# Polishcuk ex. 7 pg. 149

### Setup

E is an elliptic curve and S is the Fourier Mukai transform with the poincare as its kernel:

Let be the line bundle associated to the divisor , which is the identity of . Note which is very ample (which can be proved by the Riemann Roch theorem and criterion for a line bundle to give a closed immersion see for example, pg 163 of Miranda). Define by

Where the pullback and tensor are both derived. Let

A coherent locally free sheaf has a rank . Also if , then is a line bundle, which corresponds to a divisor which has a degree, so set . Also any sheaf on has a characteristic . These notions are extended to any by

**The Exercise**

There is a map

.

Abbreviate the name of this map to . The point of this exercise is to show the functors and induce the action of on . Most of the argument consists of

Lemma: For

proof: First replace by a finite resolution of locally free sheaves of finite rank (this is doable by, for example, prop. 3.26 pg 77 of Huybrechts). Thus the lemma can be reduced to the case where is a locally free sheaf on .

The next big result to use is that this is proved vectbund. Also there is (11.3.7) from Polischuk: and , where is the dimension of the abel. var.

In this case, , so I compute

QED.

### The action of S_L[1]

Now the action of is easy to compute. Indeed, if the lemma says . Pulling back a locally free sheaf does not change its rank and by the Grothendiek-Riemann-Roch theorem (ex. 5 Polishcuk pg. 149) , which shows . Now recalling the alternating sum definition of rank and degree it follows that shifting the complex by 1 introduces a (-1) factor hence

as required.

### The action of T_L

Its clear that has the same rank as , so it must be shown that . For an invertible sheaf corresponding to an effective divisor consisting of only smooth points, there is an exact sequence

Where is a skyscraper sheaf that has support on . In particular, the stalk . The maps are as follows. The invertible sheaf corresponds to a closed subscheme whose ideal sheaf is locally generated at by a function that vanishes to order , say the function is , in a local coordinate (since E is a curve just need one coordinate) , the function is . Then is locally generated near by . So, locally, the first have of the above sequence is

Or equivalently, since is not a zero divisor

The assumption that the points of D are smooth points says is a regular local ring of dimension 1; in particular, the maximal ideal is principle. So . From this it follows that the cokernel is a vector space of dimension with basis which give the desired surjection to , so the map of sheaves of above is exact as it is exact on stalks.

In the case at hand, and the sequence is . Then tensoring with a locally free sheaf F, I get

From which I conclude . Now for any skyscraper sheaf for (Miranda pg. 303). Thus since . Thus

Thus the action of is correct for locally free sheaves. For a general complex of locally free sheaves I have

As required.

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- Published:
- June 30, 2009 / 1:00 am

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