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Results tagged with homological-algebra
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user 11540
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
10
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1
answer
907
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Is the homotopy category of an abelian model category abelian?
A model structure on an abelian category $A$ is called an abelian model structure if the cofibrations are precisely the monomorphisms with cofibrant cokernel, and if the fibrations are precisely the e …
- 30.3k
2
votes
1
answer
252
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Does anyone have a copy of Salce's paper "Cotorsion theories for abelian groups"?
The paper "Cotorsion theories for abelian groups" by L. Salce, was published in 1979 in Symposia Math. 21, pages 1-21. According to Google Scholar, it's been cited 233 times, and I keep seeing citatio …
- 30.3k
4
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0
answers
174
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Seeking an unpublished manuscript by Tetsuro Okuyama
Several papers in representation theory attribute the notion of relatively projective modules to Tetsuro Okuyama's manuscript "A generalization of projective covers of modules over finite group algebr …
- 30.3k
12
votes
3
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Can we say anything about the Krull dimension of a localization?
I'm looking for a theorem of the form
If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.
My attempts to do t …
- 30.3k
1
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2
answers
344
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Is there a relationship between the right global dimensions of R and R[1/v]?
A few days ago I asked a similar question about Krull dimension and got fantastic answers. Unfortunately, for the application I have in mind (a question on ring spectra), Krull dimension doesn't gener …
- 30.3k
9
votes
1
answer
1k
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When does the homological dimension of a tensor product equal the sum of dimensions?
The notion of dimension I prefer most is right global dimension, but the question can also be asked for other notions (e.g. weak dimension, injective dimension, Krull dimension). Letting $d$ be whiche …
- 30.3k
10
votes
1
answer
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Is there a way to define a prime ideal object via diagrams in the category of rings?
I like to think in terms of commutative diagrams rather than referring to elements. So to me a group is really a group object, i.e. an object with some maps satisfying certain commutative diagrams. Yo …
- 30.3k
5
votes
2
answers
990
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An example where finitistic dimension does not equal right global dimension?
The (right) big finitistic dimension of a ring is Findim$(R) =$ sup{proj.dim(M) | $M$ a right $R$-module of finite projective dimension}. The (right) little finitistic dimension findim$(R)$ is the sup …
- 30.3k
52
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3
answers
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What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?
On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis hol …
- 30.3k