aa r X i v : . [ m a t h . R A ] M a y NONCOMMUTATIVE GEOMETRY AND DUAL COALGEBRAS
LIEVEN LE BRUYNA
BSTRACT . In arXiv:math/0606241v2 M. Kontsevich and Y. Soibelman argue that thecategory of noncommutative (thin) schemes is equivalent to the category of coalgebras.We propose that under this correspondence the affine scheme rep ( A ) of a k -algebra A isthe dual coalgebra A o and draw some consequences. In particular, we describe the dualcoalgebra A o of A in terms of the A ∞ -structure on the Yoneda-space of all the simplefinite dimensional A -representations. C ONTENTS rep ( A ) = A o simp ( A ) = corad ( A o )
23. the dual coalgebra A o rep ( A ) = A o Throughout, k will be a (commutative) field with separable closure k . In [3, § I.2] MaximKontsevich and Yan Soibelman define a noncommutative thin scheme to be a covariantfunctor commuting with finite projective limits X : alg fdk ✲ sets from the category alg fdk of all finite dimensional k -algebras (associative with unit) to thecategory sets of all sets. They prove [3, Thm. 2.1.1] that every noncommutative thinscheme is represented by a k -coalgebra.Recall that a k -coalgebra is a k -vectorspace C equipped with linear structural mor-phisms : a comultiplication ∆ : C ✲ C ⊗ C and a counit ǫ : C ✲ k satisfying thecoassociativity ( id ⊗ ∆)∆ = (∆ ⊗ id )∆ and counitary property ( id ⊗ ǫ )∆ = ( ǫ ⊗ id )∆ = id .By being representable they mean that every noncommutative thin scheme X has asso-ciated to it a k -coalgebra C X with the property that for any finite dimensional k -algebra B there is a natural one-to-one correspondence X ( B ) = alg k ( B, C ∗ X ) Here, for a k -coalgebra C we denote by C ∗ the space of linear functionals Hom k ( C, k ) which acquires a k -algebra structure by dualizing the structural coalgebra morphisms.They call C X the coalgebra of distributions on X and define the noncommutative algebraof functions on X to be the dual k -algebra k [ X ] = C ∗ X .Whereas the dual C ∗ of a k -coalgebra is always a k -algebra, for a k -algebra A it is nottrue in general that the dual vectorspace A ∗ is a coalgebra, because ( A ⊗ A ) ∗ A ∗ ⊗ A ∗ .Still, one can define the subspace A o = { f ∈ A ∗ = Hom k ( A, k ) | ker ( f ) contains a twosided ideal of finite codimension } and show that the duals of the structural morphisms on A determine a k -coalgebra structureon this dual coalgebra A o , see for example [5, Prop. 6.0.2]. With these definitions,
Kostant duality asserts that the functors alg k o , , coalg k ∗ l l are adjoint, [5, Thm. 6.0.5]. That is, for any k -algebra A and any k -coalgebra C , there is anatural one-to-one correspondence between the homomorphisms alg k ( A, C ∗ ) = coalg k ( C, A o ) Moreover, we have [5, Lemma 6.0.1] that for f ∈ alg k ( A, B ) , the dual map f ∗ determinesa k -coalgebra morphism f ∗ ∈ coalg k ( B o , A o ) .For a k -algebra A one can define the contravariant functor rep ( A ) describing its finitedimensional representations [3, Example 2.1.9] rep ( A ) : coalg fdk ✲ sets C alg k ( A, C ∗ ) from finite dimensional k -coalgebras coalg fdk to sets , which commutes with finite directlimits. As on finite dimensional k -(co)algebras Kostant duality is an anti-equivalence ofcategories alg fdk ∗ - - coalg fdk ∗ l l we might as well describe rep ( A ) as the noncommutative thin scheme represented by A o rep ( A ) : alg fdk ✲ sets B = C ∗ alg k ( A, B = C ∗ ) = coalg k ( C = B ∗ , A o ) the latter equality follows again from Kostant duality. Therefore, we propose Definition 1.
