By Zoque E.
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Extra info for A basis for the non-crossing partition lattice top homology
Since Rx = Re for some e = e 2 eR, we may assume x is idempotent. Let yE xR n EBejR. Then xy = 0 = y, so xR = O. Now every map f: EB ejR -7 EBejR is given by left multiplication by an element mt E R, and the preceding says mt is unique. EBjejR) contains the "diagonal" cyclic R-module. n e; Re j . 23 can be rephrased i. d. (R/ I) = 0, VI f:. R => gl. d. (R) = O. d. (R/I)';;;; I for all I, absolutely no conclusion about gl. d. (R) can be drawn without extra hypotheses on R. We will come back to this later.
M R ) = gl. d. 11) <==> w. d. (R M). = 0, \lA and VB ~=> i. d. 10 for EA)' The same proof works for weak dimensions. Definition. gl. p. d. (R M) is called the left global dimension of R (I. g1. d. (R)), gl. p. d. (M R ) is the right global dimension of R (written gl. d. (R) since in general we will work in MR ). gl. W. d. (M R ) is called the weak global dimension of R. gl. w. d. (R) is independent of sides. " 36 BARBARA L. OSOFSKY Remark. gl. w. d. (R):S;;; 1. d. (R) since KnA projective => Kn(A) is flat.
If R is regular, select x E J - J2, and set R* By induction, p. d. ((J/xR)R') =n = R/xR. - 2. 26, p. d. ((J/xR)R) J(R*) =n - = J/xR. 1= p. d. 25. If p. d. 30, Then we have an exact sequence 0 - Rx/Jx - 3x JjJx - E J - J 2 , X not a zero divisor in R. J/Rx - O. Let JjJ2 = Rx/Jx EB D, U the preimage of U in J. By Nakayama's lemma Rx + U=1. Let y E Rx n U. Then y = rx E U so rx E J2 and rE J. d. d. d. (J/Rx R ,) <: n - 1 < 00. d. d. d. d. (JR )= n - 1. d. (J/Rx R ,) = n - 2. By the induction hypothesis R/xR is regular local of dimension n - I and since x is not contained in any minimal prime, R is regular local of dimension n.
A basis for the non-crossing partition lattice top homology by Zoque E.