Questions tagged [serre-duality]

Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves)

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Explicit computations of Serre duality for elliptic curves

I have an elliptic curve $E$ defined over a ring $R$, I want to compute the pairing $$ H^1(E,\mathcal{O}_E)\times H^0(E, \Omega_E^1){\rightarrow}R. $$ Clearly we have that $H^0(E, \Omega_E^1)=R \...
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How to get a concrete description of $\pi_*\Omega_{X/S}(Z)|_Z$, when $X \supset Z \to S$ is a finite extension of Dedekind schemes?

Let $X/S$ be a proper, smooth relative curve over a Dedekind scheme $S$, for example, $X = \mathbb{P}^1_S \xrightarrow{\pi} S$. Suppose that $Z \to X$ is a horizontal effective Cartier divisor such ...
3 votes
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What does a character of a scheme mean?

Here is a soft question I met in the book Introduction to Grothendieck Duality Theory by Altman and Kleiman. In Chapter I the proposition 2.1 uses a term called "a character of $X$" where $X$...
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3 votes
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The dualising sheaf of a nodal curve by Grothendieck duality

I am trying to use Grothendieck duality (Duality) to prove that the dualising sheaf $\omega_X$ of a nodal curve $X$ can be described as the pushforward sheaf of the sheaf of differential forms on the ...
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6 votes
2 answers
357 views

Distinguished triangle of dualizing complexes and/or determinants?

Q1 : If $X \to Y \to Z$ are maps of schemes, is there a relation such as $$\omega_{X/Z} \overset{?}{=} \omega_{Y/Z}|_X \overset{L}{\otimes} \omega_{X/Y}$$ between their dualizing complexes? Or maybe ...
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4 votes
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Is the Serre dualizing complex local in the analytic topology?

There is a Serre dualizing complex $S_X\in D^b Coh(X)$ for any scheme $X$ of finite type. For proper schemes, it is characterized as the sheaf defining the right adjoint to derived global sections, ...
8 votes
1 answer
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Compactly supported sections of coherent sheaves and the dualizing complex

Suppose $U$ is a (possibly singular) scheme and $X$ is a compactification (potentially unnecessary at least in characteristic $0$). Let $\pi:X\to *$ be the map to the point (though one can consider ...
8 votes
2 answers
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Gorenstein varieties: why the two definitions are equivalent?

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 ...
2 votes
0 answers
122 views

Poincare-duality for Hochschild Homology using Weibel's Hochschild sheaf

There is a notion of Poincare-duality for Hochschild homology, which works for $k$-algebras $A$ such that there is a $d\in \mathbf{N}$ with $\mathrm{Ext}^i(A,A^e)$ is zero except for $i=d$, and $\...
7 votes
1 answer
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The Serre duality theorem intuition

It is a well known fact that proper scheme $X$ over $k$ has a up to isomorphism unique dualizing sheaf (EGA I, Hartshorne). This dualizing sheaf $\omega_X$ comes with two striking properties: (i) ...
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13 votes
1 answer
376 views

Does the $\overline{\partial}$ operator have closed image?

Let $X$ be a complex-analytic manifold, not necessarily compact. Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...
4 votes
0 answers
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Is Serre duality related to Pontryagin duality?

I am wondering if there is some relationship between Serre duality and Pontryagin duality for compact complex manifolds. In this case Serre duality reduces to the commutativity of Hodge-star operator ...
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3 votes
0 answers
191 views

Serre duality graded singularity category

Let $R$ be a local Gorenstein ring of Krull dimension $d$ with an isolated singularity. Defined $D_{sing}(R)$ as the Verdier quotient $D^b(R)/Perf(R)$). Then, a famous result of Auslander says that ...
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12 votes
0 answers
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What does deformation theory have to do with Serre duality?

The cotangent bundle $\Omega_X$ of a smooth scheme $X$ shows up in two places in my understanding of algebraic geometry. The first is deformation theory, where maps out of $\Omega_X$ control the ...
1 vote
1 answer
798 views

On the dualizing sheaf of a curve

Let $X$ be a smooth projective surface in $\mathbb{P}^n$ and $C$ be an effective curve. I know that the dualizing sheaf, $\omega_C$ of $C$ is $\mathcal{E}xt^{n-1}_{\mathbb{P}^n}(\mathcal{O}_C,K_{\...
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3 votes
1 answer
271 views

Balanced dualizing complex vs rigid dualizing complex?

In noncommutative projective geometry, there is a counterpart of dualizing complex in commutative world. It seems to me that they are called either a balanced dualizing complex or rigid dualizing ...
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8 votes
1 answer
946 views

Serre duality and Hirzebruch-Riemann-Roch in the non-projective case

Serre duality and the Hirzebruch-Riemann-Roch formula are usually stated for $X$ a smooth projective algebraic variety. Do you know of a reference which proves these results for $X$ smooth and proper? ...