Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $A$ be a finite dimensional algebra over a field $k$ and global dimension of $A$ is finite. I want to study $A$ as a bimodule i.e. as $A^e=A \otimes A^{op}$-module. It is easy to see that $pd_{A^e} A=gl(A)=n$. My question is can one construct resolution of $A$ of length $n$ in some natural way? I mean, for example, bar resolution is given by some natural construction, but it is too big, Koszul resolution is another example of natural resolution (for commutative rings), perhaps there are other resolutions that, for example, uses information about structure of indecomposable projectives or some other facts about $A$.

Second question is there a construction of injective resolution of $A$ over $A$?

share|improve this question
    
Lemma 1.5 in [Happel: Hochschild cohomology for finite-dimensional algebras] gives the projectives modules appearing in the minimal projective resolution of $A$ as an $A^e$-module (but not the maps). –  Julian Kuelshammer May 11 '13 at 8:14

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.