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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Exterior powers of the Cartan matrix and Dyck paths
(This question can be formulated purely combinatorially in terms of Dyck paths, which is done in the second part of the question. But I am more interested whether this can be explained by some sort of ...
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Additivity of group cocycles?
In Juven Wang, Zheng-Cheng Gu, and Xiao-Gang Wen - Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond, the authors calculated many ...
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On Serre's condition and singular locus of determinantal rings
Let $R$ be a Commutative Noetherian ring. Let $\mathbf X:=[X_{ij}]_{1\le i \le r, 1 \le s \le t}$ be a matrix of indeterminates. Let $t>1$ be an integer, and $I_t(\mathbf X)$ denote the ideal in $...
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Infinite dimensional homology and spectral sequences
I am new to spectral sequences, so I'm not sure about the difficulty of this question.
Suppose we have a filtration of a chain complex $\emptyset=D_{-1}\subset D_{0}\subset D_1\subset\dots D_n= C$ and ...
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3
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1
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$\Omega$ for noetherian semiperfect rings
Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-...
- 26.1k
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Decompose an unbounded (cochain) complex in the homotopy category
Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits a distinguished triangle
$$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[...
anon
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166
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Grothendieck trace formula for schemes with étale fundamental groups that have no dense cyclic subgroup
This question may be more of a philosophical rather than mathematical nature.
Assume I have a scheme $X$ and an endomorphism $F:X\longrightarrow X$. For instance, $X$ might be of finite type over $\...
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Simplicial enrichment on unbounded algebras over an operad
In his paper "Homological Algebra of Homotopy Algebras" V.Hinich introduced a simplicial structure on algebras in unbounded chain complexes over arbitrary $\Sigma$-split operads. Not to get ...
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Is there a homotopical analogue of short exact sequence?
For $R$-modules for a commutative ring $R$, submodules and quotients are put on roughly the same footing; the kernel of a quotient is an injection into the source, and the cokernel of this injection ...
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Leray spectral sequence and pullbacks
I am trying to find a reference for the following well-known result on the functoriality of the Leray spectral sequence:
Let $\pi:X\to Y$ and $\pi':X'\to Y'$ be morphisms of schemes and denote by $E_2^...
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Explicit proof that algebra is derived wild
Following the terminology of
Drozd, Yuriy A., Derived tame and derived wild algebras, Algebra Discrete Math. 2004, No. 1, 57-74 (2004). ZBL1067.16028.
let $A$ and $R$ be algebras over a field $k$. A ...
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Essentially zero inverse system of abelian groups
I am learning local cohomology from Hartshorne’s Local Cohomology book.
My question is about the notion of essentially zero inverse system of abelian groups, which is defined to be an inverse system ...
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Is the category of modules over a commutative ring the category of abelian objects in a topos?
The categories of modules over commutative rings are especially notable Abelian categories. Wanting to extend this class a bit, I thought of this question:
Let $R$ be a commutative ring with $1$. Does ...
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A question concerning cancellation of ideals
I am working on a number theory project, and at one stage, I encounter a commutative algebra problem. Vaguely speaking, my hope is to show that two ideals are equal. Now I shall explain the data I am ...
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Hochschild homology computation of certain type
I know that in general Hochschild homology is not very computable. However, this certain type of Hoshschild homology shows up and I feel like there could be existing result.
Let $k$ be a field and $A$ ...
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Kähler differentials give a left Quillen functor
Is there a reference for the fact that the functor of Kähler differentials is a left Quillen functor on the category of $\mathrm{CDGA}_k/B$? Here $k$ is a field of characteristic $0$, and $B$ is some ...
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Why does this construction not give a functorial cone in the homotopy category of cochain complexes?
I have heard the expression recently that one should be careful when constructing cones in the homotopy category - namely, that this is not functorial. However, when working through some examples in ...
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Kunneth formula for hypercohomology
Let $A_{\bullet}$ and $B_{\bullet}$ be two bounded complexes of sheaves over a variety $X$. Is there a Kunneth-like formula for the hypercohomology of the tensor product $A_{\bullet}\otimes B_{\bullet}...
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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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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 ...
anon
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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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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
$$ \mathcal{O}(\mathbb{C}^n)_0 \otimes_{\mathbb{C}} \...
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Explicit form of boundary operators of topological cones
Let $\Omega$ be a regular, finite, $n$-dimensional CW complex with chain modules $\mathscr{C}_k$ and boundary operators $\partial_k$.
For many problems in computational geometry, a key operation is to ...
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Lifting theorem for modules over a DGA
In the survey paper "Cartan's Constructions, the homology of $K(\pi,n)$'s, and some later developments" by J. C. Moore ( http://www.numdam.org/item/AST_1976__32-33__173_0/ ) he states a ...
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Applications of the Dold-Kan correspondence
The Dold-Kan correspondence says essentially that simplicial abelian groups and nonnegative chain complexes of abelian groups are equivalent objects. While this is a very natural statement, I am not ...
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Is it true that $D(\mathcal{O}_Y\text-\mathrm{mod}_X)\cong D_X(\mathcal{O}_Y\text-\mathrm{mod})$?
Is it true that $D(Y_X)\cong D_X(Y)$, where $Y$ is a (smooth) variety, $X$ is a closed subvariety of $Y$ (possibly singular)? Here $D(Y_X)$ denotes the derived category of the abelian category of ...
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Spectral sequence for two fibrations
Given maps of fibrations, i.e. commutative diagrams of smooth manifolds
$$\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 ...
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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 ...
user473423
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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 ...
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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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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 ...
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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 $\text{mod}\,A$. ...
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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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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 ...
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