The noncommutative affine scheme rep ( A ) is the noncommutative (thin)scheme corresponding to the dual k -coalgebra A o of A . simp ( A ) = corad ( A o ) The dual k -coalgebra A o is usually a huge object and hence contains a lot of informationabout the k -algebra A . Let us begin by recalling how the geometry of a commutative affine k -scheme X is contained in the dual coalgebra A o of its coordinate ring A = k [ X ] .Recall that a coalgebra D is said to be simple if it has no proper nontrivial subcoalgebras.In particular, a simple coalgebra D is finite dimensional over k and by duality is such that D ∗ is a simple k -algebra, that is, D ∗ is a central simple L -algebra where L is a finiteseparable extension of k .Hence, in case A = k [ X ] (and k is separably closed) we have that all simple subcoalge-bras of A o are one-dimensional (and hence are spanned by a group-like element), becausethey correspond to simple representations of A .That is, A o is pointed and by [5, Prop. 8.0.7] we know that any cocommutative pointedcoalgebra is the direct sum of its pointed irreducible components (at the algebra level, thissays that a semi-local commutative algebra is the direct sum of locals). Therefore, A o = ⊕ x ∈ X C x where each C x is pointed irreducible and cocommutative. As such, each C x is a subcoal-gebra of the enveloping coalgebra of the abelian Lie algebra on the tangent space T x ( X ) .That is, we recover the points of X as well as tangent information from the dual coalgebra A o .But then, the dual algebra of A o , that is the ’noncommutative’ algebra of functions A o ∗ decomposes as A o ∗ = ⊕ x ∈ X ˆ O x,X the direct sum of the completions of the local algebras at points. The diagonal embedding A = k [ X ] ⊂ ✲ A o ∗ inevitably leads to the structure scheaf O X .We will now associate a topological space associated to any k -algebra A , generalizingthe space of points equipped with the Zariski topology when A is a commutative affine ONCOMMUTATIVE GEOMETRY AND DUAL COALGEBRAS 3 k -algebra. In the next section we will describe the dual coalgebra A o when A is a noncom-mutative affine k -algebra.The coradical corad ( C ) of a k -coalgebra C is the (direct) sum of all simple subcoalge-bras of C . It is also the direct sum of all simple subcomodules of C , when C is viewed asa left (or right) C -comodule.In the example above, when A = k [ X ] , we have that corad ( A o ) = ⊕ x ∈ X k ev x wherethe group-like element ev x is evaluation in the point x . This motivates : Definition 2.
For a k -algebra A we define the space of points simp ( A ) to be the set ofdirect summands of corad ( rep ( A )) = corad ( A o ) . That is, simp ( A ) is the set of simplesubcoalgebras of rep ( A ) . By Kostant duality it follows that simp ( A ) is the set of all finite dimensional simplealgebra quotients of the k -algebra A , or equivalently, the set of all isomorphism classes offinite dimensional simple A -representations, explaining the notation.We can equip this set with a Zariski topology in the usual way, using the evaluation map A o × A ev ✲ k ( f, a ) f ( a ) when restricted to the subcoalgebra corad ( A o ) . Note that the evaluation map actuallydefines a measuring of A to k [5, Prop. 7.0.3], that is, A o ⊗ A ev ✲ k satisfies ev ( f ⊗ aa ′ ) = X ( f ) f (1) ( a ) f (2) ( a ′ ) and ev ( f ⊗
1) = ǫ ( f )1 k Definition 3.
