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Brauer-Severi varieties

Published September 27, 2004 by lievenlb

![][1]
Classical Brauer-Severi varieties can be described either as twisted
forms of projective space (Severi\’s way) or as varieties containing
splitting information about central simple algebras (Brauer\’s way). If
$K$ is a field with separable closure $\overline{K}$, the first approach
asks for projective varieties $X$ defined over $K$ such that over the
separable closure $X(\overline{K}) \simeq
\mathbb{P}^{n-1}_{\overline{K}}$ they are just projective space. In
the second approach let $\Sigma$ be a central simple $K$-algebra and
define a variety $X_{\Sigma}$ whose points over a field extension $L$
are precisely the left ideals of $\Sigma \otimes_K L$ of dimension $n$.
This variety is defined over $K$ and is a closed subvariety of the
Grassmannian $Gr(n,n^2)$. In the special case that $\Sigma = M_n(K)$ one
can use the matrix-idempotents to show that the left ideals of dimension
$n$ correspond to the points of $\mathbb{P}^{n-1}_K$. As for any central
simple $K$-algebra $\Sigma$ we have that $\Sigma \otimes_K \overline{K}
\simeq M_n(\overline{K})$ it follows that the varieties $X_{\Sigma}$ are
among those of the first approach. In fact, there is a natural bijection
between those of the first approach (twisted forms) and of the second as
both are classified by the Galois cohomology pointed set
$H^1(Gal(\overline{K}/K),PGL_n(\overline{K}))$ because
$PGL_n(\overline{K})$ is the automorphism group of
$\mathbb{P}^{n-1}_{\overline{K}}$ as well as of $M_n(\overline{K})$. The
ringtheoretic relevance of the Brauer-Severi variety $X_{\Sigma}$ is
that for any field extension $L$ it has $L$-rational points if and only
if $L$ is a _splitting field_ for $\Sigma$, that is, $\Sigma \otimes_K L
\simeq M_n(\Sigma)$. To give one concrete example, If $\Sigma$ is the
quaternion-algebra $(a,b)_K$, then the Brauer-Severi variety is a conic
$X_{\Sigma} = \mathbb{V}(x_0^2-ax_1^2-bx_2^2) \subset \mathbb{P}^2_K$
Whenever one has something working for central simple algebras, one can
_sheafify_ the construction to Azumaya algebras. For if $A$ is an
Azumaya algebra with center $R$ then for every maximal ideal
$\mathfrak{m}$ of $R$, the quotient $A/\mathfrak{m}A$ is a central
simple $R/\mathfrak{m}$-algebra. This was noted by the
sheafification-guru [Alexander Grothendieck][2] and he extended the
notion to Brauer-Severi schemes of Azumaya algebras which are projective
bundles $X_A \rightarrow \mathbf{max}~R$ all of which fibers are
projective spaces (in case $R$ is an affine algebra over an
algebraically closed field). But the real fun started when [Mike
Artin][3] and [David Mumford][4] extended the construction to suitably
_ramified_ algebras. In good cases one has that the Brauer-Severi
fibration is flat with fibers over ramified points certain degenerations
of projective space. For example in the case considered by Artin and
Mumford of suitably ramified orders in quaternion algebras, the smooth
conics over Azumaya points degenerate to a pair of lines over ramified
points. A major application of their construction were examples of
unirational non-rational varieties. To date still one of the nicest
applications of non-commutative algebra to more mainstream mathematics.
The final step in generalizing Brauer-Severi fibrations to arbitrary
orders was achieved by [Michel Van den Bergh][5] in 1986. Let $R$ be an
affine algebra over an algebraically closed field (say of characteristic
zero) $k$ and let $A$ be an $R$-order is a central simple algebra
$\Sigma$ of dimension $n^2$. Let $\mathbf{trep}_n~A$ be teh affine variety
of _trace preserving_ $n$-dimensional representations, then there is a
natural action of $GL_n$ on this variety by basechange (conjugation).
Moreover, $GL_n$ acts by left multiplication on column vectors $k^n$.
One then considers the open subset in $\mathbf{trep}_n~A \times k^n$
consisting of _Brauer-Stable representations_, that is those pairs
$(\phi,v)$ such that $\phi(A).v = k^n$ on which $GL_n$ acts freely. The
corresponding orbit space is then called the Brauer-Severio scheme $X_A$
of $A$ and there is a fibration $X_A \rightarrow \mathbf{max}~R$ again
having as fibers projective spaces over Azumaya points but this time the
fibration is allowed to be far from flat in general. Two months ago I
outlined in Warwick an idea to apply this Brauer-Severi scheme to get a
hold on desingularizations of quiver quotient singularities. More on
this next time.

