All Questions
40 questions
2
votes
2
answers
416
views
Tensor product over $\mathbb{Z}$ and p-adic integer ring $\mathbb{Z}_p$
Thanks for your reading. Suppose we have two $\mathbb{Z}_p$-modules $A,B$. Do we always have $A \otimes_{\mathbb{Z}} B \simeq A \otimes_{\mathbb{Z}_p} B$, as abelian groups or $\mathbb{Z}_p$-modules? ...
- 319
2
votes
0
answers
122
views
Quasi-isomorphisms of P-algebras
In the paper "Homotopy algebras are homotopy algebras" from Markl a notion of strong homotopy morphism between strong homotopy P-algebras is defined. The author restricts to the case where $...
- 215
5
votes
1
answer
191
views
Are module finite algebras over semiperfect rings again semiperfect?
Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is ...
- 412
5
votes
0
answers
285
views
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 ...
- 889
5
votes
1
answer
208
views
Equivalences of categories of complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R, S$ be two commutative $k$-algebras.
Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...
- 889
1
vote
1
answer
182
views
A question about surjective maps between quadratic algebras
Let $V$ be a finite-dimensional vector space and
$$
U \subseteq W \subseteq V \otimes V
$$
be a proper inclusion of vector subspaces. Then take the tensor algebra
$$
T(V) = \bigoplus_{i=1}^{\infty} V^{...
- 643
3
votes
1
answer
173
views
$\Omega$ for noetherian semiperfect rings
Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-...
- 26.5k
7
votes
1
answer
408
views
Homological dimensions of rings of smooth functions
What is the global dimension of the algebra $C^\infty\mathbb R$ of smooth functions $\mathbb R\to\mathbb R$? What is the global dimension of the algebra $(C^\infty\mathbb R)_0$ of germs of smooth ...
- 700
5
votes
1
answer
317
views
Localization of a ring and the Hom functor
Let $R=\mathbb{Z}[x,x^{-1}]$ be the ring of Laurent polynomials in $x$, $\mathfrak{p}=(1-x)$ be an ideal in $R$ and $R_\mathfrak{p}$ be the localization. I want to know what $\text{Hom}_R(R_\mathfrak{...
user340793
4
votes
1
answer
564
views
Nondegenerate pairings versus perfect pairings for finitely generated projective modules
Let $R$ be a (not necessarily commutative) ring, $M$ a left $R$-module, and $N$ a right $R$-module. We say that a pairing
$$
\langle -,-\rangle:M \otimes_R N \to R
$$
is non-degenerate if, for all $n \...
- 145
2
votes
2
answers
417
views
Transition maps in trivial direct limit
If $\{X_i, i\in I\}$ is a directed system of abelian groups such that we have
$$\varinjlim_{i\in I}X_i = 0$$
is it true that for every $i$ and large enough $j\ge i$ the transition map $f_{i,j} : X_i\...
user290895
3
votes
1
answer
153
views
Artinian Tor modules (Reference request)
I am looking for a reference for the following basic fact:
Let $R$ be a noetherian ring, let $M$ be an artinian $R$-module, let $N$ be a finitely generated $R$-module, and let $i\in\mathbb{N}$. ...
- 6,700
6
votes
1
answer
395
views
Tor functor and invertible elements
Let $A$ be a commutative ring, $a \subset A$ be an ideal. For $A$-module $M$ let $S \subset A$ be the set of elements, which are invertible in $M$, so $M$ is actually a $S^{-1}A$-module. It is not ...
- 332
5
votes
1
answer
368
views
Topological Hochschild homology of Azumaya algebra
Let $R$ be a commutative ring, let $A$ be an Azumaya algebra over $R$, does its topological Hochschild homology coincide with that of $R$? For example, let $\mathbb{H}$ be the quaternion algebra over ...
user39380
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
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
4
votes
0
answers
131
views
Injective resolution of the ring of entire functions
Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis.
I would think that the injective dimension ...
- 26.5k
2
votes
0
answers
139
views
$\omega$-categorical algebra
Let us consider a 1-category $C$. For any commutative and unital ring $k$ the the free $k$-module generated by the morphisms of $C$ can be equipped with an algebra structure by setting $fg$ to be ...
- 517
3
votes
1
answer
243
views
Few commutative algebra questions
A few commutative algebra questions for which I have no reference
For $P$ = "catenarian", "coherent", " Jacobson":
1- is an arbitrary product of rings satisfying $P$, a ring satisfying $P$?
