Xingting, I do not know the answer for your question. Nevertheless, the concepts involved in such questions are nicely developed in the following book (expressed in BibTex format):

@book {MR2640631,
AUTHOR = {Majadas, Javier and Rodicio, Antonio G.},
TITLE = {Smoothness, regularity and complete intersection},
SERIES = {London Mathematical Society Lecture Note Series},
VOLUME = {373},
PUBLISHER = {Cambridge University Press},
ADDRESS = {Cambridge},
YEAR = {2010},
PAGES = {vi+134},
ISBN = {978-0-521-12572-7},
MRCLASS = {13D03 (13B10)},
MRNUMBER = {2640631 (2011m:13028)},
MRREVIEWER = {Srikanth B. Iyengar},
DOI = {10.1017/CBO9781139107181},
URL = {http://dx.doi.org/10.1017/CBO9781139107181},
}