Questions tagged [homological-dimension]

For questions having to do with projective and injective dimensions of modules, global dimension of rings and algebras, and related concepts.

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11
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2answers
252 views

Obstruction to splitting an object in derived category into a sum of two-term complexes

Let $\mathcal{A}$ be an abelian category, and $D$ its bounded derived category. An object $M \in D$ may be described as a list of cohomology objects $H^i = H^i(M)$ together with some complicated ...
2
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1answer
102 views

Global dimension of a certain $K[x,y]$-algebra

Question. Let $K$ be a field (assume $K=\mathbb{C}$ if this simplifies the problem). What is the right global dimension of the $K[x,y]$-algebra: $$ A=\left[\begin{array}{cc} K[x,y] & xK[x,y] \\ K[...
3
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1answer
127 views

Projective dimension of graded modules

Short version: Why is the projective dimension of a graded module the same as the projective dimension of its underlying ungraded module? Longer version: Let $G$ be a commutative group, let $R$ ...
30
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1answer
810 views

If $\text{dim}(X \times X) = 2\text{dim}(X)$, does $\text{dim}(X^n) = n\text{dim}(X)$?

I have been learning some (topological) dimension theory and have gotten through most of the basic material, at this point, and am about to start looking at papers. In particular, I want to get ...
29
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2answers
4k views

What is the dimension of the mathematical universe?

Forcing construction in set theory leads to a new understanding of the mathematical (multi)universe by providing a machinery through which one can construct new models of the universe from the ...
1
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1answer
231 views

question about infinite global dimension

I ask a question about $\prod k$ in Mathematics about several days.https://math.stackexchange.com/q/2766054/453628. And I have the following question: 1.What is the global dimension about $\prod k$? ...
3
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1answer
173 views

Is there a notion of injective, projective, flat, dimension for a differential graded algebra?

Given a differential graded algebra $(A_\bullet,d)$, is there a well-defined notion of a K-injective, K-projective, K-flat dimension of a differential graded module, or even of the category of ...
6
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2answers
217 views

Generalized Stallings theorem

Well known theorem due to Stallings (finished by Swan) characterises free groups as those with $cd_{\mathbb Z} \leq 1$. We can treat it as a model case and try to extrapolate somehow to other ...
8
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3answers
810 views

Homological vs. cohomological dimension of a group/space

I have several related questions regarding homological vs. cohomological dimension of a space/group (this is not a duplicate of this). The standard definition of the cohomological dimension $cd(X)$ ...
4
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1answer
649 views

Different ways of having infinite global dimension

Is there any ring $R$ of infinite global dimension such that any $R$-module is a retract (i.e. direct summand) of some $\oplus_{i\in I}M_i$ where each $M_i$ has finite projective dimension? I ask ...
5
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0answers
111 views

Projective dimension of ring over its center

If $A$ is a ring and $Z(A)$ is its center then what is a sufficient condition for the projective dimension of $A$ over $Z(A)$ (ie: $pd_{Z(A)}(A)$) to be finite? (Assuming that $A\neq Z(A)$).
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0answers
105 views

Projective dimension of a sub-ideal

Let $\mathbf{k}$ be a field, and let $S=\mathbf{k}[x_1,x_2,\ldots,x_n]$. Let $I\subset J$ be finitely generated monomial ideals in $S$. Is it true that the projective dimension of $I$ is either ...
2
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1answer
392 views

Interpretation of Hochschild Homology groups

In all the literature I've come across there are many concrete interpretations of the first few Hochschild Cohomology groups. For example $HH^1(A,M)\cong Derivation/Inner Derivations$ etc.... In ...
0
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1answer
663 views

question on an exercise on homological algebra?

Suppose R has finite global dimension n, N is a f.g. module, F is a free module and Ext^n(N,F) is not equal to zero, then Ext^n(N, R) is not trival either. Note, HERE R is not Noetherian necessarily.
3
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2answers
363 views

Can a zerodivisor reduce both the depth and the dimension?

