Questions tagged [homological-dimension]

For questions having to do with projective and injective dimensions of modules, global dimension of rings and algebras, and related concepts.

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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 ...
user26223's user avatar
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7 votes
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What is the category of coherent sheaves on a logarithmic scheme?

I try to learn basic things on logarithmic geometry, and in particular I don't find much on the category of coherent sheaves on a logarithmic scheme: is it a notion that makes sense or differ from ...
hennlu's user avatar
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5 votes
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Homological characterization of perfect resolutions

Suppose that $R$ is a left Noetherian associative ring with unit and $M$ a finitely generated left $R$-module. It is a standard fact that if the $\mathrm{Ext}$-groups $\mathrm{Ext}^i_R(M,N)$ vanishe ...
Yonatan Harpaz's user avatar
5 votes
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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 $A\neq Z(A)$).
ABIM's user avatar
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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?
Andre 's user avatar
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4 votes
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Reflexive vs. pseudo-coherent abelian groups

Recall that a module M over some ring R is pseudo-coherent if it admits a resolution whose terms are finitely generated projective R-modules. Another notion is to ask whether M is reflexive when ...
Jakob's user avatar
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Generalization of semi-hereditarity

Let $R$ be a ring. A left $R$-module $K$ is called an $N$-th kernel if there are projective left $R$-modules $P_1, \ldots P_N$ and a short exact sequence $$ 0\rightarrow K \rightarrow P_N \rightarrow \...
nikola karabatic's user avatar
1 vote
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On the equivalence of two definitions of cohomological dimension for locally compact topological spaces

$\mathbf{The \ Problem \ is}:$ Let $X$ is a locally compact, separable metric space. Let $G$ be an abelian group. Now I came across two definitions of cohomological dimension of $X.$ One is the usual ...
Rabi Kumar Chakraborty's user avatar
1 vote
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Does this ring homomorphism have finite flat dimension?

Let $k$ be a field and consider the ring homomorphism $f:k[x,xy,xy^2]\rightarrow k$ defined by mapping $x,xy,xy^2$ to zero in $k$. I am trying to show that this ring homomorphism has finite flat ...
Boris's user avatar
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1 vote
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On infinite global dimensions of "slightly non-commutative" rings

Assume $R$ is a commutative Noetherian ring of finite Krull dimension; $R'$ is a not commutative ring that contains $R$ in its center and also finitely generated as an $R$-module. If the (left) global ...
Mikhail Bondarko's user avatar
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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 ...
todea's user avatar
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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 ...
emmy's user avatar
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