8
votes
algebraic de Rham cohomology of singular varieties
A very explicit example is given by the cusp $X=\operatorname {Spec}(A)$ where $A=\frac {\mathbb C[\xi,\eta]}{(3\eta^2-2\xi^3)}=\mathbb C[x,y]$.
Since the set of closed points $X(\mathbb C)$ in its ...
- 47.5k
4
votes
Accepted
Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism
First of all, Deligne-Illusie worked with relative Frobenius $F = F_{X/S}: X \to X'$, so the target of the composition should be $\Omega_{X'/S}^p$ instead of $\Omega_{X/S}^p$.
Your question about why ...
- 1,829
4
votes
When is the module of Kahler differentials free?
This is just expanding my comment above (which I messed up forgetting dollar signs).
For simplicity, let me assume that $X\subset\mathbb{A}^n$ be a $d$ dimensional smooth variety with $\Omega^1_X$ ...
- 6,262
4
votes
Accepted
Residue of the canonical sheaf along subvariety
I believe that what you wrote is not entirely correct and that might be the reason that it does not seem to work out. As a response to one of your comments, $K_C$ actually makes sense in this case, ...
- 42.9k
3
votes
Accepted
Is there an analogue of the module of differentials for "higher order derivations" in the Hochschild/cyclic senses?
The answer is yes and it is very simple. It helps to understand the case $n=1$ first in the way I explained in my thesis in Prop. 4.5.3. Namely, $\Omega^1_{S/R}$ can be constructed as the quotient of ...
- 63.1k
2
votes
Accepted
Kähler differentials on an Artinian local ring
I finally remembered the example (though not the reference). Take $f=x^2y^2+x^5+y^5\in R=\mathbb{C}[[x,y]]$. Then $f_x,f_y$ form a regular sequence in $R$ and thus $R/I$ where $I=(f_x,f_y)$ is an ...
- 6,262
2
votes
Accepted
Categorical Kähler differentials and the Leibniz rule
1. The Leibniz rule follows immediately from the last description
of derivations as morphisms of commutative rings X:R→u(M).
Indeed, u(M) is the square-zero extension of some R-module M'
(in the ...
- 37.8k
1
vote
Confusion about the (Grothendieck–Poincaré) double dual of reflexive differentials vs usual differentials on a normal Cohen–Macaulay scheme
I believe you are confusing local duality (on a Cohen–Macaulay loal ring) and global duality on a scheme.
On a scheme $X$ with dualizing complex $\mathcal{K}^{\bullet}$, Serre–Grothendieck duality is ...
- 7,300
1
vote
The left exactness of conormal sequence when $X$ is singular
It is easier to think in terms of smoothness instead of nonsingularity (by Grothendieck's EGA 0$_{IV}$ 22.5.8, both comcepts are the same if the base field $k$ is perfect). Then EGA 0$_{IV}$ 22.6.1, ...
- 533
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