The
Zariski topology of a k -algebra A is the set simp ( A ) equipped with thetopology generated by the basic closed sets V ( a ) = { S ∈ simp ( A ) | ev ( S ⊗ a ) = 0 , that is f ( a ) = 0 , ∀ f ∈ S } Having associated a topological space to a k -algebra, one might ask when this is a func-tor. Functoriality has always been a problem in noncommutative geometry. Indeed, a sim-ple B -representation does not have to remain a simple A -representation under restrictionof scalars via φ : A ✲ B .Still, if we define rep ( A ) = A o , we get functionality for free. If A φ ✲ B is an algebramorphism, we have seen that the dual map maps B o to A o , so we have a morphism B o = rep ( B ) φ ∗ ✲ rep ( A ) = A o A coalgebra is the direct limit of its finite dimensional coalgebras, and they correspondunder duality to finite dimensional algebras. Hence, φ ∗ is the natural map on finite dimen-sional representations by restriction of scalars.The observed failure of functoriality on the level of points translates on the coalgebra-level to the fact that for a coalgebra map B o ✲ A o the coradical corad ( B o ) does nothave to be mapped to corad ( A o ) , in general.However, when corad ( B o ) is cocommutative, we do have that φ ∗ ( corad ( B o )) ⊂ corad ( A o ) by [5, Thm. 9.1.4]. In particular, we recover the functor of points in com-mutative algebraic geometry.Clearly, we still have corad ( B o ) ✲ A o in general. This corresponds to the fact thatthere is always a map simp ( B ) ✲ rep ( A ) .Next, let us turn to the algebra of functions on rep ( A ) . By definition we have k [ rep ( A )] = A o ∗ and we can ask how this algebra relates to the algebra A .In general, it is not true that A ⊂ ✲ A o ∗ . This only holds when A o is dense in A ∗ inwhich case the k -algebra is said to be proper , see [5, § A is a finitely generated k -algebra, then A is indeedproper and this is a consequence of the Hilbert Nullstellensatz and the Krull intersectiontheorem. LIEVEN LE BRUYN
When A is noncommutative, this is no longer the case. For example, if A = A n ( k ) the Weyl algebra over a field of characteristic zero k , then A is simple whence has no twosidedideals of finite codimension. As a result A o = 0 ! As our proposal for the noncommutativeaffine scheme rep ( A ) is based on finite dimensional representations of A , it will not besuitable for k -algebras having few such representations.3. THE DUAL COALGEBRA A o In general though, A o is a huge object, so it is very difficult to describe explicitly. Inthis section, we will begin to tame A o even when A is noncommutative.In order not to add extra problems, we will assume that k is separably closed in this sec-tion. The general case can be recovered by taking Gal ( k /k ) -invariants (replacing quiversby species in the sequel).Over a separably closed field k all simple subcoalgebras are full matrix coalgebras M n ( k ) ∗ , that is, M n ( k ) ∗ = ⊕ i,j k e ij with ∆( e ij ) = P nk =1 e ik ⊗ e kj and ǫ ( e ij ) = δ ij .Hence, corad ( A o ) = ⊕ S M n S ( k ) ∗ where the sum is taken over all finite dimensionalsimple A -representations S , each having dimension n S .In algebra, one can resize idempotents by Morita-theory and hence replace full matricesby the basefield. In coalgebra-theory there is an analogous duality known as Takeuchiequivalence , see [6].The isotypical decomposition of corad ( A o ) as an A o -comodule is of the form ⊕ S C ⊕ n S S ,the sum again taken over all simple A -representations. Take the A o -comodule E = ⊕ S C S and its coendomorphism coalgebra A † = coend A o ( E ) then Takeuchi-equivalence (see for example [1, § §
5] and the references contained in thispaper for more details) asserts that A o is Takeuchi-equivalent to the coalgebra A † which is pointed , that is, corad ( A † ) = k simp ( A ) = ⊕ S k g S with one group-like element g S forevery simple A -representation. Remains to describe the structure of the full basic coalgebra A † .For a (possibly infinite) quiver ~Q we define the path coalgebra k ~Q to be the vectorspace ⊕ p k p with basis all oriented paths p in the quiver ~Q (including those of length zero, corre-sponding to the vertices) and with structural maps induced by ∆( p ) = X p = p ′ p ” p ′ ⊗ p ” and ǫ ( p ) = δ ,l ( p ) where p ′ p ” denotes the concatenation of the oriented paths p ′ and p ” and where l ( p ) denotesthe length of the path p . Hence, every vertex v is a group-like element and for an arrow (cid:31)(cid:30)(cid:29)(cid:28)(cid:24)(cid:25)(cid:26)(cid:27) v a / / (cid:31)(cid:30)(cid:29)(cid:28)(cid:24)(cid:25)(cid:26)(cid:27) w we have ∆( a ) = v ⊗ a + a ⊗ w and ǫ ( a ) = 0 , that is, arrows are skew-primitiveelements.For every natural number i , we define the ext i -quiver −−→ ext iA to have one vertex v S forevery S ∈ simp ( A ) and such that the number of arrows from v S to v T is equal to thedimension of the space Ext iA ( S, T ) . With ext iA we denote the k -vectorspace on the arrowsof −−→ ext iA .The Yoneda-space ext • A = ⊕ ext iA is endowed with a natural A ∞ -structure [2], defininga linear map (the homotopy Maurer-Cartan map , [4]) µ = ⊕ i m i : k −−→ ext A ✲ ext A from the path coalgebra k −−→ ext A of the ext -quiver to the vectorspace ext A , see [2, § Theorem 1.
