All Questions
Tagged with ra.rings-and-algebras homological-algebra
285 questions
34
votes
1
answer
5k
views
Freyd-Mitchell's embedding theorem
Freyd–Mitchell's embedding theorem states that: if $A$ is a small abelian category, then there exists a ring R and a full, faithful and exact functor $F\colon A \to R\text{-}\mathrm{Mod}$.
I have been ...
- 3,004
32
votes
7
answers
4k
views
"Sums-compact" objects = f.g. objects in categories of modules?
Hello,
Let us call an object of an additive category sumpact (contraction of "sums" and "compact") if taking $Hom$ from it (considered as functor from the category to $Ab$) commutes with coproducts. ...
- 5,562
28
votes
0
answers
527
views
What algebraic structure characterizes all natural operations between differential operators and differential forms?
On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms:
the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$;
...
- 37.8k
24
votes
1
answer
2k
views
Sums of injective modules, products of projective modules?
Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension?
Analogously, under what assumptions on R does a ...
- 15.6k
22
votes
1
answer
531
views
Interpretations for higher Tor functors
Let's work in the category $R$-${\sf mod}$, for concreteness. I know that one can see the modules ${\rm Ext}^n_R(M,N)$ as modules of equivalence classes of $n$-extensions of $M$ by $N$ (Yoneda ...
- 1,163
21
votes
2
answers
4k
views
intuition for hochschild homology
According to this post Intuition for group homology, I wonder what is the intuition for Hochschild homology.
The Hochschild homology is defined as the homology of this complex
chain.
Given a ...
- 416
20
votes
3
answers
2k
views
Can a module be an extension in two really different ways?
(Edit: I've realized that there was an error in my reasoning when I was convincing myself that these two formulations are equivalent. Hailong has given a beautiful affirmative answer to my first ...
19
votes
3
answers
13k
views
Künneth formula for cohomology
$\DeclareMathOperator\Hom{Hom}$Is there an algebraic Künneth formula for cohomology?
More precisely assume $A_{*}, B_{*}$ are chain complexes of free $R$-modules ($R$ is a $PID$) and $M, N$ are $R$-...
- 1,357
17
votes
3
answers
2k
views
Characterising categories of vector spaces
Consider the category $FdVect_k$ of finite dimensional $k$-vector spaces, for some given field. It is abelian, semisimple, in that each object is a finite sum of simple objects (of which there is only ...
- 35.4k
15
votes
2
answers
2k
views
When is bar-cobar duality an equivalence?
Let $A$ be an augmented differential graded algebra over a field $k$. I will write $BA$ for its bar construction (whose homology is $Tor^A(k, k)$). This is a co-augmented differential graded ...
- 4,133
15
votes
1
answer
516
views
Non isomorphic two term complexes with isomorphic kernel, image and cokernel
Let $R$ be a ring. Can we have two $R$-module maps $A, B: R^n \to R^m$ such that $\mathrm{Ker}(A) \cong \mathrm{Ker}(B)$, $\mathrm{Im}(A) \cong \mathrm{Im}(B)$ and $\mathrm{CoKer}(A) \cong \mathrm{...
- 156k
15
votes
1
answer
1k
views
Countable Hom/Ext implies finitely generated
Today I learned this interesting fact from Jerry Kaminker: If $A$ is an abelian group such that $\mathrm{Hom}(A,\mathbb{Z})$ and $\mathrm{Ext}(A,\mathbb{Z})$ are both countably generated, then in fact ...
- 56.6k
14
votes
2
answers
770
views
Global homological dimension of group rings
In all that follows, let $k$ be a field and $G$ be a finite group.
It is well-known that the order of $G$ is invertible in $k$ iff the group ring $k[G]$ is semisimple, which is equivalent, inter alia, ...
- 32.4k
14
votes
1
answer
699
views
Who introduced the abstract definition of a DGA?
Differential graded algebras, or DGAs, are a basic object of study in many areas of modern mathematics. While they were present (implicitly at least) since the start of modern differential geometry, I ...
- 219
13
votes
4
answers
3k
views
What is a "block" in an abelian category?
In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which ...
- 52.9k
13
votes
1
answer
5k
views
What are tame and wild hereditary algebras?
What are tame and wild hereditary algebras?
Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. ...
- 2,992
13
votes
1
answer
669
views
Is a "smooth" finite-dimensional algebra separable modulo its radical?
Let $k$ be a field, and let us write the "unadorned" tensor $\otimes$ in place of $\otimes_k$. For a unital finite-dimensional $k$-algebra $A$, let $A^e = A \otimes A^{op}$ denote the enveloping ...