[1]: http://www.neverendingbooks.org/DATA/brauer.jpg
[2]: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.html
[3]: http://www.cirs-tm.org/researchers/researchers.php?id=235
[4]: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Mumford.html
[5]: http://alpha.luc.ac.be/Research/Algebra/Members/michel_id.html

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curvatures

Published September 16, 2004 by lievenlb

[Last
time][1] we saw that the algebra $(\Omega_V~C Q,Circ)$ of relative
differential forms and equipped with the Fedosov product is again the
path algebra of a quiver $\tilde{Q}$ obtained by doubling up the arrows
of $Q$. In our basic example the algebra map $C \tilde{Q} \rightarrow
\Omega_V~C Q$ is clarified by the following picture of $\tilde{Q}$
$\xymatrix{\vtx{} \ar@/^/[rr]^{a=u+du} \ar@/_/[rr]_{b=u-du} & &
\vtx{} \ar@(u,ur)^{x=v+dv} \ar@(d,dr)_{y=v-dv}} $ (which
generalizes in the obvious way to arbitrary quivers). But what about the
other direction $\Omega_V~C Q \rightarrow C \tilde{Q}$ ? There are two
embeddings $i,j : C Q \rightarrow C \tilde{Q}$ defined by $i : (u,v)
\rightarrow (a,x)$ and $j : (u,v) \rightarrow (b,y)$ giving maps
$\forall a \in C Q~:~p(a) = \frac{1}{2}(i(a)+j(a))~\quad~q(a) =
\frac{1}{2}(i(a)-j(a))$ Using these maps, the isomorphism $\Omega_V~C
Q \rightarrow C \tilde{Q}$ is determined by $ a_0 da_1 \ldots da_n
\rightarrow p(a_0)q(a_1) \ldots q(a_n)$ In particular, $p$ gives the
natural embedding (with the ordinary multiplication on differential
forms) $C Q \rightarrow \Omega_V~C Q$ of functions as degree zero
differential forms. However, $p$ is no longer an algebra map for the
Fedosov product on $\Omega_V~C Q$ as $p(ab) = p(a)Circ p(b) + q(a) Circ
q(b)$. In Cuntz-Quillen terminology, $\omega(a,b) = q(a) Circ q(b)$ is
the _curvature_ of the based linear map $p$. I\’d better define
this a bit more formal for any algebra $A$ and then say what is special
for formally smooth algebras (non-commutative manifolds). If $A,B$ are
$V = C \times \ldots \times C$-algebras, then a $V$-linear map $A
\rightarrow^l B$ is said to be a _based linear map_ if $ l | V = id_V$.
The _curvature_ of $l$ measures the obstruction to $l$ being an algebra
map, that is $\forall a,b \in A~:~\omega(a,b) = l(ab)-l(a)l(b)$ and
the curvature is said to be _nilpotent_ if there is an integer $n$ such
that all possible products $\omega(a_1,b_1)\omega(a_2,b_2) \ldots
\omega(a_n,b_n) = 0$ For any algebra $A$ there is a universal algebra
$R(A)$ turning based linear maps into algebra maps. That is, there is a
fixed based linear map $A \rightarrow^p R(A)$ such that for every based
linear map $A \rightarrow^l B$ there is an algebra map $R(A) \rightarrow
B$ making the diagram commute $\xymatrix{A \ar[r]^l \ar[d]^p & B
\\\ R(A) \ar[ru] &} $ In fact, Cuntz and Quillen show that $R(A)
\simeq (\Omega_V^{ev}~A,Circ)$ the algebra of even differential forms
equipped with the Fedosov product and that $p$ is the natural inclusion
of $A$ as degree zero forms (as above). Recall that $A$ is said to be
_formally smooth_ if every $V$-algebra map $A \rightarrow^f B/I$ where
$I$ is a nilpotent ideal, can be lifted to an algebra morphism $A
\rightarrow B$. We can always lift $f$ as a based linear map, say
$\tilde{f}$ and because $I$ is nilpotent, the curvature of $\tilde{f}$
is also nilpotent. To get a _uniform_ way to construct algebra lifts
modulo nilpotent ideals it would therefore suffice for a formally smooth
algebra to have an _algebra map_ $A \rightarrow \hat{R}(A)$ where
$\hat{R}(A)$ is the $\mathfrak{m}$-adic completion of $R(A)$ for the
ideal $\mathfrak{m}$ which is the kernel of the algebra map $R(A)
\rightarrow A$ corresponding to the based linear map $id_A : A
\rightarrow A$. Indeed, there is an algebra map $R(A) \rightarrow B$
determined by $\tilde{f}$ and hence also an algebra map $\hat{R}(A)
\rightarrow B$ and composing this with the (yet to be constructed)
algebra map $A \rightarrow \hat{R}(A)$ this would give the required lift
$A \rightarrow B$. In order to construct the algebra map $A
\rightarrow \hat{R}(A)$ (say in the case of path algebras of quivers) we
will need the Yang-Mills derivation and its associated flow.

