The cohomology structure of string algebras

More documents

Recommendations

Info

4 J. C. BUSTAMANTE addition we identify to Hom E e(radA ⊗n , A). In a similar way we define Cmin n (A, A), and, moreover, the obtained differentials will be denoted by b • , and δ • . The products in the cohomology HH ∗ (A) are induced by products defined using the Bar resolution of A (see [9]). Keeping in mind that our algebras are assumed to be triangular (so that the elements of Cmin n (A, A) take values in radA), these products are easily carried to K rad • (A) (see section 1.5). However, the spaces involved in that resolution are still "too big" to work with. We wish to carry these products from K rad • (A) to K• min (A). In order to do so, we need explicit morphisms µ • : K min • (A) K rad • (A), and ω • : K rad • (A) K min • (A) (compare with [13] p.736). Define a morphism of A−A−bimodules µ n : A ⊗ kΓ n ⊗ A A ⊗ radA ⊗n ⊗ A by the rule µ n (1⊗α 1 · · · α n ⊗1) = 1⊗α 1 ⊗· · ·⊗α n ⊗1. A straightforward computation shows that µ • = (µ n ) n≥0 is a morphism of complexes, and that each µ n splits. On the other hand, an element 1 ⊗ p 1 ⊗ · · · ⊗ p n ⊗ 1 of A ⊗ radA ⊗n ⊗ A, can always be written as 1 ⊗ p ′ 1α 1 ⊗ p 2 ⊗ · · · ⊗ α n p ′ n ⊗ 1 with α 1 , α n arrows in Q. Define a map ω n : A ⊗ radA ⊗n ⊗ A A ⊗ kΓ n ⊗ A by the rule { ω n (1⊗p ′ 1α 1 ⊗· · ·⊗α n p ′ p ′ n⊗1) = 1 ⊗ α 1 p 2 · · · p n−1 α n ⊗ p ′ n if α 1 p 2 · · · p n−1 α n ∈ Γ n 0 otherwise. Again, ω • = (ω n ) n≥0 defines a morphism of complexes, which is an inverse for µ • . Summarizing what precedes, we obtain the following lemma. 1.4. Lemma. With the above notations, a) µ • : K min • (A) K rad • (A), and ω • : K rad • (A) K min • (A) are morphisms of complexes, and b) ω • µ • = id, and thus, since the involved complexes have the same homology, these morphisms are quasi-isomorphisms. 1.5. Products in cohomology. Following [9], given f ∈ Crad n (A, A), g ∈ Crad m (A, A), and i ∈ {1, . . . , n}, define the element f ◦ i g ∈ C n+m−1 rad (A, A) by the rule f◦ i g(r 1 ⊗· · ·⊗r n+m−1 ) = f(r 1 ⊗· · ·⊗r i−1 ⊗g(r i ⊗· · ·⊗r i+m−1 )⊗r i+m ⊗· · ·⊗r n+m−1 ). The composition product f ◦ g is then defined as ∑ n i=1 (−1)(i−1)(m−1) f ◦ i g. Let f ◦ g = 0 in case n = 0, and define the bracket [f, g] = f ◦ g − (−1) (n−1)(m−1) g ◦ f. On the other hand, the cup-product f ∪ g ∈ C n+m (A, A) is defined by the rule rad □ (f ∪ g)(r 1 ⊗ · · · ⊗ r n+m ) = f(r 1 ⊗ · · · ⊗ r n )g(r n+1 ⊗ · · · ⊗ r n+m ). Recall that a Gerstenhaber algebra is a graded k−vector space A endowed with a product which makes A into a graded commutative algebra, and a bracket [−, −] of degree −1 that makes A into a graded Lie algebra, and such that [x, yz] = [x, y]z + (−1) (|x|−1)|y| y[x, z], that is, a graded analogous of a Poisson algebra. The cup product ∪ and the bracket [−, −] define products (still denoted ∪ and [−, −]) in the Hochschild cohomology ∐ i≥0 HHi (A), which becomes then a Gerstenhaber algebra (see [9]).
THE COHOMOLOGY STRUCTURE OF STRING ALGEBRAS 5 Using µ • and ω • we can define analogous products in C min • (A, A). We study these products later. 2. Preparatory lemmata Recall that all our algebras are assumed to be triangular, monomial and quadratic. The spaces Cmin n (A, A) = Hom E e(kΓ n, A) have natural bases B n that we identify to the sets {(α 1 · · · α n , p)| α 1 · · · α n ∈ Γ n , p is a path with s(p) = s(α 1 ), t(p) = t(α n ), p ∉ I}. More precisely the map of E − E−bimodules f : kΓ n A corresponding to (α 1 · · · α n , p) is defined by { p if γ1 · · · γ f(γ 1 · · · γ n ) = n = α 1 · · · α n , 0 otherwise. Given a path p in Q, denote by the two sided ideal of kQ generated by p. We distinguish three different kinds of basis elements in Cmin n (A, A), which yield three subspaces C−, n C+, n and C0 n of Cmin n (A, A): (1) C− n is generated by elements of the form (α 1 · · · α n , α 1 p), (2) C0 n is generated by elements of the form (α 1 · · · α n , w) such that w ∉< α 1 >, w ∉< α n >, and (3) C+ n is generated by elements of the form (α 1 · · · α n , qα n ) such that q ∉< α 1 >. Clearly, as vector spaces we have C n min (A, A) = Cn − ∐ C n 0 ∐ C n +. Moreover, let B − , B 0 , and B + be the natural bases of C n −, C n +, and C n 0 . Consider an element f = (α 1 · · · α n , α 1 p), ∈ B n −. First of all, since α 1 α 2 ∈ I, then p ∉< α 2 >, and, since A is assumed to be triangular, then p ∉< α 1 >. The following figure illustrates this situation: α e 1 α 0 e 2 1 α · · · e n α n−1 n+1 e n e n+1 p Now, define f + : kΓ n A in the following way: { (−1) f + (γ 1 · · · γ n ) = n+1 pα n+1 if γ 1 · · · γ n = α 2 · · · α n+1 ∈ Γ n , 0 otherwise. Analogously, given g = (α 1 · · · α n , qα n ) ∈ B n +, we define g − : kΓ n { α0 q if γ g − (γ 1 · · · γ n ) = 1 · · · γ n = α 0 · · · α n−1 ∈ Γ n , 0 otherwise. It is easily seen that f + ∈ C n +, and g − ∈ C n −. A by 2.1. Lemma. Let A = kQ/I be a string triangular algebra, f ∈ B n −, and g ∈ B n +. a) f − f + ∈ Imδ n−1 , and b) g − g − ∈ Imδ n−1 . Proof : We only prove statement a). Let h = (α 2 · · · α n , p) ∈ C n−1 min (A, A). We show that f + = f − δ n−1 h, indeed:
Page 1 and 2: The cohomology structure of string
Page 3: THE COHOMOLOGY STRUCTURE OF STRING
Page 7 and 8: THE COHOMOLOGY STRUCTURE OF STRING
Page 9 and 10: THE COHOMOLOGY STRUCTURE OF STRING

The cohomology structure of string algebras

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?