- 5,407
12
votes
1
answer
922
views
Does this algebra have finite global dimension ? (Human vs computer)
Usually computers can calculate the global dimension of a finite dimensional quiver algebra much faster than humans. But in this case a high end computer (calculating for 3 weeks) was not able to ...
- 26.5k
12
votes
0
answers
533
views
Does there exist a Noetherian ring of finite injective dimension but higher Krull dimension?
Definition: a (not necessarily commutative) left and right Noetherian ring $R$ is said to be Auslander-Gorenstein if
(i) $R$ has finite left and right injective dimension (in which case it turns out ...
- 221
11
votes
1
answer
1k
views
Motivation behind the definition of hochschild cohomology
For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as ...
- 595
11
votes
0
answers
841
views
Is "Determinant" a Hochschild coboundary?
Assume that $n>2$.
Is there an associative unital algebra structure on $\mathbb{C}^{n}$ such that $D$, the determinant as a $n-\text{form} $ on $\mathbb{C}^{n}$,
would be a Hochschild ...
- 356
10
votes
3
answers
1k
views
Dual of a bimodule
For a noncommutative ring $R$, and an $R$-$R$-bimodule $B$, is there a "correct/natural" notion of a dual bimodule? I am interested, really, when $B$ is projective as a left $R$-module.
Note: ...
10
votes
1
answer
1k
views
Is there a way to define a prime ideal object via diagrams in the category of rings?
I like to think in terms of commutative diagrams rather than referring to elements. So to me a group is really a group object, i.e. an object with some maps satisfying certain commutative diagrams. ...
- 30.3k
10
votes
1
answer
1k
views
Equivalent descriptions of Coherent Groups
Attending a series of lectures, I have recently been exposed to the notion of Coherent groups, defined as following:
Def: A group $G$ is called Coherent if every finitely generated subgroup $H$ of $G$...
- 493
10
votes
1
answer
307
views
Rings where all indecomposable projective modules are finitely generated
Let $X$ be the class of (unital, associative and not necessarily commutative) rings $R$ where every indecomposable projective $R$-module is finitely generated.
Question 1: Is there a nice equivalent ...
- 26.5k
9
votes
3
answers
1k
views
Poincaré duality for (co)homology of Lie algebras?
Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements.
In Algebra, Geometry, and Software Systems by Joswig & Takayama on p.200, it ...
- 1,589
9
votes
2
answers
2k
views
Global dimensions of non-commutative rings
This is related to my previous question: When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau? I now would like to know the global dimension of the ring $R=\mathbb{C}\langle x_1,\dots,x_n\rangle/...
- 1,663
9
votes
1
answer
236
views
Formal smoothness of path algebras and connections
Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if
$$
\Omega^1_kA = \operatorname{Ker}(\...
- 985
9
votes
2
answers
622
views
Is there any transitivity for separable algebras?
If $R$ is a commutative ring (with $1$), then an $R$-algebra $A$ is said to be separable if $A$ is projective as an $A$-$A$-bimodule. (The notion of an "$A$-$A$-bimodule" includes the requirement that ...
- 33.8k
9
votes
2
answers
1k
views
Extensions of an infinite product of copies of Z by Z
The question is simple:
Let $P$ be an infinite direct product of copies of $\mathbb Z$. Do there exist any nontrivial extensions
$$0 \to \mathbb Z \to E \to P \to 0$$
in the category of commutative ...
- 4,015
9
votes
1
answer
1k
views
When does the homological dimension of a tensor product equal the sum of dimensions?
The notion of dimension I prefer most is right global dimension, but the question can also be asked for other notions (e.g. weak dimension, injective dimension, Krull dimension). Letting $d$ be ...
- 30.3k
9
votes
0
answers
365
views
A characterisation of symmetric algebras using Hochschild (co)homology
A finite dimensional (connected if needed) $K$-algebra $A$ over a field $K$ is called symmetric when $A \cong Hom_K(A,K)$ as $A$-bimodules. Symmetric algebras are Frobenius algebras and include for ...
- 26.5k
8
votes
2
answers
1k
views
Algebra Counterexample Request: Linear Quotients
A result of Herzog, Hibi, and Zheng in "Monomial ideals whose powers have a linear resolution" states that:
Theorem: Let $I\subseteq\Bbbk[x_1,\ldots,x_n]$ be a monomial ideal generated in degree 2. ...
