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.
144 questions from the last 365 days
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Coxeter matrix of Dyck path
I am trying to understand Gjergji Zaimi's answer in What are the periodic Dyck paths?. In the third paragraph he claims that
Next, we define the matrix $X_D$
similarly to the Cartan matrix except we ...
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0
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107
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Derived tensor products and regular sequences
Let $R \to A$ be a homomorphism of commutative rings, and let $x\in R$ be an element (or a sequence of elements in $R$, if you prefer) that is both $R$-regular and $A$-regular. Then we have
$$
A\...
- 115
3
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1
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163
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Theory of $n$-truncated $A_\infty$ categories/functors?
One can define certain higher categorical truncations. For example, a discussion from the quasi-categories point of view can be found in HTT 5.5.6.
On the other hand, as a model of linear $\infty$-...
- 169
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124
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Generalization of Koszul cohomology for wedge product with a fixed q-form: literature and references?
Let $A$ be a commutative ring. For an odd positive integer $q$ and an integer $p$ in the range $1 \leq p \leq q$, consider the cochain complex:
$0 \to \Lambda^p(A^N) \to \Lambda^{p+q}(A^N) \to \cdots \...
- 21
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209
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Tate cohomology as a derived functor
I am trying to understand in what sense Tate cohomology may be regarded as the co-fibre of the norm map, from the perspective of derived functors.
Say $G$ is a finite group and $M$ is a $G$-module ...
- 2,907
2
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1
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148
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Baer sums of extensions
Apologies in advance if this question is too basic, I looked briefly through Weibel and the stacks project and couldn't find any relevant reference.
Let $\mathcal{A}$ denote an abelian category, and ...
- 2,907
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163
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Problem understanding the cup-products for the modified cohomology in David Harari's book "Galois Cohomology and Class Field Theory"
I'm recently reading the very beginning of David Harari's book Galois Cohomology and Class Field Theory, and I met a problem with the definition of cup-products for the modified cohomology.
Before ...
- 525
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114
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Selmer complex and total complex
Thanks for your reading. I'm studying Selmer complex book by Jan Nekovar. For the definition of Selmer complex I meet a problem.
In the introduction(page 9, 0.8.0) the author gives us a definition of ...
- 319
2
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0
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137
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Leray spectral sequence for étale homology
Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. ...
- 658
3
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1
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179
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Left exact functor $F$ preserves quasi-isomorphism between $F$-acyclics
In this math overflow page, the poster gives a proof of the statement "an additive left exact functor $F$ preserves quasi-isomorphisms between $F$-acyclic objects." I'm having trouble ...
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133
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Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139
4
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1
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227
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Literature Request: The derived category is Krull-Schmidt
I am looking for literature where it is proven that the derived category of bounded complexes over a finite-dimensional algebra is Krull-Schmidt. I found this question
Literature request: $K^b(\text{...
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66
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Projective cover (minimal) for (derived)complete modules over Noetherian local rings exist?
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $M$ be an $R$-module which is $\mathfrak m$-adically derived complete. Then, does there exist a free $R$-module $F$ and a surjective $...
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Homotopy between posets
This is entirely a new area for me and I apologise in advance if the questions are silly.
In Quillen's paper "Homotopy properties of the posets of non-trivial $p$-subgroup of a group" (see ...
- 139
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168
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Theorems related to Chevalley's theorem
Recently I have read Chevalley's theorem of a complete local ring which basically says that if $(R,\mathfrak{m})$ is a complete local ring and if $\{b_n\}$ be a sequence of ideals such that $b_n \...
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1
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An lower bound for the injective dimension of a module
Let $A$ be a finite dimensional algebra and $M$ a non-injective $A$-module.
Question: Do we have idim $M>$domdim $\tau^{-1}(M)$?
Here idim $N$ denotes the injective dimension of a module $N$ and ...
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Determine equivalences in the generated collection of subgroups and quotients
Let $A$ be an abelian group, and $B_1, B_2, \dots, B_m$ be subgroups of $A$. Define the family of subgroups $\mathcal{D}_0 = \{ \{0\}, A, B_1, B_2, \dots, B_m \}$.
Let $\mathcal{C}_1$ be the ...
- 807
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117
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Tilting complexes arising from homotopy equivalences
Let $k$ be a field and let $A$ and $B$ be finite-dimensional selfinjective $k$-algebras. Suppose we have an isomorphism of homotopy categories $F: K^b(A-mod) \cong K^b(B-mod)$ that descends to a ...
