Gorenstein ring
In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense.
Gorenstein rings were introduced by Grothendieck in his 1961 seminar (published in (Hartshorne 1967)). The name comes from a duality property of singular plane curves studied by Gorenstein (1952) (who was fond of claiming that he did not understand the definition of a Gorenstein ring[citation needed]). The zero-dimensional case had been studied by (Macaulay 1934). (Serre 1961) and (Bass 1963) publicized the concept of Gorenstein rings.
Frobenius rings are noncommutative analogs of zero-dimensional Gorenstein rings. Gorenstein schemes are the geometric version of Gorenstein rings.
For Noetherian local rings, there is the following chain of inclusions.
- Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings
Definitions
A Gorenstein ring is a commutative Noetherian ring such that each localization at a prime ideal is a Gorenstein local ring, as defined below. A Gorenstein ring is in particular Cohen–Macaulay.
One elementary characterization is: a Noetherian local ring R of dimension zero (equivalently, with R of finite length as an R-module) is Gorenstein if and only if HomR(k, R) has dimension 1 as a k-vector space, where k is the residue field of R. Equivalently, R has simple socle as an R-module.[1] More generally, a Noetherian local ring R is Gorenstein if and only if there is a regular sequence a1,...,an in the maximal ideal of R such that the quotient ring R/( a1,...,an) is Gorenstein of dimension zero.
For example, if R is a commutative graded algebra over a field k such that R has finite dimension as a k-vector space, R = k ⊕ R1 ⊕ ... ⊕ Rm, then R is Gorenstein if and only if it satisfies Poincaré duality, meaning that the top graded piece Rm has dimension 1 and the product Ra × Rm−a → Rm is a perfect pairing for every a.[2]
Another interpretation of the Gorenstein property as a type of duality, for not necessarily graded rings, is: for a field F, a commutative F-algebra R of finite dimension as an F-vector space (hence of dimension zero as a ring) is Gorenstein if and only if there is an F-linear map e: R → F such that the symmetric bilinear form (x, y) := e(xy) on R (as an F-vector space) is nondegenerate.[3]
For a commutative Noetherian local ring (R, m, k) of Krull dimension n, the following are equivalent:[4]
- R has finite injective dimension as an R-module;
- R has injective dimension n as an R-module;
- The Ext group [math]\displaystyle{ \operatorname{Ext}^i_R(k,R) = 0 }[/math] for i ≠ n while [math]\displaystyle{ \operatorname{Ext}^n_R(k,R) \cong k; }[/math]
- [math]\displaystyle{ \operatorname{Ext}^i_R(k,R) = 0 }[/math] for some i > n;
- [math]\displaystyle{ \operatorname{Ext}^i_R(k,R) = 0 }[/math] for all i < n and [math]\displaystyle{ \operatorname{Ext}^n_R(k,R) \cong k; }[/math]
- R is an n-dimensional Gorenstein ring.
A (not necessarily commutative) ring R is called Gorenstein if R has finite injective dimension both as a left R-module and as a right R-module. If R is a local ring, R is said to be a local Gorenstein ring.
Examples
- Every local complete intersection ring, in particular every regular local ring, is Gorenstein.
- The ring R = k[x,y,z]/(x2, y2, xz, yz, z2−xy) is a 0-dimensional Gorenstein ring that is not a complete intersection ring. In more detail: a basis for R as a k-vector space is given by: [math]\displaystyle{ \{1,x,y,z,z^2\}. }[/math] R is Gorenstein because the socle has dimension 1 as a k-vector space, spanned by z2. Alternatively, one can observe that R satisfies Poincaré duality when it is viewed as a graded ring with x, y, z all of the same degree. Finally. R is not a complete intersection because it has 3 generators and a minimal set of 5 (not 3) relations.
- The ring R = k[x,y]/(x2, y2, xy) is a 0-dimensional Cohen–Macaulay ring that is not a Gorenstein ring. In more detail: a basis for R as a k-vector space is given by: [math]\displaystyle{ \{1,x,y\}. }[/math] R is not Gorenstein because the socle has dimension 2 (not 1) as a k-vector space, spanned by x and y.
Properties
- A Noetherian local ring is Gorenstein if and only if its completion is Gorenstein.[5]
- The canonical module of a Gorenstein local ring R is isomorphic to R. In geometric terms, it follows that the standard dualizing complex of a Gorenstein scheme X over a field is simply a line bundle (viewed as a complex in degree −dim(X)); this line bundle is called the canonical bundle of X. Using the canonical bundle, Serre duality takes the same form for Gorenstein schemes as in the smooth case.
