Questions tagged [global-dimension]

For questions about or related to the global dimension of a ring A, which is defined to be the supremum of the set of projective dimensions of all A-modules. To be used with (ring-theory), (homological-algebra) and (dimension-theory-algebra).

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Is the global dimension of a polynomial ring in infinitely many indeterminates over a field infinite?

As is well known, the global dimension of $k[x_1,...,x_n]$, where $k$ is a field, is $n$ . Similarly, the projective dimension of $k$ as a module over $k[x_1,...,x_n]$ is also $n$ . However, in the ...
Liang Chen's user avatar
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1 answer
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Do the finite-dimensional algebras in the Beilinson's Theorem have finite gloabl dimensions?

It's a well-known theorem by Beilinson [1] that states that for each $n$ there exists a finite dimensional-algebra $B_n$ such that there is an equivalence of triangulated categories $D^\text{b}(\text{...
Noto_Ootori's user avatar
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25 views

Tor-dimension of $A/(a)\otimes_k B$, where $A$ and $B$ are Dedekind domains

Let $A$ and $B$ be two Dedekind domains which contain a field $k$ which is algebraically closed in both $A$ and $B$. Let $a$ be a non zero element in $A$. What is the Tor-dimension of $A\otimes_k B$? ...
Stabilo's user avatar
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Flat dimension of a module

We say an $R$-module $M$ has finite flat dimension (finite projective dimension) and write $fd_RM<\infty$ (resp, $pd_RM<\infty$ ), provided there is a finite flat (resp, projective)resolution $...
Saeed Yazdani's user avatar
2 votes
1 answer
128 views

Change of rings of scalars and projecivity?

Let $f:R\to S$ be a ring homomorphism and M be a left S-module. We can consider M as an R-module via $ r.m := f(r)m $. I know that if M is a flat S-module and S is flat as R-module then M is a flat R-...
Mourad Khattari's user avatar
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94 views

Semisimplicity and global dimension

I know this could be a dumb question, but I've been studying for hours and I might be too tired to see why: A ring R is semisimple if and only if its global dimension is zero. We define the global ...
PAB's user avatar
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what is the projective dimension of $ (x,y)\mathbb{C}[x,y]_{(x,y)}$?

For the local ring $R = \mathbb{C}[x,y]_{(x,y)}$ and its maximal ideal $M = (x,y)\mathbb{C}[x,y]_{(x,y)}$. What is the projective dimension $\operatorname{pd}_R(M)$ of M? My thought: I tried to ...
NEMO's user avatar
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2 votes
1 answer
204 views

Socle and minimal number of generators of an ideal in a regular local ring

Let $(R, \mathfrak m, k)$ be a regular local ring of dimension $2$. Note that Hom$_R (k, M)$ is a finite dimensional $k$-vector space for any finitely generated $R$-module $M$ and Hom$_R (k, R/I) \...
user521337's user avatar
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2 answers
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How to calculate "next" point on line which is 50 meters from current point on earth? (autonomous boat)

I'm currently building an autonomous boat for which I define a path to follow. This path consists of multiple way points which are connected by straight lines. The boat doesn't need to be exactly on ...
kramer65's user avatar
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Noetherian rings whose prime ideals have projective dimension bounded above

For a module $M$ over a commutative Noetherian ring $R$, let $pd_R (M)$ denote the projective dimension of $M$ as an $R$-module. Now let $R$ be a commutative Noetherian ring such that $\sup \{ pd_R (Q)...
user521337's user avatar
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1 vote
1 answer
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On a special type of Noetherian regular rings

Let $R$ be a commutative Noetherian ring having the property that for every $R$-module $M$ that has finite projective dimension, every submodule of $M$ also has finite projective dimension. Then $R$ ...
user521337's user avatar
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1 vote
1 answer
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Valuation ring, of infinite global dimension, with principal maximal ideal

Does there exist a Valuation ring $(R, \mathfrak m)$ , with principal maximal ideal, of infinite global dimension ? Corollary 2 of the following paper by Osofsky has an example of Valuation ring of ...
user521337's user avatar
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2 votes
1 answer
539 views

Geometric intuition of the dimension of Grassmannians and flag manfolds [duplicate]

I wish to understand geometrically (not just algebraically) why the dimension of the Grassmanian $G(k,n)$ is $k(n-k)$ and the dimension of a flag manifold $F(k_{1},k_{2},...,k_{n},N)$ is $\sum_{i=1}^{...
Martin Hurtado's user avatar
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Corollary 4.19 from "Homological methods in commutative algebra"

I would like to show the following result: for a noetherian local ring $A$, we have $\mathrm{gl.dim}_A=\mathrm{hd}_A (k)$. Notice that the left side term of the equality is the global dimension of $A$,...
Mamadness's user avatar
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Global dimension of $\prod k$

Let $k$ be a field. Consider an infinite direct product of rings $\prod k$. This is an example of Von-Neumann regular ring (name also absolutely flat), that is, every module is flat. I think this ring ...
Jian's user avatar
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1 vote
0 answers
338 views

