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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Mapping cone and chain homotopy inverse?

I am reading this from "Generalized Lefschetz Numbers" by Husseini, 1982: My issue is the first sentence of the proof. I know essentially nothing about mapping cones in homological algebra (...
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Is the tensor product of noncontractible chain complexes of bimodules also noncontractible?

Let $A,B,C$ be rings and let $X$ and $Y$ be bounded chain complexes of $(A,B)$-bimodules and $(B,C)$-bimodules, respectively. Suppose that $X$ and $Y$ are not contractible. Here are my questions: Is ...
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Explicit proof of Quillen's connectivity theorem

Definition Let $A$ be a commutative ring. An ideal $I \triangleleft A$ is called quasiregular if $I/I^2$ is flat over $A/I$ and there is a canonical isomorphism of algebras $$ \Lambda_A I/I^2\...
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Is the Schofield semi invariant defined at $V/IV$?

Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...
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When does a local Cohen–Macaulay ring admit a non-zero finitely generated maximal Cohen–Macaulay module of finite injective dimension?

Let $(R,\mathfrak m)$ be a local Cohen–Macaulay ring. Then, it is well- nown that there exists a non-zero finitely generated $R$-module of finite injective dimension; for instance $\operatorname{Hom}...
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On inverse limits of $\pi$-adically complete algebras

Consider the following situation, let $\mathcal{R}$ be a discrete valuation ring with uniformizer $\pi$ (say the valuation ring of a finite extension $K$ of $\mathbb{Q}_{p}$. Let $\{ A_{n}\}_{n\in\...
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1 answer
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Can information theory characterise a (topological) space?

Consider an objective function f: $\mathbb{R}^n\rightarrow\mathbb{R}$, with a vector of variables $\theta$, i.e. $f(\theta)$, $\theta \in \mathbb{R}^n$. Depending on $f$, there can be interesting ...
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dg-natural transformation between FM functors and Hom between kernels

The question is related to Morphism between Fourier-Mukai functors implies the morphism between kernels? Consider a complex smooth projective variety $X$ and the bounded derived category $D^b(X)$, it ...
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Rings of weak dimension ≤ 1 vs. semihereditary rings?

Rings in this question are assumed to be commutative. I am asking "natural" examples of rings of weak dimension $\le1$ which are not semihereditary. It would be better if there are integral ...
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Finitistic dimension conjecture — why artin algebras?

As I understand it, the finitistic dimension conjecture says that if a ring $A$ is finitely generated over a commutative artinian ring $K$, then the finitistic dimension of $A$ is finite. My question ...
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Homomorphism group of the additive group of rational numbers $\mathbb{Q}$ into quasi-cyclic group $\mathbb{Z}(p^\infty)$ [duplicate]

I read somewhere that the group of homomorphisms from the additive group of rational numbers $\mathbb{Q}$ into the quasi-cyclic group $\mathbb{Z}(p^\infty)$ is isomorphic to the additive group of ...
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base change property of Topological Hochschild homology

What is the "base change property" of topological Hochschild homology? In Proposition 11.10 of Bhatt-Morrow-Scholze's paper "Topological Hochschild homology and integral p-adic Hodge ...
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Flatness of tensor products of analytic germs

Let $\mathcal{O}(\mathbb{C}^n)_0$ denote the local ring of germs at the origin of holomorphic functions on $\mathbb{C}^n$. Consider the obvious map $$ \mathcal{O}(\mathbb{C}^n)_0 \otimes_{\mathbb{C}} \...
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Explicit form of boundary operators of topological cones

Let $\Omega$ be a regular, finite, $n$-dimensional CW complex with chain modules $\mathscr{C}_k$ and boundary operators $\partial_k$. For many problems in computational geometry, a key operation is to ...
4 votes
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Lifting theorem for modules over a DGA

In the survey paper "Cartan's Constructions, the homology of $K(\pi,n)$'s, and some later developments" by J. C. Moore ( http://www.numdam.org/item/AST_1976__32-33__173_0/ ) he states a ...
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Applications of the Dold-Kan correspondence

The Dold-Kan correspondence says essentially that simplicial abelian groups and nonnegative chain complexes of abelian groups are equivalent objects. While this is a very natural statement, I am not ...
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Is it true that $D(\mathcal{O}_Y\text-\mathrm{mod}_X)\cong D_X(\mathcal{O}_Y\text-\mathrm{mod})$?

