There are two definitions of Gorenstein singularities in the literature. Using Grothendieck's (or Serre's) duality, one defines the "dualizing sheaf" an object $\hat K_M$ of derived category of coherent sheaves such that $H^n(\hat K_M)=C$ and the multiplication $$ H^i(M, B) \times H^{n-i}(RHom^*(B, \hat K_M))\to H^n(\hat K_M)=C $$ defines a perfect pairing on the cohomology. Then $M$ is Gorenstein if $\hat K_M$ is a line bundle. This is the definition used in Hartshorne, Wikipedia, Stacks Project, Kollar-Mori and so on.

This is a stronger form of Cauhen-Macaulay property, which assumes that the complex of sheaves $\hat K_M$ is a coherent sheaf, that is, a complex of coherent sheaves quasi-isomorphic to a complex concentrated in one degree.

In their book "Toric varieties" Cox, Little and Schenck define a Gorenstein variety as the one with Cartier canonical bundle. The canonical bundle of a normal variety is defined as the reflexization of the sheaf of top-degree differential forms. In particular, this implies the Gorenstein property for all normal varieties with trivial canonical bundle outside of singularities. This definition is used in some other papers on toric geometry, and in papers on Sasakian geometry, for example Gauntlett, Martelli, Sparks, Yau "Obstructions to the Existence of Sasaki-Einstein Metrics".

Curiously enough, Cox, Little and Schenck use the usual definition of Cohen-Macaulay singularities (in another chapter) and never mention that Gorenstein singularities are a special case of Cohen-Macaulay. They also say (when defining the Gorenstein singularities) that there are other definitions around, and give a reference to Hartshorne. I suppose that they meant to say that their definition is equivalent to the standard, in particular, implies the Cohen-Macaulay property, but they never claim or prove it, as far as I can see.

I want a reference to the equivalence of these two notions. I think I have a vague idea how to prove it, but I am sure it is written up somewhere, maybe not in full generality (I need it only for isolated canonical singularities anyway).


2 Answers 2


I would say that definition of Gorenstein given in Hartshorne etc. is the correct one; it certainly doesn't require normality. For example, a singular plane curve is Gorenstein in this sense, but not in the second "CLS" sense. However, if $M$ is Cohen-Macaulay and normal, then the dualizing sheaf agrees with the canonical sheaf as defined in your second paragraph, i.e. $(\Omega_M^{\dim M})^{**}$. Under these assumptions, the two definitions of Gorenstein agree. It's conceivable that there may exist a normal variety satisfying the second version of Gorenstein and not the first. I don't have a counterexample.

(I will try to dig up some references later.)

  • $\begingroup$ Many thanks! I just need a reference to the most restrictive form of the statement, with isolated, normal, canonical singularity of Calabi-Yau type. However, the definition in Cox-Little-Schenk already assumes normality, so (even if there are non-normal examples) there are no contradictions. $\endgroup$ Nov 3, 2021 at 12:47
  • $\begingroup$ However, Cox-Little-Schenk and Gauntlett, Martelli, Sparks, Yau nowhere assume that the singularity is Cohen-Macaulay. I thought it should follow from the rest of the assumptions, but in afterthought I am no longer sure it follows. $\endgroup$ Nov 3, 2021 at 12:54
  • $\begingroup$ It is implicit in Gauntlett, Martelli, Sparks, Yau that the singularity is rational (they consider the cone of the anticanonical embedding of a smooth Fano manifold), and I suppose that in Cox-Little-Schenk it is toric, so maybe they factor this in, and silently assume the Cohen-Macaulay property. I am not sure if Cohen-Macaulay follows from toric or from rational, though. $\endgroup$ Nov 3, 2021 at 12:59
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    $\begingroup$ Toric implies rational implies Cohen--Macaulay; this is in Kempf--Knudsen--Mumford--Saint-Donat. There do exist normal varieties that are Gorenstein in the CLS sense but not Cohen--Macaulay, such as cones over abelian varieties of dimension at least 2. For some other beautiful examples, see Freitag and Kiehl's paper on cusps of Hilbert modular varieties, Invent. Math. 24 (1974), 121–148. $\endgroup$
    – inkspot
    Nov 3, 2021 at 13:16
  • $\begingroup$ many thanks! I would try to find the reference by googling Kempf--Knudsen--Mumford--Saint-Donat and in Freitag and Kiehl. $\endgroup$ Nov 3, 2021 at 16:15

As Donu mentioned, Gorenstein can be defined as Cohen-Macaulay and such that the canonical=dualizing sheaf is a line bundle. The point is that the dualizing complex is quasi-isomorphic to a sheaf if and only if the underlying scheme is Cohen-Macaulay and then that sheaf is isomorphic to the canonical sheaf. Being normal is actually not a requirement. Any plane curve is Gorenstein. An interesting simple example of a CM but not Gorenstein scheme is the union of three lines intersecting in a single point that do not lie in a single plane. Then you can project this to a plane so that it is an isomorphism on each line, but not on the union even though it is a 1-1 map. (It is not an isomorphism, because the image is Gorenstein)

There are many examples of singularities that have a trivial canonical sheaf but which are not CM and hence not Gorenstein. Perhaps the simplest is a cone over an abelian variety of dimension at least 2. The cone minus the vertex is an affine bundle over the abelian variety and hence has a trivial canonical sheaf and hence so is the canonical sheaf of the cone (The canonical sheaf is always $S_2$, so it is determined in codimension $1$). The reason it is not CM is that the middle cohomologies of the structure sheaf of the abelian variety will make the local cohomology modules non-zero at the vertex starting at $i=2$ pushing the depth down to $2$. (This is why you need the assumption on the dimension; if the abelian variety is a curve, then the cone is a surface, so depth $2$ is still CM).


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