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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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