- 1,324
8
votes
2
answers
2k
views
Algebraic Morse theory
In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
- 1,589
8
votes
1
answer
346
views
If a faithfully flat extension of dg/A_$\infty$-algebra is formal, is the original algebra formal (over positive characteristic)?
Proposition 6.2 of Formality of DG algebras (after Kaledin) by Lunts reads (with a few additions to clarify notation):
Let $k$ be a field of characteristic 0. Let $A$ be an $A_\infty$ algebra ...
- 44.7k
8
votes
0
answers
313
views
Fractal homological algebra
The usual definition of a chain complex requires for the indices to be integer numbers. However, taking inspiration from the theory of Hausdorff dimension, one can think of 'fractal' chain complexes (...
- 327
8
votes
0
answers
4k
views
Kunneth spectral sequence
In Rotman's Homological Algebra, 1st edition, there is written:
Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also hold ...
- 1,589
8
votes
0
answers
234
views
Depth for non-commutative rings
The depth of a ring or module is one of the most basic invariants in commutative ring theory.
Q1: Is there also a powerful notion of depth for non-commutative rings ?
By a search in mathscinet, I ...
- 16.2k
7
votes
2
answers
1k
views
A question on curved algebras, papers by Positselski and E. Segal
I am trying to understand something about curved dg algebras as studied by Positselski, E. Segal. These come up in mirror symmetry and when one wants to study Kozsul duality for algebras that are more ...
- 3,758
7
votes
1
answer
592
views
Example of a ring where every module of finite projective dimension is free?
I'm interested in seeing an example of a ring which is not self-injective where every module admitting a finite projective resolution is free, or at least projective.
Note that self-injectivity says ...
- 63.9k
7
votes
1
answer
624
views
$\mathbb{Z}$-graded algebras and tensor products
Let $A = \bigoplus_{k \in \mathbb{Z}} A_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is strongly graded:
$$
A_kA_l = A_{k+l}.
$$
...
- 145
7
votes
1
answer
370
views
Gorenstein symmetric conjecture for arbitrary rings
The Gorenstein symmetric conjecture states that for Artin algebras $A$ one has the the regular module has finite injective dimension as a right module if and only if it has finite injective dimension ...
- 26.5k
7
votes
2
answers
747
views
Ostensibly different products on Ext-groups
The following is presumably not the greatest generality in which this question makes sense.
Given a ring $k$, graded-commutative if it helps, and a Hopf-algebra $A$ over $k$, there is a Yoneda ...
- 163
7
votes
1
answer
408
views
Homological dimensions of rings of smooth functions
What is the global dimension of the algebra $C^\infty\mathbb R$ of smooth functions $\mathbb R\to\mathbb R$? What is the global dimension of the algebra $(C^\infty\mathbb R)_0$ of germs of smooth ...
- 700
7
votes
1
answer
246
views
Rings in which every module has an injective image
Consider the class of rings $R$ with identity such that any left $R$-module has a non-zero injective homomorphic image. Any such ring is clearly a left V-ring. Is it true that any such ring must be ...
- 157
7
votes
1
answer
398
views
Why can't one modify Kaplansky's proof to conclude that every projective module is a direct sum of its finitely generated projetive submodules?
Due to the examples given in the answer to this question, I know that the conclusion is of course incorrect. But by reading Kaplansky's proof of theorem 1 in this paper and replacing every occurrence ...
- 960
7
votes
1
answer
909
views
Algebra structure $Tor(A,A)$
This is a question i asked on math.stackexchange but i didn't get any answer.
Let $A$ be algebra over commutative ring $k$ and $P_{\bullet}=(P_i,d_i)\rightarrow A$, $k$ projective resolution. Then we ...
- 173
7
votes
2
answers
484
views
Using Dunwoody's results on cohomological dimension to learn about a von Neumann regular group ring
Just recently I've stumbled across Warren Dicks' book Groups, trees and projective modules (1980) and I was pretty stunned. I know nothing of group cohomology, but I gather the "tree" ...
- 1,507
7
votes
1
answer
288
views
Skew differential graded algebra
A sigma, or skew, derivation is a natural generalisation of the
notion of derivation depending on an algebra automorphism $\sigma$ which
when equal to $id = \sigma$ reduces to the usual notion of a
...
- 1,144
7
votes
1
answer
555
views
Quadratic algebras, quadratic operads, quadratic categories and quantum cohomology
Motivated by the quantisation of the symmetric laws in physics, the category of quadratic algebras has been endowed with two tensor products by Manin in his Montreal lectures notes. These products ...
- 3,652