2- if a ...
user95222
3
votes
1
answer
394
views
Localization of the injective hull
Let $R$ be a Noetherian commutative ring. Let $E(M)$ denote the injective hull of $M$. I want to show that $E(M)_\mathfrak{p}\simeq E(M_\mathfrak{p})$ for any $\mathfrak{p}\in \text{Spec}(R)$.
To do ...
- 31
4
votes
0
answers
74
views
self-cogenerator rings
Let $\mathbb{U}$ be a non-empty set (class) of objects of a
category $C$. An object $B$ in $C$ is said to be cogenerated by
$\mathbb{U}$ or $\mathbb{U}$-cogenerated if, for every pair of
distinct ...
- 41
3
votes
1
answer
457
views
Milnor patching for general modules
The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings
$$
\begin{array}{}
R & \xrightarrow{f_2} & R_2 \\
\downarrow{f_1} & &...
- 197
2
votes
0
answers
868
views
depth of ideal in polynomial ring
Let $R$ be a Noetherian local ring. Then the "depth" of an ideal $I$ measures the maximal length of regular sequence inside $I$. And $depth (R/I)$ measures the maximal length of regular sequence in ...
- 183
6
votes
1
answer
2k
views
A Hom-Tensor identity - $\text{Hom}_{R}(P,B)\otimes _SC \cong \text{Hom}_{R}(P,B \otimes_S C) $
let $R,S$ be associative algebras over $\mathbb{C}$. Let $\mathcal{C} \subseteq$ $R$-Mod be a full abelain subcategory of $R$-Mod which is the category of $R$-modules. Let $B$ and $C$ be, a $(R,S)$-...
- 61
-2
votes
1
answer
572
views
Tensor products of simple modules over algebras [closed]
Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules.
We can regard $M\otimes_{\mathbb C}N$ as $A\otimes_{\mathbb C} B$-modules in natural ...
7
votes
0
answers
228
views
Terminology for vanishing of Hochschild homology with symmetric coefficients?
In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then
it should hopefully be understood by most readers as saying $H^n(A,M)=0$...
- 25.8k
3
votes
1
answer
216
views
Global dimension of graded Lie algebra
The rational global dimension of a graded algebra $A=(A_k)_{k\geq 0}$, with $A_0=\mathbb Q$, denoted here ${\rm gl}\dim A$ is defined to be the greatest integer $k$ (or $\infty$) such that ${\rm Ext}^...
- 189
4
votes
1
answer
381
views
Classical deformation of algebras
Given a complex manifold (or a smooth scheme) $X$, the classical (infinitesimal) deformation theory is parametrized by the first cohomology with coefficients in the tangent sheaf $H^1 (X, T_X)$.
...
- 1,211
1
vote
0
answers
199
views
Criterion for global dimension of subring
All rings are assumed to be associative and unital.
If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...
- 5,407
1
vote
0
answers
246
views
Global dimension of a subalgebra with all units
(All rings here are always assumed to be unital and associative).
Setup
Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying:
If $u$ is a unit in $...
- 5,407
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
5
votes
1
answer
415
views
Inverse limit of Gorenstein local rings is again Gorenstein?
If we have the system of surjective ring homomorphisms
$f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$
for an arbitrary $i \geq 0$ such that all $R_i$ are Gorenstein local ring. Let us put
$R^{\...
- 51
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
5
votes
1
answer
197
views
Localizations of hereditary rings
It is known that if a commutative Noetherian ring $R$ is hereditary then for any maximal ideal $M$ the localization $R_M$ is also hereditary. Is the Noetherian assumption necessary?
- 447
3
votes
0
answers
450
views
Ext groups of affine scheme
Let $A$ be a commutative ring. $\textbf{Spec}(A)$ is the the
spectrum of $A$. $M$ and $N$ are $A$-modules. $\tilde{M}$ and
$\tilde{N}$ are sheaves associated on $\textbf{Spec}(A)$
respectively.
I ...
- 491
3
votes
0
answers
271
views
About free resolutions of graded commutative algebras
Hi, I'm having troubles in adapting certain algebraic constructions to graded cases.
We know that if $A$ is a commutative ring and $a_1,...,a_k$ are elements on $A$, there is a construction of the ...
-1
votes
1
answer
803
views
Question on an exercise on homological algebra?
Suppose $R$ has finite global dimension $n$, $N$ is a finitely generated module, $F$ is a free module, and $\operatorname{Ext}^n(N, F) \neq 0$. Then $\operatorname{Ext}^n(N, R)$ is also non-trivial.
...
- 77
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
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
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