Is it true that $D(Y_X)\cong D_X(Y)$, where $Y$ is a (smooth) variety, $X$ is a closed subvariety of $Y$ (possibly singular)? Here $D(Y_X)$ denotes the derived category of the abelian category of ...
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Spectral sequence for two fibrations

Given maps of fibrations, i.e. commutative diagrams of smooth manifolds $$\begin{matrix} \ F & \to & E &\to & B \\\ \downarrow & & \downarrow & & \downarrow \\\ \ F'...
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Regular ring is smooth when the field is perfect

Take $A$ a (not necessarily local) commutative algebra over a field $k$ which is essentially of finite type (i.e. a localization of a finitely generated algebra). In simple words, I just want to know ...
8 votes
1 answer
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Is the category of chain complexes a reflexive or coreflexive subcategory of the category of functors?

Let $A$ be an abelian category (you can assume additional conditions for its goodness). Let $\mathrm{Seq}(A) = \mathrm{Func}(\mathbb{Z}, A)$, where $\mathbb{Z}$ is the standard order category on ...
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Construction of a certain long exact sequence

Let $A$ be a noetherian ring (not necessarily commutative) or for simplicity even a finite dimensional algebra over a field. Let $X$ and $U$ be finitely generated $A$-modules and let $add(U)$ be the ...
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Quotients of Koszul algebras

Let $A$ be a noncommutative Koszul algebra (see here for a definition of Koszul) and let $c \in A$ be a central element. Will the quotient of $A$ by the ideal generated by $c$ again be Koszul. If not ...
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Double complex of simplicial resolution

In his lectures on condensed mathematics on page 30 Peter Scholze speaks of the double complex of a simplicial resolution. How is this defined? In the next line, he writes that if $A_\bullet$ is a ...
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Cohomology of a countable directed union of groups

It's puzzled me for a long time why two arguments in group cohomology look connected but no immediate visible connection is available. First, it is a theorem that if a group $G$ is the union of a ...
3 votes
1 answer
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Hilbert's Syzygy Theorem in the bigraded case

I've been recently wondering how to prove the existence of a Hilbert polynomial for finitely generated bigraded modules $M$ over a polynomial ring $R=k[X_0,...,X_n,Y_0,...,Y_m]$ with the usual ...
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1 answer
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Unbounded acyclic resolutions

Let $\mathscr A$ be a Grothendieck abelian category. Then every object in $\operatorname{Ch}(\mathscr A)$ is quasi-isomorphic to a $K$-injective object [Stacks, Tag 079P]. In particular, for any left ...
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Example of a periodic free resolution over a hypersurface

I'm reading "HOMOLOGICAL ALGEBRA ON A COMPLETE INTERSECTION, WITH AN APPLICATION TO GROUP REPRESENTATIONS" by David Eisenbud I'm wondering what would be a nice example illustrating Theorem 6....
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Indecomposable modules over a noncommutative noetherian ring

Let $R$ be a noncommutative noetherian ring. Can I say that every indecomposable injective right module appears as a direct summand of a term in the minimal injective resolution of $R_R$? I know this ...
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Cyclic homology can be recovered from topological cyclic homology?

Let $R$ be a commutative ring, $S$ a $R$-algebra of finite type. By an equivalence of ring spectra $$ \operatorname{HH}(S/R) \simeq \operatorname{THH}(S)\otimes_{\operatorname{THH}(R)} HR, $$ ...
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What do we know about a sheaf $M$ if we know its derived fibers $\mathsf{L}x^* M$, for $x\in X(k)$?

Let $X$ be a scheme over a field $k$. (Feel free to assume that $X$ is an algebraic variety, if needed.) Also, let $M^\bullet$ be a complex in the derived category of quasi-coherent sheaves $\mathsf{D}...
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What properties do the categories $\mathbf{GrpMod}$ and $\mathbf{GrpMod}^*$ of compatible pairs have? Can we do homological algebra with them?

Consider the following category $\mathbf{GrpMod}^*$ of compatible pairs, that is: an object is a pair $(G,M)$, where $G$ is a group and $M$ is a left $\Bbb Z[G]$-module. A morphism $(G,M) \to (H,N)$ ...
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Homological characterization of perfect resolutions

Suppose that $R$ is a left Noetherian associative ring with unit and $M$ a finitely generated left $R$-module. It is a standard fact that if the $\mathrm{Ext}$-groups $\mathrm{Ext}^i_R(M,N)$ vanishe ...
3 votes
1 answer
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For a finite dimensional $k$-Algebra $A$ does infinite representation type imply $(\text{rad}_A^{\omega})^2 \neq 0$?

Let $k$ be a field, $A$ a finite dimensional $k$-Algebra, $\text{mod}\,A$ the category of finite dimensional left $A$-modules and $\text{rad}_A$ the collection of radical morphisms in $\text{mod}\,A$. ...
2 votes
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Invariants of objects in $\operatorname{Ch}(\mathrm{Ab})$ up to chain homotopy

$\newcommand\Ab{\mathrm{Ab}}\newcommand\ab{\mathrm{ab}}\DeclareMathOperator\Ch{Ch}\DeclareMathOperator\Kom{Kom}\newcommand\ho{\mathrm{ho}}$Let $\Ab$ be the category of finitely generated abelian ...
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7 votes
1 answer
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Are perfect complexes the same as compact objects in D(R) for noncommutative rings?

