# Spirit of homological algebra

Hi, I would like to start using homological algebra in Poisson geometry (to characterize the flatness of the contravariant curvatures). Would you help me with an appropriate literature for this specific problem? following the Gerstanhaber conjecture : ''Every restrict theory of deformation generates its proper cohomology theory'' Is the construction of the differential operator a standard procedure as the cobord of the singular cohomology? More precisely, if $V=\oplus_{n\in\mathbb{Z}}V_n$ is a graded algebra, which is a module on $A$ (ring or algebra), we can define the spaces of $C^k(V,V)$ and $C^k(V,A)$ of $k$-multilinear maps (or forms), then how to define the differential $d : C^k(V,.)\to C^{k+1}(V,.)$, to have $d^2=0$? Is there a standard definition? Sorry I am just a beginner in homological algebra! Thanks for any help.

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## 1 Answer

It sounds like you're asking for a reference. Based on the latter half of your question (from "more precisely" onwards), I recommend Chuck Weibel's Introduction to Homological Algebra. He is very good at including all the details when it comes to explicitly defining the spaces $C^k(V,V)$ and the differentials $d$.

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Thanks David, for your useful answer! I will also look for the book by Gelfand & Manin [Methods of Homological Algebra][1] I would like to study first the covariant flat connections. In this case, one have a flat covariant connection $\nabla$ on a differentiable manifold $M$ which makes the $A=C^\infty(M)$-module of vector fields $V=\mathfrak{X}(M)$ an algebra for the multiplication $X\cdot Y=\nabla_XY$. Then my problem is to define a differential $d : C^k(V,\cdot)\to C^{k+1}(V,\cdot)$ such that $d\circ d=0$. What (co)-homology should I look: Hoschschild or Koszul-Vinberg? Thank you. – amine Jul 31 '12 at 12:02
I'm sorry, but I don't know much about the geometry side of things (e.g. flat connections). I do remember that when I took differential topology the instructor used Hilton and Stammbach for any and all homological algebra which came up. Perhaps that's because they take a more geometric/topological view? I don't know. We used Hochschild. I have never heard of Koszul-Vinberg cohomology. By the way, there's another MO question about homological algebra texts here: mathoverflow.net/questions/2533/homological-algebra-texts. – David White Jul 31 '12 at 19:42
Many thanks, David! – amine Jul 31 '12 at 21:57
Thanks David, for your useful answer! I will also look for the book by Gelfand & Manin [Methods of Homological Algebra][1] I would like to study first the covariant flat connections. In this case, one have a flat covariant connection $\nabla$ on a differentiable manifold $M$ which makes the $A=C^\infty(M)$-module of vector fields $V=\mathfrak{X}(M)$ an algebra for the multiplication $X\cdot Y=\nabla_XY$. Then my problem is to define a differential $d : C^k(V,\cdot)\to C^{k+1}(V,\cdot)$ such that $d\circ d=0$. What (co)-homology should I look: Hoschschild or Koszul-Vinberg? Thank you. – amine Nov 4 '14 at 11:09