zariski opens in corresponding Grassmannian) quadratic algebras $R$ with $\operatorname{dim} R_2 < \frac{3}{4}(\...
- 4,299
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A depth version of a conjecture of Yamagata
Let $A$ be a finite-dimensional $K$-algebra.
Recall that the grade of an $A$-module $M$ is defined as the smallest $i$ such that $\operatorname{Ext}_A^i(M,A) \neq 0$ and the depth of $A$ is defined ...
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Quotient of a polynomial ring with a prime ideal is Cohen$-$Macaulay
[Bruns-Herzog, Exercise 2.1.17] Let $k$ be a field and $R = k[x_1, . . . , x_n]$. Suppose $\mathfrak{p} \subset R$ is a prime ideal, $ht\mathfrak{p} \in \{0, 1, n − 1, n\}$. Show that $R/\mathfrak{p}$ ...
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Multiplication map by a ring element on an object vs. all its suspensions in singularity category
Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...
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Exact sequence for relative cohomology + normal crossing divisors
Let $X$ be smooth algebraic variety over $\mathbb C$ and $D_1, D_2$ are snc divisors such that $D_1\cup D_2$ is also snc.
Is it true that there is an exact sequence
$$H^*(X, D_1\cup D_2)\to H^*(X, D_1)...
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Vanishing of self-hom in Spanier–Whitehead stabilization category
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SW{SW}$Let $R$ be a commutative Noetherian ring. For $R$-modules $M,N$, let $\mathcal I_R(M,N)$ be the collection of all $f\in \text{Hom}_R(M,N)$ ...
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Cohen–Macaulayness of $k[[x^2, x^3, xy, y]]$ over $k[[x^2, y]]$
Let $k$ be a field and $R = k[[x^2, y]]$ and $S = k[[x^2, x^3, xy, y]]$. Since $R \subset S$, is $S$ Cohen–Macaulay as $R$-module?
To check this, what I have observed is that in $S$, the maximal ...
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DG algebra structure on minimal free resolution of modules over regular local ring
Let $(Q, \mathfrak n, k)$ be a regular local ring. Let $I\subseteq \mathfrak n^2$ be an ideal, and fix a minimal generating set $\mathbb f= f_1,\cdots, f_n$ of $I$. The Koszul complex $E:= Q[e_1,...,...
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Homotopy equivalence of chain complexes from subcomplexes and quotient complexes
Let $C_1$ be a finite-dimensional chain complex over $\mathbb{C}$ coefficients. Let $S_i$ be a subcomplex of $C_1$ and let $Q_1$ be the quotient complex. Suppose $S_1$ and $Q_1$ are chain homotopy ...
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Generalization of $H^*(\Gamma; \mathbb{Z\Gamma}) \cong H^*_c(X; \mathbb{Z})$?
Let $\Gamma$ be a group acting freely and cocompactly on an acyclic space $X$. I know that there is a isomorphism $H^*(\Gamma, \mathbb{Z\Gamma}) \cong H^*_c(X; \mathbb{Z})$ of $\Gamma$-modules. The ...
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An $E_{\infty}$-algebra is a $C_{\infty}$-algebra?
Past this question in MO have raised the following questions for me.
Question
In characteristic $0$, it is well-known that a Kadeishvili‘s $C_{\infty}$-algebra is an $E_{\infty}$-algebra.
However, do ...
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For which subgroups the transfer map kills a given element of a group?
$\newcommand{\ab}{{\rm ab}}
\newcommand{\ord}{{\rm ord}}
$Let $G$ be a finite or profinite group. Consider the abelianized group
$$G^\ab=G/G'$$
where $G'$ is the commutator subgroup of $G$.
Let $H\...
- 13.4k
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Simplicial resolution for commutative group scheme
Let $X$ be a quasi-projective $k$-variety. In this case the symmetric power $S^d(X)$ is well-defined. If $S^\bullet(X)=\bigsqcup_{n>0}S^d(X)$, where we suppose $S^0(X)=\operatorname{spec}(k)$, then ...
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Ext for commutative Gorenstein algebras
Let $A$ be a finite dimensional commutative Gorenstein $K$-algebra over a field $K$.
Question 1: Is there an easy example of $A$-modules $M$ and $N$ such that $\mathrm{Ext}_A^1(M,N)=0$ but $\mathrm{...
- 25.4k
10
votes
2
answers
661
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Spectral sequences and short exact sequences
Suppose I take a short exact sequence of filtered chain complexes:
$$0\to A\xrightarrow{p} B\xrightarrow{q} C\to 0$$
We assume that $p$ and $q$ are filtration-preserving, so that $p(F_rA)\subseteq ...
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164
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Preservation of (co)limits under taking derived categories
Let $R$ be a commutative ring. Let $\{A_i\}_{i \in I}$ be a diagram of $R$-linear $1$-categories, indexed by a finite poset $I$. (If this matters, assume that the $A_i$ have finitely many objects).
...
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270
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Serre subcategories of the category of chain complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R$ be a commutative $k$-algebra.
We denote by $\operatorname{Mod}(R), C(R), $ and $ D(R)$ the category of $R$-modules, the category ...
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On the invariance of the Kaledin class
In Formality of DG algebras (after Kaledin), Lunts introduces an $A_\infty$-Hochschild cohomology class, called the Kaledin class, controlling formality of an $A_\infty$-algebra up to a certain order. ...
- 6,679
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Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings
Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
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