2
votes
Accepted
Two examples for projective resolutions for finite dimensional algebras
There is no example satisfying (a).
Suppose $(P_\bullet,d_\bullet)$ is a minimal projective resolution of $M$, so that $\Omega^i(M)=\operatorname{im}d_i=\ker d_{i-1}=N\oplus N'$.
Then the composition ...
- 26.6k
2
votes
Accepted
Examples of abelian categories satisfying AB3 (but not AB4) and AB4 (but not AB5)
There are well-known and naturally occurring examples for the dual questions, so you can just take the opposite categories.
Categories of sheaves of abelian groups are $AB3^*$ but typically not $AB4^*$...
- 26.6k
2
votes
Induced filtration on polynomial ring with coefficients in a filtered associative algebra
It depends.
If you really consider $t$ to have degree $0$ and therefore to have $k[t]$ unfiltered (or rather with trivial filtration), then the induced filtration on $A[t]$ should be the first one.
If ...
- 23.6k
1
vote
Freeness of $I/I^2$ implies that of $\dfrac{I}{I^2+xR}$ over $R/I$?
Counterexample: let $k$ be a field, $A=k[[t]], B=k[[t^3, t^7, t^8]], \mathfrak{m}=t^3B+t^7B+t^8B, I= t^6A\subset B, x=t^{11}.$
$(B,\mathfrak{m})$ is a noetherian local ring and $\sqrt{I}=\mathfrak{m}$ ...
- 514
1
vote
What is the connection between the two definitions of homology in 1.1 of Rotman's An Introduction to Homological Algebra?
Don't worry, Rotman just wants to give examples of the use of the term "homology" in mathematics. It should be a motivational introduction, but I am afraid that it is not really a felicitous ...
- 67.2k
1
vote
Tensor product over several modules
For each $p \in P$, the map
\begin{align}
M \times N &\to M \otimes N \otimes P \\
(m,n) & \mapsto m \otimes n \otimes p
\end{align}
is bilinear, and so there is a unique linear map
$$
f_p \...
- 19.5k
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