In this question $R$ is a commutative noetherian local ring with unity. One can construct examples of rings $R$ and zerodivisors $z$ such that $\dim R/(z)=\dim R-1$, e.g., $S\colon=k[a,b,c],\ \...
8
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0answers
370 views

Reference/ elementary proof of a result about projective dimension in group rings

Hello- I've had to use a result that sounds like it should be well-known, but I couldn't find any references, and my proof is rather unsatisfactory. I was hoping someone here could help! The problem ...
5
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0answers
329 views

Homological dimension of completed path algebras.

Let A = c[Q]/I be a finite dimensional quotient of a path algebra over a quiver Q, with I being the ideal of relations. Is it true that the I-adic completion of A has finite homological dimension?
8
votes
2answers
1k 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/...
0
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0answers
329 views

finite homological dimension

Hi, I found the following in the proof of a theorem: $ Z \subset Y \times M$ where $M$ is a smooth projective variety over $\mathbb{C}$, $Y$ is a scheme and $Z$ is a subscheme of the product, flat ...
1
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0answers
105 views

The full linear ring is of finite projective dimension over the enevelopping algebra?

It is known that if $R=End_k(V)$, with $V$ a finite dimension $k$-vector space then $R$ is projective as $R^e$-module, thus of projective dimension $pd_{R^e}(R)=0$. If $V$ is of infinite dimension ...
1
vote
1answer
563 views

Recollements and global dimension

Let $A, B, C$ be algebras. Suppose that $D^b(A)$ (the bounded derived category of $A$) admits a recollement relative to $D^b(B)$ and $D^b(C)$. Then, by a result of Alfred Wiedemann's paper "On ...
5
votes
1answer
792 views

An example where finitistic dimension does not equal right global dimension?

The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup ...
9
votes
1answer
824 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 ...
1
vote
2answers
232 views

Is there a relationship between the right global dimensions of R and R[1/v]?

A few days ago I asked a similar question about Krull dimension and got fantastic answers. Unfortunately, for the application I have in mind (a question on ring spectra), Krull dimension doesn't ...
45
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3answers
4k views

What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?

On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis ...
43
votes
7answers
5k views

Does $SL_3(R)$ embed in $SL_2(R)$?

Is there any non-trivial ring such that $SL_{3}(R)$ is isomorphic to a subgroup of $SL_{2}(R)$? $SL_{3}(\mathbb{Z})$ is not an amalgam, and has the wrong number of order $2$ elements to be a subgroup ...
5
votes
3answers
2k views

Do the homological dimension and cohomological dimension for a group agree?

Or equivalently, if $G$ is a group, do the projective and injective dimension of $Z$ (viewed as a $ZG$-module) agree? Thanks!
10
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3answers
1k views

Projective dimension of zero module

Is there any consensus on what the projective dimension of the zero module should be? Here are three statements one commonly encounters in textbooks, sometimes with or without the condition $M\neq 0$: ...
5
votes
1answer
342 views

Behavior of the projective dimension of modules in a continuous chain of extensions

Let $R$ be an arbitrary ring. Let $D$ be the class of $R$-modules of projective dimension less than or equal to a natural number $n$. If $L$ is the direct union of a continuous chain of submodules ${...
6
votes
2answers
1k views

Projective & injective dimensions

$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. I'm not quite sure about the relation between finiteness of projective and injective dimensions of $M$. Does the finiteness (or infiniteness)...
3
votes
2answers
872 views

Depth and dimension

$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. Then is it true that $\mbox{depth }M\le\mbox{depth }A$ just like $\mbox{dim }M\le\mbox{dim }A$? I don't see any relation between an $M$-...
6
votes
1answer
209 views

Injective dimension of cyclic modules

Let $R$ be a non-Noetherian ring. Is its left global dimension ${\rm{lD}}(R)$ equal to $\sup \{ {\rm{id}}(M) \mid M \text{ is a cyclic $R$-module} \}$? Here $\rm{{id}}(M)$ denotes the injective ...