## Tokyo-Nagoya Algebra Seminar

Seminar information archive ～05/22｜Next seminar｜Future seminars 05/23～

Organizer(s) | Noriyuki Abe, Aaron Chan, Osamu Iyama, Yasuaki Gyoda, Hiroyuki Nakaoka, Ryo Takahashi |
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### 2021/05/06

16:00-17:30 Online

Please see the URL below for details on the online seminar.

Derived quotients of Cohen-Macaulay rings (English)

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Liran Shaul**(Charles University)Derived quotients of Cohen-Macaulay rings (English)

[ Abstract ]

It is well known that if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is an $A$-regular sequence, then the quotient ring $A/(a_1,\dots,a_n)$ is also a Cohen-Macaulay ring. In this talk we explain that by deriving the quotient operation, if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is any sequence of elements in $A$, the derived quotient of $A$ with respect to $(a_1,\dots,a_n)$ is Cohen-Macaulay. Several applications of this result are given, including a generalization of Hironaka's miracle flatness theorem to derived algebraic geometry.

[ Reference URL ]It is well known that if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is an $A$-regular sequence, then the quotient ring $A/(a_1,\dots,a_n)$ is also a Cohen-Macaulay ring. In this talk we explain that by deriving the quotient operation, if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is any sequence of elements in $A$, the derived quotient of $A$ with respect to $(a_1,\dots,a_n)$ is Cohen-Macaulay. Several applications of this result are given, including a generalization of Hironaka's miracle flatness theorem to derived algebraic geometry.

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html