Origin Aim of the Thesis Structure of Minimal Injective Gorenstein Rings Examples References

Gorenstein Rings

Chau Chi Trung Bachelor Thesis Defense Presentation Supervisor: Dr. Tran Ngoc Hoi

University of Science - Vietnam National University Ho Chi Minh City

May 2018

Email: [email protected] Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References Content

1 Origin

2 Aim of the Thesis

3 Structure of Minimal Injective Resolution

4 Gorenstein Rings

5 Examples

6 References Origin

∙ Grothendieck introduced the notion of Gorenstein variety in algebraic geometry. ∙ Serre made a remark that rings of finite injective dimension are just Gorenstein rings. The remark can be found in [9]. ∙ Gorenstein rings have now become a popular notion in and given birth to several definitions such as nearly Gorenstein rings or almost Gorenstein rings. Aim of the Thesis

This thesis aims to 1 present basic results on the minimal injective resolution of a module over a , 2 introduce Gorenstein rings via Bass number and 3 answer elementary questions when one inspects a type of ring (e.g. Is a subring of a Gorenstein?). Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References Structure of Minimal Injective Resolution

Unless otherwise specified, let 푅 be a Noetherian with 1 ̸= 0 and 푀 be an 푅-module.

Theorem (E. Matlis) Let 퐸 be a nonzero injective 푅-module. Then we have a direct sum ∼ decomposition 퐸 = ⊕푖∈퐼 푋푖 in which for each 푖 ∈ 퐼, 푋푖 = E푅(푅/푃 ) for some 푃 ∈ Spec(푅). For each 푄 ∈ Spec(푅), we set ∼ Λ(푄, 퐸) = {푋푖|퐼 ∈ 퐼, 푋푖 = E푅(푅/푄)}.

Definition Let 푖 ∈ Z and 푄 ∈ Spec(푅). We set 휇푖(푄, 푀) = dim Ext푖 (푅 /푄푅 , 푀 ) 푅푄/푄푅푄 푅푄 푄 푄 푄 and call it the 푖-th Bass number of 푀. Structure of Minimal Injective Resolution

Theorem Let 푖 ∈ Z, 푄 ∈ Spec(푅) and

휕0 0 휕1 1 푖 휕푖+1 푖+1 0 → 푀 −→ E푅(푀) −→ E푅(푀) → · · · → E푅(푀) −−−→ E푅 (푀) → · · · be a minimal injective resolution of 푀. Then 휇푖(푄, 푀) is equal to the cardinality of the 푅-modules of the form E푅(푅/푄) which 푖 appear in E푅(푀) as direct summands, that is 푖 푖 휇 (푄, 푀) = |Λ(푄, E푅(푀))|. Therefore,

푖 휇푖(푃,푀) E푅(푀) = ⊕푃 ∈Spec(푅) E푅(푅/푃 ) . A Small Remark

Proposition

If 푅 is local and id푅(푅) < ∞, then

id푅(푅) = dim(푅) = depth푅(푅). Gorenstein Rings

Definition

Suppose that 푅 is local. 푅 is Gorenstein if id푅(푅) < ∞. Generally, 푅 is Gorenstein if 푅푃 is Gorenstein for each 푃 ∈ Spec(푅).

A question naturally arises: Are Gorenstein property and finite injective dimension equivalent? As a matter of fact, we have the following (Bass proved it in [9]). Proposition

id푅(푅) < ∞ if and only if 푅 is Gorenstein and dim(푅) < ∞. Gorenstein Rings and Regular Sequence

Proposition

Let (푅, m) be a and 푓1, . . . , 푓푡 be an 푅-. Then 푅 is Gorenstein if and only if 푅/(푓1, . . . , 푓푡)푅 is Gorenstein.

Proposition (new?)

Let 푓1, ··· , 푓푡 be an 푅-regular sequence and 푓푖 ∈ Jac(푅) for all 푖. Then 푅 is Gorenstein if and only if 푅/(푓1, . . . , 푓푡)푅 is Gorenstein. Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References Gorenstein Rings and Bass Number

Theorem Let (푅, m) be a local ring and set 푑 = dim(푅). TFAE: 1 푅 is Gorenstein. 2 휇푖(m, 푅) = 0 for some 푖 > 푑. 3 휇푖(m, 푅) = 0 for every 푖 > 푑. {︃ 1 if 푖 = 푑 4 휇푖(m, 푅) = . 0 otherwise {︃ 푖 1 if 푖 = dim(푅푄) 5 휇 (푄, 푅) = for every 푖 ∈ Z and 0 otherwise 푄 ∈ Spec(푅). 6 휇푖(m, 푅) = 0 for every 푖 < 푑 and 휇푑(m, 푅) = 1. Gorenstein Rings of Zero

Theorem Let (푅, m) be local and suppose that dim(푅) = 0 (equivalently, 푙푅(푅) < ∞). TFAE: 1 푅 is Gorenstein. 2 0 is irreducible in 푅, that is if 0 = 퐼 ∩ 퐽 for some ideals 퐼 and 퐽 of 푅, then 퐼 = 0 or 퐽 = 0.

