**20**

votes

**6**answers

4k views

### Heuristic behind the Fourier-Mukai transform

What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform?
Moreover, could someone recommend a concise introduction to the subject?

**30**

votes

**7**answers

3k views

### Simplicial objects

How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a ...

**24**

votes

**2**answers

3k views

### How do I know the derived category is NOT abelian?

I have heard the claim that the derived category of an abelian category is in general additive but not abelian. If this is true there should be some toy example of a (co)kernel that should be there ...

**11**

votes

**3**answers

2k views

### Examples for Decomposition Theorem

There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.
Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth ...

**11**

votes

**5**answers

3k views

### Sheaves without global sections

The line bundle $O(-1)$ on a projective space or $O(-\rho)$ on a flag variety has a property that all its cohomology vanish. Is there a story behind such sheaves?
Here are more precise questions. Let ...

**24**

votes

**2**answers

525 views

### Complete the following sequence: point, triangle, octahedron, . . . in a dg-category

Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish).
An object $X$ in $\mathcal C$ gives a "point":
$$X$$
A morphism $X\xrightarrow f Y$ in $\mathcal C$ ...

**4**

votes

**2**answers

1k views

### Why is the derived tensor product only defined for bounded above derived categories?

In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has
$$\otimes: D^{-}(X) \times D^{-}(X) \to ...

**11**

votes

**0**answers

348 views

### Reconstructing the Chow ring from the derived category

Let $X$ be a smooth projective variety and write $\mathbf{D}(X)$ for its triangulated category of perfect complexes of quasi-coherent sheaves.
Recall that $\mathbf{D}(X)$ determines the Grothendieck ...

**0**

votes

**2**answers

309 views

### Injective resolution for right derived functor

This question is base on my previous question, and I repeat it here:
Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of ...

**3**

votes

**2**answers

604 views

### Two basic questions on derived categories

Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold).
Let $f_*\colon ...

**3**

votes

**0**answers

451 views

### unique enhancement for derived categories

I have two questions concerning the existence and uniqueness of enhancements in the following cases:
i.) Let $A$ be a finite dimensional $k$ algebra of finite global dimension. Does the triangulated ...

**3**

votes

**2**answers

333 views

### Earliest/most standard reference for derived categories of hereditary algebras

Let $A$ be a hereditary algebra and let $\mathcal{D}$ be the derived category of bounded complex of finitely generated $A$-modules. Then, for any complex $C_{\bullet}$ in $\mathcal{D}$, we have ...

**2**

votes

**1**answer

217 views

### Is $K^0(X)\to K_0(X)$ monomorphic for a noetherian scheme $X$?

This question is related to the MO questions What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes? and Does a fully faithful functor between triangulated categories induce ...