TITLE: Cohomology of associative algebras
QUESTION [16 upvotes]: Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is projective over $k$, the Hochschild cohomology, say, can be written as $Ext^*_{A \otimes A^{op}}(A,A)$, where $A^{op}$ is the opposite algebra, i.e., $A$ with $xy$ redefined to be $yx$.
On the other hand, when $A$ is augmented, the ext-group $Ext^*_A(k,k)$ is also referred to as the cohomology of $A$. What is the difference between these two notions of cohomology, and why would I choose one over the other?

REPLY [5 votes]: In my view Hochschild cohomology is the most interesting cohomology on associative (and I dare say commutative algebras).  So far all that has been said is about different methods of computation.  But there are also many applications and ways of viewing it.  
Skip the following paragraph if you want, it's just a side point.

The one that sticks in my mind is the
  application to deformation theory. 
  The Hochschild cochain complex is
  actually the object of interest in
  deformation theory, its homology is
  just one invariant of it and captures
  the infinitesimal deformations.  The
  cochain complex carries a specific
  algebraic structure; it's an algebra
  for the braces operad.  And then
  there's the celebrated (and many times
  proved ;-)) Deligne conjecture which
  says that it may be viewed as a
  homotopy Gerstenhaber algebra. 
  Finally there's Kontsevich's formality
  result which says that for smooth
  commutative algebras that looking at
  homology and its Gerstenhaber algebra
  structure actually does capture all
  information about the Hochschild
  cochains and hence the deformation
  theory of the algebra.

Anyway I didn't mean the write that, but just got overexcited, my point in writing this answer was to say that there are other homology theories.  
For example there's the bar homology.  This homology is little known which is a big pity because it's actually rather special!  There's a very good reason why it's not studied though and that's because for a unital algebra its homology is always zero, but it is still interesting because it the chain complex a coalgebra and we're not interested in its homotopy type as a complex and so shouldn't be taking its homology at all!  The coalgebra actually gives generators and relations for the algebra, it's the derived functor of
$A \mapsto A/(A.A)$
from the category of associative algebras to vector spaces.
But you guys like taking homology, so I should give you a better reason for studying the bar homology.  Suppose you have an augmented algebra, so we can split the identity off and write
$A = k\oplus A'$
Then the bar homology of $A'$ is not necessarily zero and gives interesting invariants of the algebra.  In the char 0 commutative case this is well studied, you guys might know it as part of rational homotopy theory.  The commutative bar homology of the cohomology ring of a nice space is the rational homotopy of the space.