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284
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Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?
Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
- 412
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216
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Compute the singular homology group modulo barycentric subdivision
Let $X$ be a topological space, and let $C(X)$ denote its singular chain complex with boundary operator $\partial$ and $n$-th chain group $C_n$. We know there exists a barycentric subdivision operator ...
- 807
3
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169
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equivalence of two categories
I am new to algebraic geometry and category theory. I am wondering about the following functor is equivalence of categories or not.
Let $X$ be irreducible scheme and $x$ be its unique generic point. ...
- 613
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112
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Finitely generated projective modules over Noetherian endomorphism ring
Let $\mathcal A$ be a locally Noetherian Grothendieck abelian category and $M\in \mathcal A$ be a Noetherian object. Set $B:=\text{End}_{\mathcal A}(M)$. Let $B$-mod be the category of finitely ...
- 413
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Equivalence of triangulated categories defined by tilting objects
I asked this question on stack exchange (https://math.stackexchange.com/questions/4983535/equivalence-of-triangulated-categories-defined-by-tilting-objects), but got no responses, so I ask it here.
...
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181
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Levelled trees and the homotopy transfer theorem
In section 10.3.12 of Loday-Vallette's book "Algebraic operads", given a $P_\infty$-algebra $(A,d,\alpha)$ the Homotopy Transfer Theorem applied to $H_*(A,d)$ is studied. There, because the ...
- 215
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124
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Minimal model for $A_\infty$-categories
Is there a reference for existence and construction of the minimal model of an $A_\infty$-category? Most references I found ultimately refer to Lefèvre-Hasegawa's thesis but there doesn't seem to be a ...
- 489
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80
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Selforthogonal modules and finitistic dimension
Algebras $A$ are always finite dimensional over a field here.
A module $M$ is called selforthogonal if $\operatorname{Ext}_A^i(M,M)=0$ for all $i>0$.
Define the orthogonal finitistic dimension $\...
- 26.5k
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513
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When does derived tensor product commute with arbitrary products?
Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
- 412
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86
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Lifting maps on the spectral sequence of a double complex to the derived category
Question
The differentials on the $(r+1)$th page of a spectral sequence are maps on the cohomologies of the complexes on $r$th page. So, between two adjacent complexes $K^\bullet,L^\bullet$ on the $r$...
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120
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Derived tensor by perfect complex preserves exact triangle in singularity category?
Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
- 413
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207
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When is the derived category of a ring generated by injective modules
Are there any equivalent conditions on a ring to the condition that the localizing subcategory of $D(R)$ generated by injective modules is the entire category? Are there any non-examples in Boolean ...
- 2,346
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176
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Example of a Boolean Ring with infinite injective dimension over itself
It is known that Boolean rings can have infinite global dimension (free Boolean algebra on a large enough number of generators) [ see The Global Dimension of Boolean Rings by Pierce]. Are there any ...
- 2,346
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168
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When do we have $\bigoplus_{i + j = n} R^i f_* \mathbb{Q}_\ell \otimes_{\mathbb{Q}_\ell} R^j g_* \mathbb{Q}_\ell \cong R^{i + j} h_* \mathbb{Q}_\ell$?
Milne, Étale Cohomology, theorem 8.5 states the following version of the Künneth formula (in slightly greater generality). Let $\Lambda$ be a finite commutative ring. Let $X, Y, S$ be schemes with $S$ ...
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126
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full strong exceptional collection
I am wondering whether, if a triangulated category $\mathcal{D}$ has a full strong exceptional collection (infinite), it is triangle-equivalent to the bounded derived category of finitely generated ...
- 359
1
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0
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135
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Local cohomology and image of $1$ under the canonical map from Ext to local cohomology
Let $R$ be a commutative Noetherian local ring, and $S$ be an $R$-algebra. Let $x_1,\dots,x_t$ be elements, in the maximal ideal of $R$, which is a regular sequence on both $R$ and $S$, and let $I$ be ...
- 412
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307
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Serre functors and global dimensions
Let $k$ be a field.
Let $\mathcal{C}$ be an abelian category (over $k$).
We say that $\mathcal{C}$ has a finite global dimension if there exists integer $n > 0$ such that
$$
\operatorname{Ext}^i(M,...
- 889
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0
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151
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Can Harrison cohomology be written using Ext?
