## Formal deformations of algebras over not necessarily commutative rings

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In Iain Gordon's survery article "Symplectic reflection algebras" the concept of formal deformations of algebras over semisimple artinian (not necessarily commutative) rings is summarized (chapter 2). Unfortunately, deformations are needed in this generality and there are a few general things I don't understand:

1. An algebra over a semisimple artinian $\mathbb{C}$-algebra $k$ is "defined" as a $k$-bimodule $A$ with a $k$-bimodule morphism $A \otimes_k A \rightarrow A$. Is this a standard definition and is it correct that there is no associativity or unity assumption? I could not find a single book defining an algebra over a not necessarily commutative ring.

2. In the definition of a formal deformation given in the survey there also seems to be no associativity or unity assumption. Is this a standard definition and does Hochschild cohomology also work in this setting with the same interpretation? My problem is that when I don't restrict my deformations to be associative or have a unit, I might get a lot more deformations.

Is there some literature discussing this in more detail?

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## 1 Answer

I recall your question on a related topic...

I am sure that associativity is assumed here (and just omitted because the audience is unlikely to think of any other algebras); as for unitality, won't it be preserved automatically exactly because of that yoga "when we deform A we don't want to deform k"?

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I remember my question, but there it was about algebras over commutative rings. That's fine! – S1 Apr 19 2010 at 21:35

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For associative rings there is some noncommutative deformation theory of Laudal but also a nice unpublished preprint of Michael Artin from 1994. Without associativity you would tend to have to big things to have useful artinianess; besides even in the commutative case the existence of the unit element in a ring is responsible for the fact that every artinian unital ring is also noetherian. – Zoran Škoda Apr 20 2010 at 4:03
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