**0**

votes

**0**answers

48 views

### Injective Dimension of a quotient of the quantum plane

I am wondering what the injective dimension of the following ring is:
$$\frac{A}{(ax, bx)}$$
where $A$ is the so-called quantum plane $k\langle x, y \rangle/(xy + yx)$ with $k$ a field, and $a, b$ are ...

**6**

votes

**3**answers

544 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**

votes

**1**answer

213 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 ...

**3**

votes

**0**answers

77 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 $A\neq Z(A)$).

**1**

vote

**0**answers

72 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**

votes

**1**answer

301 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**

votes

**1**answer

626 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**

votes

**2**answers

345 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],\ ...

**7**

votes

**0**answers

318 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**

votes

**0**answers

280 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

**2**answers

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 ...

**0**

votes

**0**answers

245 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**

vote

**0**answers

102 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 ...

**0**

votes

**1**answer

419 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 ...

**4**

votes

**1**answer

563 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 ...

**8**

votes

**1**answer

664 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

**2**answers

177 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 ...

**38**

votes

**3**answers

3k 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 ...

**41**

votes

**7**answers

4k 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 ...

**4**

votes

**3**answers

1k 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!

**9**

votes

**3**answers

701 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

**1**answer

287 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 ...

**3**

votes

**2**answers

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 ...

**3**

votes

**2**answers

688 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 ...

**6**

votes

**1**answer

197 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 ...