# Tag Info

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
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

### 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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