The dual coalgebra A o is Takeuchi-equivalent to the pointed coalgebra A † which is the sum of all subcoalgebras contained in the kernel of the linear map µ = ⊕ i m i : k −−→ ext A ✲ ext A determined by the A ∞ -structure on the Yoneda-space ext • A . ONCOMMUTATIVE GEOMETRY AND DUAL COALGEBRAS 5
We can reduce to finite subquivers as any subcoalgebra is the limit of finite dimensionalsubcoalgebras and because any finite dimensional A -representation involves only finitelymany simples. Hence, the statement is a global version of the result on finite dimensionalalgebras due to B. Keller [2, § S , . . . , S r be distinct simplefinite dimensional A -representations and consider the semi-simple module M = S ⊕ . . . ⊕ S r which determines an algebra epimorphism π M : A ✲✲ M n ( k ) ⊕ . . . ⊕ M n r ( k ) = B If m = Ker ( π M ) , then the m -adic completion ˆ A m = lim ← A/ m n is an augmented B -algebra and we are done if we can describe its finite dimensional (nilpotent) representations.Again, consider the A ∞ -structure on the Yoneda-algebra Ext • A ( M, M ) and the quiver on r -vertices −−→ ext A ( M, M ) and the homotopy Mauer-Cartan map µ M = ⊕ i m i : k −−→ ext A ( M, M ) ✲ Ext A ( M, M ) Dualizing we get a subspace Im ( µ ∗ M ) in the path- algebra k −−→ ext A ( M, M ) ∗ of the dualquiver. Ed Segal’s main result [4, Thm 2.12] now asserts that ˆ A m is Morita-equivalent to ˆ A m ∼ M ( k −−→ ext A ( M, M ) ∗ ) ˆ ( Im ( µ ∗ M )) where ( k −−→ ext A ( M, M ) ∗ ) ˆ is the completion of the path-algebra at the ideals generated bythe paths of positive length. The statement above is the dual coalgebra version of this.R EFERENCES[1] William Chin,
A brief introduction to coalgebra representation theory , in ”Hopf Algebras” M. Dekker Lect.Notes in Pure and Appl. Math. (2004) 109-132. Online at http://condor.depaul.edu/ ∼ wchin/crt.pdf[2] Bernhard Keller, A-infinity algebras in representation theory ∼ keller/publ/art.dvi[3] Maxim Kontsevich and Yan Soibelman, Notes on A ∞ -algebras, A ∞ -categories and non-commutative ge-ometry I , arXiv:math.RA/0606241 (2006)[4] Ed Segal, The A ∞ deformation theory of a point and the derived category of local Calabi-Yaus ,math.AG/0702539 (2007)[5] Moss E. Sweedler, Hopf Algebras , monograph, W.A. Benjamin (New York) (1969)[6] M. Takeuchi,
Morita theorems for categories of comodules , J. Fac. Sci. Univ. Tokyo 24 (1977) 629-644D
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