[1]: http://www.matrix.ua.ac.be/index.php?p=354

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differential forms

Published September 14, 2004 by lievenlb

The
previous post in this sequence was [(co)tangent bundles][1]. Let $A$ be
a $V$-algebra where $V = C \times \ldots \times C$ is the subalgebra
generated by a complete set of orthogonal idempotents in $A$ (in case $A
= C Q$ is a path algebra, $V$ will be the subalgebra generated by the
vertex-idempotents, see the post on [path algebras][2] for more
details). With $\overline{A}$ we denote the bimodule quotient
$\overline{A} = A/V$ Then, we can define the _non-commutative
(relative) differential n-forms_ to be $\Omega^n_V~A = A \otimes_V
\overline{A} \otimes_V \ldots \otimes_V \overline{A}$ with $n$ factors
$\overline{A}$. To get the connection with usual differential forms let
us denote the tensor $a_0 \otimes a_1 \otimes \ldots \otimes a_n =
(a_0,a_1,\ldots,a_n) = a_0 da_1 \ldots da_n$ On $\Omega_V~A =
\oplus_n~\Omega^n_V~A$ one defines an algebra structure via the
multiplication $(a_0da_1 \ldots da_n)(a_{n+1}da_{n+2} \ldots da_k)$$=
\sum_{i=1}^n (-1)^{n-i} a_0da_1 \ldots d(a_ia_{i+1}) \ldots da_k$
$\Omega_V~A$ is a _differential graded algebra_ with differential $d :
\Omega^n_V~A \rightarrow \Omega^{n+1}_V~A$ defined by $d(a_0 da_1 \ldots
da_n) = da_0 da_1 \ldots da_n$ This may seem fairly abstract but in
case $A = C Q$ is a path algebra, then the bimodule $\Omega^n_V~A$ has a
$V$-generating set consisting of precisely the elements $p_0 dp_1
\ldots dp_n$ with all $p_i$ non-zero paths in $A$ and such that
$p_0p_1 \ldots p_n$ is also a non-zero path. One can put another
algebra multiplication on $\Omega_V~A$ which Cuntz and Quillen call the
_Fedosov product_ defined for an $n$-form $\omega$ and a form $\mu$ by
$\omega Circ \mu = \omega \mu -(-1)^n d\omega d\mu$ There is an
important relation between the two structures, the degree of a
differential form puts a filtration on $\Omega_V~A$ (with Fedosov
product) such that the _associated graded algebra_ is $\Omega_V~A$ with
the usual product. One can visualize the Fedosov product easily in the
case of path algebras because $\Omega_V~C Q$ with the Fedosov product is
again the path algebra of the quiver obtained by doubling up all the
arrows of $Q$. In our basic example when $Q$ is the quiver
$\xymatrix{\vtx{} \ar[rr]^u & & \vtx{} \ar@(ur,dr)^v} $ the
algebra of non-commutative differential forms equipped with the Fedosov
product is isomorphic to the path algebra of the quiver
$\xymatrix{\vtx{} \ar@/^/[rr]^{a=u+du} \ar@/_/[rr]_{b=u-du} & &
\vtx{} \ar@(u,ur)^{x=v+dv} \ar@(d,dr)_{y=v-dv}} $ with the
indicated identification of arrows with elements from $\Omega_V~C Q$.
Note however that we usually embed the algebra $C Q$ as the degree zero
differential forms in $\Omega_V~C Q$ with the usual multiplication and
that this embedding is no longer an algebra map (but a based linear map)
for the Fedosov product. For this reason, Cuntz and Quillen invent a
Yang-Mills type argument to “flow” this linear map to an algebra
embedding, but to motivate this we will have to say some things about
[curvatures][3].

[1]: http://www.neverendingbooks.org/index.php?p=352
[2]: http://www.neverendingbooks.org/index.php?p=349
[3]: http://www.neverendingbooks.org/index.php?p=353

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