# 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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### 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 ...
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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 $A$,...
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### Global dimension of $\prod k$

Let $k$ be a field. Consider 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 ...
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### 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 ...
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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 ...
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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 ...
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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 :)
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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 ...
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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 ...
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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 ...
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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 ...
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