The Stacks Project proves Thomason's insight that compact objects of the derived category $\simeq$ bounded complexes of finitely generated projective modules in Section 15.78, but the running ...
4 votes
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For a local complete intersection ring $(R,\mathfrak m)$ with $\mathfrak m^3=0\ne \mathfrak m^2$, $\mathfrak m$ can be generated by two elements

Let $(R,\mathfrak m,k)$ be a local complete intersection ring with $\mathfrak m^3=0\ne \mathfrak m^2$. As $0\ne \mathfrak m^2 \subseteq \text{soc}(R)$ and $R$ is Gorenstein, so we get $\mathfrak m^2 =\...
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What can be said about the derived functor of a composition between unbounded derived categories?

Let $\mathcal A, \mathcal B,\mathcal C$ be abelian categories and let $F:\mathcal A \to \mathcal B,G: \mathcal B \to \mathcal C$ be left exact functors such that $RF:D(\mathcal A) \to D(\mathcal B), ...
2 votes
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Do Weil cohomology theories for schemes over arbitrary rings exist, and do the standard theorems (Lefschetz fixed point, Tr. Formula etc.) still hold?

A Weil cohomology theory is a functor that assigns to a smooth projective variety $X$ of dimension $d$ over a field $k$ a graded ring of cohomology groups with values in a field $K$ of characteristic $...
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f.g. module $M$ over a complete local CM ring of dimension 1 such that $M, \text{Hom}_R(M,M), \text{Ext}^1_R(M,M)$ have finite injective dimension

Let $(R,\mathfrak m)$ be a local, $\mathfrak m$-adically complete, Cohen-Macaulay ring of dimension $1$. Assume that there exists a finitely generated $R$-module $M$ of depth $0$ such that $M$, $\text{...
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5 votes
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Bounding the projective dimension of modules by the number of points and arrows

Let $A=KQ/I$ be a finite dimensional connected quiver algebra with admissible ideal $I$ with n points and m arrows and $M$ an $A$-module. Question: Is the projective dimension of $M$ bounded by $n+m$ ...
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11 votes
1 answer
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Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper

In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...
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On $\text{depth}_S\left(\dfrac {S}{JS+xS}\right)$, when $\text{depth}_R(R/J)=0$, and $R\to S$ is a certain flat map of local rings

Let $(R, \mathfrak m) \xrightarrow{\phi} (S,\mathfrak n) $ be a flat homomorphism of local rings such that $\mathfrak n=\mathfrak m S +xS$ for some $x\in \mathfrak n \setminus \mathfrak n^2$. Let $J$ ...
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Commutative local rings which satisfy Krull-Remak-Schmidt

Question 1: Can the class of local (always noetherian and commutative) rings be classified for which the Krull-Remak-Schmidt theorem (KRS) holds for finitely generated modules? They contain for ...
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10 votes
1 answer
563 views

Reference request: infinity categories for the commutive algebraist/algebraic geometer

In a survey article Algebraic geometry in mixed characteristic, B. Bhatt writes For instance, given a commutative ring $R$ with a finitely generated ideal $I$, the assignment carrying $R$ to the $\...
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pullback square in abelian category and derived categories

Let $\mathcal{A}$ be an Abelian category. Take objects $A,B,C$ and $D$ in $\mathcal{A}$, and morphisms $b:B\to A$, $b':B\to A$, $c:C\to A$, $e:D\to C$, $e':D\to C$ and $f:D\to B$ such that diagrams $\...
6 votes
1 answer
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How does the Balmer spectrum fail to describe the algebraic geometry of categories of non-compact objects?

In trying to understand the higher algebraic geometry of the stable homotopy category, one thing I've come across repeatedly is the claim that one should only consider the Balmer spectrum of a tt-...
3 votes
2 answers
313 views

Is there an elementary reason that this colocalisation map of complexes is a quasi-isomorphism?

A fact about triangulated categories is that (exact) localisation functors and so-called colocalisation functors come in pairs, making an exact localisation triangle. I've tried to come up with less ...
2 votes
0 answers
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On the generalization of a Cech-to-sheaf type spectral sequence

Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...
1 vote
1 answer
112 views

Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?

Let $R$ be a commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...
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1 vote
1 answer
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Isomorphism passes on to a direct summand?

Let $\mathcal{T}$ be an additive category (or Abelian category, triangulated category). Take objects $A_1, A_2, B_1$ and $B_2$ in $\mathcal{T}$ such that there are morphisms $a_1:A_1 \to B_1$ and $a_2:...

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