By Rafael Villarreal

ISBN-10: 0824705246

ISBN-13: 9780824705244

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Xn, denote by H*(x) the homology of the ordinary Koszul complex built on the sequence x_ = {xi, . . , xn}. 2 (i) (SD) I satisfies sliding depth if depth Hi(x) > dim(R) -n + i, Vi > 0. (ii) (SCM) I is strongly Cohen-Macaulay if Hi(x) are C-M, Vi > 0. (Depths are computed with respect to maximal ideals. 3 (a) The (SD) condition localizes [156], (b) If / satisfies (SD) with respect to some generating set, then it will satisfy (SD) with respect to any other generating set of /. ,xn,Q}), where y € (x).

5 Let M be an R-module and I an ideal of R such that IM ^ M . If 0_ = 6*1 , . . , 6r is an M-regular sequence in I, then 9_ can be extended to a maximal M-regular sequence in I . Proof. By induction assume there is an M-regular sequence Q\ , . . , Oi in / for some i > r. Set_M = M/(0i, . . ,(9 i )M. If 7 £ Z(M] pick Oi+l in 7 which is regular on M. Since (0i) C (0i,0 2 ) C • • • C (0i, . . ,0i) C (0i, . , this inductive construction must stop at a maximal M-regular sequence in 7. 6 Let M be a module over a local ring (R,m).

0i) C (0i, . , this inductive construction must stop at a maximal M-regular sequence in 7. 6 Let M be a module over a local ring (R,m). If GI, . . ,0r is an M-regular sequence in m, then r < dim(M). Proof. By induction on dim(M). If dim(M) = 0, then m is an associated prime of M and every element of m is a zero divisor of M. 10 one has dim(M/0iM) < dim(M). Since 9%, . . ,9r is a regular sequence on M/OiM by induction one derives r < dim(M). 7 Let M be an R-module and let I be an ideal of R. (a) HomR(R/I, M) = (0) iff there is x 6 7 which is regular on M.

### Monomial Algebras Villarreal by Rafael Villarreal

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