- In the context of graded rings R, the canonical module of a Gorenstein ring R is isomorphic to R with some degree shift.[6]
- For a Gorenstein local ring (R, m, k) of dimension n, Grothendieck local duality takes the following form.[7] Let E(k) be the injective hull of the residue field k as an R-module. Then, for any finitely generated R-module M and integer i, the local cohomology group [math]\displaystyle{ H^i_m(M) }[/math] is dual to [math]\displaystyle{ \operatorname{Ext}_R^{n-i}(M,R) }[/math] in the sense that:
- [math]\displaystyle{ H_m^i(M) \cong \operatorname{Hom}_R(\operatorname{Ext}_R^{n-i}(M,R), E(k)). }[/math]
- Stanley showed that for a finitely generated commutative graded algebra R over a field k such that R is an integral domain, the Gorenstein property depends only on the Cohen–Macaulay property together with the Hilbert series
- [math]\displaystyle{ f(t) = \sum\nolimits_j \dim_k(R_j)t^j. }[/math]
- Namely, a graded domain R is Gorenstein if and only if it is Cohen–Macaulay and the Hilbert series is symmetric in the sense that
- [math]\displaystyle{ f\left(\tfrac{1}{t} \right)=(-1)^n t^s f(t) }[/math]
- for some integer s, where n is the dimension of R.[8]
- Let (R, m, k) be a Noetherian local ring of embedding codimension c, meaning that c = dimk(m/m2) − dim(R). In geometric terms, this holds for a local ring of a subscheme of codimension c in a regular scheme. For c at most 2, Serre showed that R is Gorenstein if and only if it is a complete intersection.[9] There is also a structure theorem for Gorenstein rings of codimension 3 in terms of the Pfaffians of a skew-symmetric matrix, by Buchsbaum and Eisenbud.[10]
Notes
- ↑ Eisenbud (1995), Proposition 21.5.
- ↑ Huneke (1999), Theorem 9.1.
- ↑ Lam (1999), Theorems 3.15 and 16.23.
- ↑ Matsumura (1989), Theorem 18.1.
- ↑ Matsumura (1989), Theorem 18.3.
- ↑ Eisenbud (1995), section 21.11.
- ↑ Bruns & Herzog (1993), Theorem 3.5.8.
- ↑ Stanley (1978), Theorem 4.4.
- ↑ Eisenbud (1995), Corollary 21.20.
- ↑ Bruns & Herzog (1993), Theorem 3.4.1.
References
- Bass, Hyman (1963), "On the ubiquity of Gorenstein rings", Mathematische Zeitschrift 82: 8–28, doi:10.1007/BF01112819, ISSN 0025-5874
- Bruns, Winfried; Herzog, Jürgen (1993), Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, ISBN 978-0-521-41068-7
- Eisenbud, David (1995), Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 978-0-387-94268-1
- Gorenstein, Daniel (1952), "An arithmetic theory of adjoint plane curves", Transactions of the American Mathematical Society 72: 414–436, doi:10.2307/1990710, ISSN 0002-9947
- Local Cohomology. A seminar given by A. Grothendieck, Harvard University, Fall 1961, Lecture Notes in Mathematics, 41, Berlin-New York: Springer-Verlag, 1967
- Hazewinkel, Michiel, ed. (2001), "Gorenstein ring", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
- Huneke, Craig (1999), "Hyman Bass and ubiquity: Gorenstein rings", Algebra, K-Theory, Groups, and Education, American Mathematical Society, pp. 55–78, doi:10.1090/conm/243/03686
- Lectures on modules and rings, Graduate Texts in Mathematics, 189, Berlin, New York: Springer-Verlag, 1999, doi:10.1007/978-1-4612-0525-8, ISBN 978-0-387-98428-5
- Macaulay, Francis Sowerby (1934), "Modern algebra and polynomial ideals", Mathematical Proceedings of the Cambridge Philosophical Society 30 (1): 27–46, doi:10.1017/S0305004100012354, ISSN 0305-0041, Bibcode: 1934PCPS...30...27M
- Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6
- Serre, Jean-Pierre (1961), Sur les modules projectifs, Séminaire Dubreil. Algèbre et théorie des nombres, 14, pp. 1–16, http://www.numdam.org/item?id=SD_1960-1961__14_1_A2_0
- "Hilbert functions of graded algebras", Advances in Mathematics 28 (1): 57–83, 1978, doi:10.1016/0001-8708(78)90045-2
See also
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