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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).

The global dimension of a ring $A$ (denoted gl dim $A$) is defined to be the supremum of the set of projective dimensions of all $A$-modules.

When the ring $A$ is noncommutative, one initially has to consider two versions of this notion, the right global dimension that arises from consideration of the right $A$-modules and the left global dimension that arises from consideration of the left $A$-modules. For an arbitrary ring $A $ the right and left global dimensions may differ. However, if $A$ is a Noetherian ring, both of these dimensions turn out to be equal to the weak global dimension, whose definition is left-right symmetric.

Reference: https://en.wikipedia.org/wiki/Global_dimension

To be used with , and .

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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 example if $R$ is a local ring?
Zac
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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 is nice! I have the following question: 1. Is…
Jian
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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_{\mathfrak m\in\operatorname{SpecMax}A}…
karparvar
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What is the global dimension of the ring $\prod_I k$, where $k$ is a field and $I$ is an infinite index set?

Let $k$ be a field. If $I$ is a finite index set, then the global dimension of the ring $\prod_I k$ is zero because $\prod_I k$ is a semisimple ring. Now, the question arises: When $I$ is an infinite index set, what is the global dimension of the…
Liang Chen
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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 asking the question is, when is $R$ regular?
BillScroggs
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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 every left $R$-module has finite flat dimension.…
Aimin Xu
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on projectivity of $\text{Hom}_R(A,R)$ as $A$-module where $A$ is a matrix ring over $R$

Let $R$ be a regular local ring and $n>0$ be an integer. Consider the matrix ring $A:=M_n(R)$. I am wondering if $\text{Hom}_R(A,R)$ is a projective $A$-module? By Lemma 2.15 it is projective as a left $A$-module if and only if it is projective as a…
uno
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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-module. My question is about projectivity and…
Mourad Khattari
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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 equal to $n$. But I don't know how to get it less…
Andylang
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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 $A$, while the right side term is the homological…
Mamadness
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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 of is whether…
user43198
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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
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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 two notions?
Math137
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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 finitely generated $A$-algebra. (All rings are not…
user237522
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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 $k$-algebra $A$, which is $k$-flat and there is an ideal…
AB_IM
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