Just like Hochschild cohomology for associative algebras and Chevalley-Eilenberg cohomology for Lie algebras, it'll be nice (or disappointing?) if Harrison cohomology can be expressed in terms of Ext'...
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131
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Is there anything like a Čech complex for calculating local cohomology over *noncommutative* rings?
$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\colim{colim}$Let $R$ be a ring, and consider a two-sided ideal $I = (r_1, r_2, \dots, r_j)$ in $R$. The corresponding $n$th local cohomology functor ...
- 1,335
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107
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Dimension of hom spaces between indecomposable modules
Undergraduate-Level Background
Let $A$ be an Artin algebra over an algebraically closed field $k$, and let $C = Rep(A)$ denotes the category of $k$-linear, $k$-finite dimensional representations of $A$...
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If a bimodule is "generated" by single elements, must the elements be conjugate?
Let $A$ and $B$ be Artin $k$-algebras for a commutative artinian ring $k$ (e.g. $A$ and $B$ are finite dimensional $k$-algebras for a field $k$). Let $M$ be an $A$-$B$-bimodule of finite length over $...
- 696
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Reference request: locally erasable delta-functor is universal
It is well-documented that an erasable delta-functor $(F^n,\delta)$ is universal. However, in 'small' abelian categories (in the technical sense or otherwise), there aren't always enough objects to ...
- 28.7k
4
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1
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184
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FI-homology of a spectral sequence of rational FI-modules
Let $(E^r_{p,q})$ be a spectral sequence of rational $\mathsf{FI}$-modules. Call $H^{\mathsf{FI}}$ the $\mathsf{FI}$-homology (see here) and $t_k X= \deg H^{\text{FI}}_k X$ the $k$-th generation ...
- 275
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339
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If the Hom-space of finite length modules is generated by single elements, must the elements be conjugate?
Let $A$ be an Artin $k$-algebra for a commutative artinian ring $k$ (e.g. $A$ is a finite dimensional algebra over a field $k$). Let $X,Y$ be finite length left $A$-modules. If $\text{Hom}_A(X,Y)$ is ...
- 696
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2
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228
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Arithmetic application: Complete group ring and group ring for infinite group
Let $G$ be a profinite(infinite) group, $\Lambda$ be the complete group ring(Iwasawa algerbra) of $G$ over a unity ring $R$. My first question is that do we know something about the relation with $\...
- 319
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1
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Does a crossed extension with trivial Postnikov invariant admit a section?
This question is part of my effort to understand higher cohomology of groups in terms of extensions.
Recall an $n$-fold crossed extension of a group $G$ by an abelian group $K$ is an exact sequence of ...
- 607
3
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241
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Could I get an interpretation for application of Euler characteristics in number theory？
As a beginner who just get in touch with Euler characteristics in this field, could I get some intuition for the arithmetic meaning of Euler characteristics of bounded complexes, for example Selmer ...
- 319
4
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0
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112
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Differentials on free algebras over operads
I am currently reading Cyclic Operads and Cyclic Homology by Getzler-Jones and have some confusions.
I am under the impression that given an (associative, say) algebra $A$ that an almost-free ...
- 195
3
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0
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85
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Simplifying sums with CAS or theorem prover
To verify that a certain map is a chain homotopy I could reduce it to an evaluation of
$S = S_1 + S_2$ where
$$
S_1 = \sum_{a=0}^p \sum_{b=0}^{p+1} (-1)^{a+b} n_b \cdot e_a
$$
$$
S_2 = \sum_{b=0}^p \...
- 708
3
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0
answers
107
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rational homology of SO(2,1) over number fields
Let $\mathrm{SO}(2,1)$ be the special orthogonal group defined by the quadratic from $q(x,y,z)=x^2+y^2-z^2$.
This is a connected non-simpy connected algebraic group.
Now, let $F$ be a number field, ...
1
vote
1
answer
104
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Finiteness of Krull dimension of commutative Noetherian ring for which maximal length of regular sequence in maximal ideals have a uniform upper bound
$\DeclareMathOperator\grade{grade}$Let $R$ be a commutative Noetherian ring. For an ideal $I$ of $R$, let $\grade(I,R)$ be the maximal length of an $R$-regular sequence in $I$.
My question is: If $\...
- 413
6
votes
2
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442
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Existence of functorial (K-)flat resolutions?
I am wondering about the following: Suppose $X$ is a reasonable scheme or stack with the resolution property. (So, all quasi-coherent sheaves admit a surjection from a flat sheaf.) Then I believe ...
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