All Questions
Tagged with algebraic-k-theory homological-algebra
16 questions
12
votes
1
answer
2k
views
Why does K-theory need schemes to be Noetherian?
The definition of K-theory of a scheme $X$ is defined as
$G_i(X):=K_i(\mathrm{Coh}(X))$ or $K_i(X):=K_i(\mathrm{Vec}(X))$.
But usually the schemes are required to be (at least locally) Noetherian, and ...
- 449
11
votes
1
answer
578
views
Are projective modules over a certain localised Laurent polynomial ring free?
Let $R=\mathbb{Z}[t^{\pm 1}]$ be the ring of Laurent polynomials, and let $S \subset R$ be the multiplicative subset generated by the polynomial $t-1$. I am interested in the ring $S^{-1}R=\mathbb{Z}[...
- 1,033
10
votes
1
answer
1k
views
Quasi-isomorphisms in exact categories
I am trying to understand quasi-isomorphisms in an exact category as defined via the mapping cylinder. I would like to know whether these form a category of weak equivalences in the sense of ...
- 770
9
votes
6
answers
4k
views
Differences between reflexives and projectives modules
Let R be a normal noetherian domain.
What is the difference between a finitely generated reflexive module and a finitely generated projective module?
Can anybody recommend any references or make ...
7
votes
2
answers
2k
views
Local property of split exact sequence
In the module category of a ring $A$, is a short exact sequence split if and only if the localization of this sequence is split for every prime ideal?
Thanks!
- 496
7
votes
0
answers
374
views
Split exact categories arising naturally
If you're interested in the $K$-theory of rings, a useful feature of the exact category of finitely generated projective (or free) modules is that it is split exact, i.e. every short exact sequence is ...
- 770
5
votes
1
answer
571
views
Perfect chain complexes
In Thomason-Trobaugh in Remark 2.4.4 it is written: "On a general scheme, the perfect complexes are locally finitely presented objects in the "homotopy stack" of derived categories."
I was wondering ...
- 595
4
votes
1
answer
259
views
Induced map in K-theory by a "trivial" bimodule
Let $R$ be a ring (not necessary commutative) and let $P_{\bullet}$ be a perfect $R$-bimodule (chain complex). I will denote the category of perfect right $R$-chain complexes by $\textbf{Perf}(R)$. ...
- 43
4
votes
0
answers
135
views
$K$-group of category of bounded chain complexes of Projective modules with finite length homologies
For a Commutative Noetherian local ring $(R, \mathfrak m)$, let $K_0^{\mathfrak m}(R)$ denote the abelian Group generated by isomorphism classes of bounded chain complexes of finitely generated free ...
- 1,209
3
votes
2
answers
353
views
Morphisms between $K_0$
I suppose this is a question with a well known answer. Suppose $A$ and $B$ are two algebras over some field and there is a map
$$
f: \operatorname{K_0}(A) \to \operatorname{K_0}(B)
$$
is it ...
- 1,545
3
votes
1
answer
463
views
Endomorphism Ring of Indecomposable MCM Modules
Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...
- 161
3
votes
1
answer
185
views
Spherical objects and K-theory
My question goes as follows: given a ring $R$ (with $1\neq 0$). Define $\mathbf{Perf}_{R}$ the the category of Prefect complexes over $R$. I want to prove that the Waldhausen $K$-theory of the ...
- 511
3
votes
0
answers
107
views
rational homology of SO(2,1) over number fields
Let $\mathrm{SO}(2,1)$ be the special orthogonal group defined by the quadratic from $q(x,y,z)=x^2+y^2-z^2$.
This is a connected non-simpy connected algebraic group.
Now, let $F$ be a number field, ...
2
votes
1
answer
120
views
Does the inclusion functor induce an injection in this case?
Notations :
$R$ is a commutative ring with unity. $P(R)$ is the category of finitely generated projective $R-$ modules, $Ch^{b}(P(R))$ is the the category of bounded chain complexes on $P(R)$ and $C^q(...
user139827
1
vote
1
answer
177
views
When the restriction of derived equivalence to a summand is a derived equivalence as well
I have a question about the equivalence of derived categories. Let $\mathcal{A} = \mathcal{A}'\oplus \mathcal{A}''$ and $\mathcal{B} = \mathcal{B}' \oplus \mathcal{B}''$ are direct sum of abelian ...
- 55
1
vote
0
answers
179
views
Injective envelope in the category of left exact functors
Let $\mathcal{A}$ be an Abelian category. $\mathcal{L}$ is the category of
absolutely pure objects of $\mathcal{A}$ and $\mathcal{L}(\mathcal{A})$ is the category of the exact left functors of $\...
- 31