Dimension of division rings extension

Let $A \subseteq B$ be two division $k$-algebras, where $k$ is a field of characteristic zero. I am not sure if I wish to further assume that $B$ is affine over $A$, namely, if $B$ is finitely ...
user237522's user avatar
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The global dimension of a $k$-algebra depends only on the simple modules

I read about a theorem a while ago that said something along the lines that the global dimension of a finite dimensional, associative unitary $k$-algebra is the maximum of the projective dimension of ...
Werner Germán Busch's user avatar
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1 answer
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Weak dimension of Rings

I was wondering if there is a characterization of rings of weak dimension one. For example, we know that a ring of weak dimension zero is a von Neumann regular ring. Is there a similar result for ...
Alopiso's user avatar
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Global Dimension 1 [closed]

I have a question about rings of global dimension one. I read that these rings are hereditary rings, that is, every right ideal is projective. How can I prove this fact ? Thanks a lot :)
Alopiso's user avatar
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4 votes
1 answer
171 views

When does an integral group ring have finite global dimension?

Let $G$ be a finite group and $R=\mathbb{Z}[G]$ the integral group ring. If $G$ is such that $R$ is Noetherian (so $G$ polycyclic-by-finite) when does $R$ have finite global dimension? Another way of ...
BillScroggs's user avatar
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1 answer
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On pullback of global sections of invertible sheaves

Let $f:X \to Y$ be a dominant/surjective morphism of projective schemes and $\mathcal{L}$ an invertible sheaf. Is it true that $H^0(\mathcal{L})=H^0(f^*\mathcal{L})$? The fact I am not totally sure ...
user43198's user avatar
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1 answer
483 views

Finite dimensional algebras with finite global dimension.

Let $A$ be a finite dimensional $k$-algebra, $k$ is a field, with a finite global dimension. I wonder if that implies $A$ is tame or finite type? or more generally is there a relation between these ...
Math137's user avatar
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Category of Morphisms Between Modules

Let $A$ be a connected finite dimensional basic $k$-algebra with $k$ an algebraically closed field, and denote by $mod(A)$ the category of finite dimensional left $A$-modules. We define the category ...
Marco Armenta's user avatar
6 votes
1 answer
423 views

Global Dimension of a Ring and its Localizations

Why is the following true? The global dimension of a noetherian ring $A$ is the supremum of the global dimension of its localizations at its maximal ideals: $$\operatorname{gldim}(A)=\sup_{\...
karparvar's user avatar
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3 votes
1 answer
279 views

Global dimension of an intermediate ring

Assume $A \subseteq B \subseteq C$ are noetherian integral domains, where $A$ and $C$ have the same finite global dimension $n$. Also assume that $C$ is a finitely generated $B$-algebra and $B$ is a ...
user237522's user avatar
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2 votes
1 answer
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Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative $k$-...
ABIM's user avatar
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3 votes
1 answer
469 views

global dimension of rings and projective (flat) dimension of modules

Let $R$ be ring such that every left $R$-module has finite projective dimension ( resp. finite injective dimension). Is the left global dimension of $R$ finite? Similarly, Let $R$ be ring such that ...
Aimin  Xu's user avatar
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2 votes
1 answer
386 views

Global dimension.

What is the global dimension of $\mathbb{Z}_{(p)}$ and $\mathbb{Z}_{(p)}/t\mathbb{Z}_{(p)}$, where $\mathbb{Z}_{(p)}$ is the local ring, $p$ prime and $p \mid t$? What is the global dimension of $\...
R2D2's user avatar
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3 votes
1 answer
593 views

How to prove the global dimension of the polynomial ring $F[x_1,...,x_n]$ is $n$?

I am trying to prove that the global dimension of the polynomial ring $F[x_1,\dots,x_n]$, where $F$ is a field , is exactly $n$. By Koszul complex, I know its global dimension is greater than or ...
Andylang's user avatar
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3 votes
5 answers
634 views

Global dimension of free algebra.

Is there any easy way to see the global dimension of a free algebra $$ A=k\langle x_{1},\dots,x_{n} \rangle $$ is 1?
Michel's user avatar
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2 votes
1 answer
496 views

The global dimension of $\mathbb Z/4\mathbb Z$

I read that the global dimension of $\mathbb Z/4\mathbb Z$ is not finite. I think that it's because that $4=2\cdot 2$ and $(2,2)\neq 1$, hence $\mathbb Z/2\mathbb Z\oplus \mathbb Z/2\mathbb Z$ is not $...
Belgi's user avatar
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10 votes
2 answers
236 views

Weak Global Dimension and Global Dimension

Let $R$ be a commutative unit ring (not necessarily Noetherian). Is there an example such that weak global dimension of $R$ is finite but the global dimension of $R$ is infinite? Can we find such an ...
Zac's user avatar
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