3 푙푅((0 :푅 m)) = 1. Gorenstein Rings and Other Types of Ring

Definition Suppose (푅, m) is local. 푅 is regular if m can be generated by dim(푅) elements. Generally, 푅 is regular if 푅푃 is a regular for each 푃 ∈ Spec(푅).

Definition Suppose 푅 is local. 푅 is Cohen-Macaulay if depth푅(푅) = dim(푅). Generally, 푅 is Cohen-Macaulay if 푅푃 is Cohen-Macaulay for each 푃 ∈ Spec(푅).

Proposition regular rings ⊂ Gorenstein rings ⊂ Cohen-Macaulay rings. Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References Examples

Example Let 푘 be a field.

1 The formal power series ring 푘[[푥1, . . . , 푥푛]] and the polynomial ring 푘[푥1, . . . , 푥푛] are regular. 2 푘[[푥, 푦]]/(푥푦) is Gorenstein but not regular. 3 푘[[푥, 푦]]/(푥2, 푥푦, 푦2) is Cohen-Macaulay but not Gorenstein. 4 푘[[푥, 푦]]/(푥2, 푥푦) is not even Cohen-Macaulay. 5 A quotient of a Gorenstein ring is not necessarily Gorenstein: 푘[[푥, 푦, 푧]]/(푥3 − 푧2, 푦2 − 푥푧, 푧3) is Gorenstein but 푘[[푥, 푦, 푧]]/(푥, 푦, 푧)2 is not. 6 A subring of a Gorenstein ring is not necessarily Gorenstein: for 푎 ≥ 3, 푘[[푡푎, 푡푎+1, . . . , 푡2푎−2]] is Gorenstein while 푘[[푡푎, 푡푎+1, . . . , 푡2푎−1]] is not. Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References ExamplesI

Example 7 Finite direct product of Gorenstein rings is Gorenstein. 8 Nagata ([13]) constructed the following ring. Let 푅 = 푘[푥1, 푥2,... ]. We set 퐼1 = {1} and 퐼푛 = {1 + 푛(푛 − 1)/2, . . . , 푛(푛 + 1)/2} for each 푛 ≥ 2. Let 푃푖 = (푥푗|푗 ∈ 퐼푖) be prime ideals of 푅. Set 푆 = 푅∖ ∪푖≥1 푃푖. Then the ring 푆−1푅 is Noetherian and has infinite Krull dimension. It is in fact regular and hence it is Gorenstein and has infinite injective dimension. Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References

References [1] Shiro Goto. Homological Methods in Commutative Algebra, Graduate Lecture Series, VIASM 2016. [2] F. F. Atiyah; I. D. MacDonald. Introduction to Commutative Algebra, Addison-Wesley Publishing Company, 1969. [3] Joseph J. Rotman. An Introduction to Homological Algebra, Academic Press Inc, 1979. [4] Robert B. Ash. A Course In Commutative Algebra, 2003. [5] Rodney Y. Sharp. Steps in Commutative Algebra, Cambridge University Press, 2000. [6] W. Bruns, J. Herzog. Cohen-Macaulay Rings, Cambridge University Press, 1993. Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References

[7] D. Eisenbud. Commutative Algebra With a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer, Berlin, 1994. [8] H. Matsumura. Commutative Ring Theory, Cambridge University Press, 1987. [9] H. Bass. On The Ubiquity of Gorenstein Rings, Math. Z., 82 (1963), 8-28. [10] H. Bass. Injective Dimension in Noetherian Rings, Trans. Amer. Math. Soc. 102 (1962), 18-29. [11] Maiyuran Arumugan. A Theorem of Homological Algebra: The Hilbert-Burch Theorem, Bachelor’s Thesis, 2005. [12] E. L. Lady. A Course in Homological Algebra - Chapter **: Gorenstein Rings and Modules, 1998. Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References

[13] M. Nagata. Local Rings, J. Wiley Interscience, New York, 1962. [14] K. Fujita. Infinite Dimensional Noetherian Hilbert Domains, Hiroshima Math. J., 5(1975), 181-185. [15] http://www.mathreference.com/mod-acc,hbt.html [16] Stacks Project Authors. The Stacks Project. http://stacks.math.columbia.edu.