9
votes

Accepted

### Can a real analytic set be expressed as the zero set of a single real analytic function?

Let me expand on my comment. In the terrific book "Topics on real analytic spaces" by Guaraldo, Macri and Tancredi one can find a discussion of such questions. See page 64 & 65 and in ...

6
votes

Accepted

### Extending IC sheaves across smooth divisors with normal crossings

The local monodromy around $D_i$ can be obtained by taking a $\eta$ a geometric generic point of $D_i$, $R$ the etale local ring of $X$ at $\eta$ with uniformizer $\pi$, then pulling $\mathcal L$ back ...

6
votes

Accepted

### How to compute the higher $K$-theory of a triangulated category having a semi-orthogonal decomposition?

Assume $\mathcal{T} = D^b(X)$ and $X$ is smooth. Then $K_0(X \times X)$ has an algebra structure (with the convolution product) and $K_\bullet(X)$ is a graded module over this algebra. So, the idea is ...

4
votes

Accepted

### Inclusion of (pulling back of) dualizing sheaves under normalization

$X$ is Gorenstein, so $\omega_X$ is a line bundle and hence so is $\nu^*\omega_X$. In particular, it is torsion-free and hence $\mu(\mathcal T)=0$, so $\mu$ indeed factors through $\omega_{\widetilde ...

4
votes

### How to compute the higher $K$-theory of a triangulated category having a semi-orthogonal decomposition?

This is also an immediate consequence of the Waldhauden additivity property of algebraic K-theory, which says (in one formulation) that any semi-orthogonal decomposition $T=\langle A,B\rangle$ gives ...

4
votes

Accepted

### Complement of plane curve and knot

If you have a weighted homogeneous polynomial $f(z_1,z_2)$ then that
means there's a $\mathbb{C}^\times$ action which preserves both
the curve $C=\{f=0\}$ and its complement. Take a small sphere
$S$ ...

3
votes

### Krull dimension of the smooth locus

$\DeclareMathOperator\Spec{Spec}$
Maybe I'm misreading this, but I don't see why you need dimension $\geq 4$, normal, domain, etc.
EDIT: I originally wrote this requiring R1 but I don't think we need ...

2
votes

Accepted

### Conjugation of the quotient of $\mathrm{SL}(n,\mathbb{C})$ by a finite subgroup

$\DeclareMathOperator\SL{SL}
\DeclareMathOperator\GL{GL}
\DeclareMathOperator\PSL{PSL}
\DeclareMathOperator\Aut{Aut}
\DeclareMathOperator\Out{Out}
\DeclareMathOperator\Ad{Ad}
$I can only address ...

1
vote

### A bestiary of topologies on Sch

Since it is a long time down the road, it turns out that now Wikipedia has its own page giving examples and links to dedicated pages for each (except the canonical topology):
https://en.wikipedia.org/...

1
vote

### Removing quasi-projective assumption in the formalism of four operations

This is explained pretty clearly in Cisinski and Deglise’s book. The input from Ayoub’s thesis is the purity property for the projections $\mathbb{P}^n_S \to S$, see Theorem 2.4.28.
The way they ...

1
vote

### The weight of a weighted filtration is given (for large $m$) by a polynomial

I'm aware this is an old question, but I'm answering it for the benefit of anyone who comes across this question in the future.
There are not one but two proofs of this result in the paper Uniform $K$-...

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