By Michael Jollenbeck, Volkmar Welker

ISBN-10: 0821842579

ISBN-13: 9780821842577

Quantity 197, quantity 923 (end of volume).

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**Sample text**

We now consider the exterior algebra: 34 5. 5. Let E = k[x1 , . . , xn ]/ x2i , xi xj + xj xi be the exterior algebra. The following complex is a minimal free resolution of E as E ⊗ E op -module: (l ) (l ) E ⊗ E op ei11 . . eirr Fi := 1 ≤ i1 < . . < ir ≤ n l 1 , . . , lr ∈ Nn with → r (l ) (l ) ei11 . . eirr (l ) j=1 (l −1) (xij ⊗ 1) + (1 ⊗ xij ) ei11 . . eijj (l ) . . eirr . In particular, we have : ⊗E dimk (TorE (k, E))α xα ti i HilbHH(E,k) (x, t) = i,α n = 1 . 1 − xi t i=1 Let S be the commutative polynomial ring in n indeterminates, then we have the following duality: HilbHH(E,k) (x, t) = HilbS (x, t), HilbHH(S,k) (x, t) = HilbE (x, t).

Lr ∈ N i = rt=1 lt r (l ) (l ) ei11 . . eirr (l ) A ei11 . . eirr := (l ) → (l −1) xit ei11 . . eitt t=1 (l ) . . eirr 4. EXAMPLES 27 deﬁnes a minimal free resolution of k as A-module, called the Cartan complex. Proof. For the exterior algebra A = k(x1 , . . 6 minimal. The set of reduction rules is given by R := {x2i → −1 0, xi xj −→ xj xi for i < j}. Then the fully attached tuples are exactly the words (xi1 , . . , xi1 , xi2 , . . , xi2 , . . , xir , . . , xir ) with 1 ≤ i1 < . . < ir ≤ n.

Ydk ) ∈ Rdk | yi ≤ γi , i = 1, . . , dk is contractible or empty for all γ ∈ Zdk (see [6]). Furthermore, (Cdβk1 )≤γ is convex. Therefore, the intersection of (Cdβk1 )≤γ with the convex set H≤β does not change homology. It follows Hi SC β• ≤γ = 0 for all γ ∈ Rdk . 3 of [6] implies the assertion. Using Forman’s theory, In [6] the authors deﬁne a Morse-matching for the complex Cdn such that the resulting Morse complex is a cellular minimal free multigraded resolution of x1 , . . 2 of [6]). 2) Cα,J → Cα+emax J ,J\{max J} ∈ M for all α ∈ Ndκ , J ⊂ [dκ ], such that 2 ≤ n − |α| ≤ |J| − 1 and max J ≥ max α := max{i ∈ [dκ ] | αi = 0}.

### Minimal resolutions via algebraic discrete Morse theory by Michael Jollenbeck, Volkmar Welker

by Brian

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