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Segal’s formal neighbourhood result

By lieven

Yesterday, Ed Segal gave a talk at the Arts. His title “Superpotential algebras from 3-fold singularities” didnt look too promising to me. And sure enough it was all there again : stringtheory, D-branes, Calabi-Yaus, superpotentials, all the pseudo-physics babble that spreads virally among the youngest generation of algebraists and geometers.

Fortunately, his talk did contain a general ringtheoretic gem. After a bit of polishing up this gem, contained in his paper The A-infinity Deformation Theory of a Point and the Derived Categories of Local Calabi-Yaus, can be stated as follows.

Let A be a \mathbb{C}-algebra and let M = S_1 \oplus \hdots \oplus S_k be a finite dimensional semi-simple representation with distinct simple components. Let \mathfrak{m} be the kernel of the algebra epimorphism A \rightarrow S to the semi-simple algebra S=End(M). Then, the \mathfrak{m}-adic completion of A is Morita-equivalent to the completion of a quiver-algebra with relations. The nice thing is that both the quiver and relations come in a canonical way from the A_{\infty}-structure on the Ext-algebra Ext^{\bullet}_A(M,M). More precisely, there is an isomorphism

\hat{A}_{\mathfrak{m}} \simeq \frac{\hat{T}_S(Ext^1_A(M,M)^{\ast})}{(Im(HMC)^{\ast})}

where the homotopy Maurer-Cartan map comes from the A_{\infty} structure maps

HMC = \oplus_i m_i~:~T_S(Ext_A^1(M,M)) \rightarrow Ext^2_A(M,M)

and hence the defining relations of the completion are given by the image of the dual of this map.

For ages, Ive known this result in the trivial case of formally smooth algebras (where Ext^2_A(M,M)=0 and hence there are no relations to divide out) and where it is a consequence of a special case of the Cuntz-Quillen “tubular neighborhood” result. Completions of formally smooth algebras at semi-simples are Morita equivalent to completions of path algebras. This fact motivated all the local-quiver technology that was developed here in Antwerp over the last decade (see my book if you want to know the details).

Also for 3-dimensional Calabi-Yau algebras it states that the completions at semi-simples are Morita equivalent to completions of quotients of path algebras by the relations coming from a superpotential (aka a necklace) by taking partial noncommutative derivatives. Here the essential ingredient is that Ext^2_A(M,M)^{\ast} \simeq Ext^1_A(M,M) in this case.

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1 comment
Posted in geometry, rants
Written on Sat, 08 December 2007 at 4:53 pm
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One Response to “Segal’s formal neighbourhood result”

  1. Doug Says:
    December 8th, 2007 at 6:01 pm

    Hi Lieven,

    I am somewhat familiar with Tropical Algebras and Geometry, more specifically the Max-Plus Algebra variant. These have been used successfully in applications by a variety of engineers and theoretical mathematicians. http://en.wikipedia.org/wiki/Tropical_geometry and Grigory Mikhalkin, ‘Tropical Geometry and its applications’, “One of the greatest advantage of Tropical Geometry is that most classic problems become much simpler after tropicalization (if such a tropicalization exists!), which does reference three papers by M Kontsevich [12-14]. http://arxiv.org/abs/math.AG/0601041

    How are A-infinity Algebras and Geometry more advantageous than Tropicals?

    I have only read: Bernhard Keller, ‘A Brief Introduction …’ http://www.institut.math.jussieu.fr/~keller/publ/IntroAinfEdinb.pdf and Maxim Kontsevich, Yan Soibelman, ‘Notes On … I.’ http://arxiv.org/abs/math/0606241 and Wiki ‘Algebraic geometry’, one of 431 non-specific Wiki references for ‘A-infinity’. http://en.wikipedia.org/wiki/Algebraic_geometry

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