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Gysin maps in Balmer-Witt theory. (English) Zbl 1140.11024
The Balmer-Witt theory is a (periodic with period $$4$$) sequence of groups $$W^{i}(C)$$ associated to a triangular category $$C$$ with a duality. The most important case is the derived category $$C=K(X)$$ of coherent sheaves of $$\mathcal{O}_{X}$$-modules on a regular scheme $$X$$. Duality is usually extension of the functor $$\mathcal{H}om_{\mathcal{O}_{x}}(-,L)$$ defined on locally free sheaves, where $$L$$ is a line bundle. Among the arising Witt groups $$W^{i}(X,L)=W^{i}(K(X))$$ there is $$W^{0}(X,\mathcal{O}_{X})$$, which coincides with the usual Witt group $$W(X)$$ of a scheme $$X$$. For an open $$j:U\rightarrow X$$ with closed complement $$u:Z\rightarrow X$$ the machinery of derived categories produces Witt groups with supports $$W_{Z}^{i}(X,L)$$ so there is a long exact localization sequence
$\cdots \rightarrow W_{Z}^{i}(X,L)\rightarrow W^{i}(X,L)\rightarrow W^{i}(U,L)\rightarrow W_{Z}^{i+1}(X,L)\rightarrow \cdots .$
Using local algebra Stephan Gille proved that in many cases $$W_{Z}^{i+r}(X,L)\cong W^{i}(Z,u^{\ast }L)$$.
In an earlier preprint {http://www.math.uiuc.edu/K-theory/0696} the author considered a partial case when $$p:E\rightarrow Z$$ is a vector bundle of rank $$r$$ and $$u:Z\rightarrow E$$ is the zero section, producing the so-called twisted Thom isomorphism $$W_{Z}^{i+r}(E,p^{\ast }L)\cong W^{i}(Z,\det E\otimes L)$$. This isomorphism is a multiplication by a Witt class of a geometric meaning. This enabled the author to compute derived Witt groups of projective spaces and split quadrics.
Now if $$N$$ is the normal bundle of $$u:Z\rightarrow X$$, there is a Thom isomorphism $W^{i}(Z,\det N\otimes u^{\ast }L)\rightarrow W_{Z}^{i+r}(N,p^{\ast }u^{\ast }L)$ and a newly defined deformation to normal cone homomorphism $$d(X,Z):W_{Z}^{i+r}(N,p^{\ast }u^{\ast }L)\rightarrow W_{Z}^{i+r}(X,L)$$. The author calls a Gysin map $$W^{i}(Z,\det N\otimes u^{\ast }L)\rightarrow W^{i+r}(X,L)$$ the composition of Thom isomorphism, deformation to normal cone homomorphism and the map $$W_{Z}^{i+r}(X,L) \rightarrow W^{i+r}(X,L)$$ from localization exact sequence. An imposing number of good functorial properties of this Gysin map is proved. This may indicate that the notion of Gysin map is a right one.

##### MSC:
 11E81 Algebraic theory of quadratic forms; Witt groups and rings 18E30 Derived categories, triangulated categories (MSC2010) 19G12 Witt groups of rings
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