# 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 ...
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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-...
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### 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 ...
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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 ...
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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 $R$ ...
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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 ...
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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 $\sum_{i=1}^{... • 1,793 3 votes 0 answers 72 views ### Corollary 4.19 from "Homological methods in commutative algebra" I would like to show the following result: for a noetherian local ring$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$,... • 127 5 votes 2 answers 345 views ### Global dimension of$\prod k$Let$k$be a field. Consider an infinite direct product of rings$\prod k$. This is an example of Von-Neumann regular ring (name also absolutely flat), that is, every module is flat. I think this ring ... • 2,450 1 vote 0 answers 338 views ### Dimension of division rings extension Let$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$is finitely ... • 6,431 0 votes 1 answer 690 views ### The global dimension of a$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$-algebra is the maximum of the projective dimension of ... 1 vote 1 answer 134 views ### Weak dimension of Rings I was wondering if there is a characterization of rings of weak dimension one. For example, we know that a ring of weak dimension zero is a von Neumann regular ring. Is there a similar result for ... • 352 0 votes 1 answer 297 views ### Global Dimension 1 [closed] I have a question about rings of global dimension one. I read that these rings are hereditary rings, that is, every right ideal is projective. How can I prove this fact ? Thanks a lot :) • 352 4 votes 1 answer 171 views ### When does an integral group ring have finite global dimension? Let$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$have finite global dimension? Another way of ... • 257 3 votes 1 answer 714 views ### On pullback of global sections of invertible sheaves Let$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})$? The fact I am not totally sure ... • 445 3 votes 1 answer 483 views ### Finite dimensional algebras with finite global dimension. Let$A$be a finite dimensional$k$-algebra,$k$is a field, with a finite global dimension. I wonder if that implies$A$is tame or finite type? or more generally is there a relation between these ... • 1,839 2 votes 0 answers 166 views ### Category of Morphisms Between Modules Let$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$-modules. We define the category ... 6 votes 1 answer 423 views ### Global Dimension of a Ring and its Localizations Why is the following true? The global dimension of a noetherian ring$A$is the supremum of the global dimension of its localizations at its maximal ideals: $$\operatorname{gldim}(A)=\sup_{\... • 5,730 3 votes 1 answer 279 views ### Global dimension of an intermediate ring Assume A \subseteq B \subseteq C are noetherian integral domains, where A and C have the same finite global dimension n. Also assume that C is a finitely generated B-algebra and B is a ... • 6,431 2 votes 1 answer 133 views ### Example of a ring of finite global dimension with flat qu0tient I've been thinking about this for quite a while but I cannot seem to find an example of If k is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative k-... • 6,687 3 votes 1 answer 469 views ### global dimension of rings and projective (flat) dimension of modules Let R be ring such that every left R-module has finite projective dimension ( resp. finite injective dimension). Is the left global dimension of R finite? Similarly, Let R be ring such that ... • 147 2 votes 1 answer 386 views ### Global dimension. What is the global dimension of \mathbb{Z}_{(p)} and \mathbb{Z}_{(p)}/t\mathbb{Z}_{(p)}, where \mathbb{Z}_{(p)} is the local ring, p prime and p \mid t? What is the global dimension of \... • 383 3 votes 1 answer 593 views ### How to prove the global dimension of the polynomial ring F[x_1,...,x_n] is n? I am trying to prove that the global dimension of the polynomial ring F[x_1,\dots,x_n], where F is a field , is exactly n. By Koszul complex, I know its global dimension is greater than or ... • 1,623 3 votes 5 answers 634 views ### Global dimension of free algebra. Is there any easy way to see the global dimension of a free algebra$$ A=k\langle x_{1},\dots,x_{n} \rangle$$is 1? • 383 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$...
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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 ...