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The Grothendieck group of \(GL(F)\times GL(G)\)-equivariant modules over the coordinate ring of determinantal varieties. (English) Zbl 0945.13006

From the introduction: Let \(K\) be a field and \(F,G\) two vector spaces over \(K\) of dimensions \(m,n\) respectively. Consider the affine space \(X= \operatorname{Hom}_K(F,G)\) of linear maps from \(F\) to \(G\). We identify \(X\) with the space \(F^* \otimes G\). The coordinate ring \(A\) of \(X\) is naturally identified with the symmetric algebra \(A=\text{Sym}(F\otimes G^*)\). Under this identification, for fixed bases \(\{f_1,\dots, f_m\}\), \(\{g_1, \dots,g_n\}\) of \(F,G\) respectively, the tensor \(f_i\otimes g^*_j\) corresponds to the \((j,i)\)-th entry function \(t_{i,j}\) on \(X\). For each \(r\) with \(0\leq r\leq\min(m,n)\) we denote by \(X_r\) the determinantal variety of maps of rank \(\leq r\): \(X_r=\{\varphi: F\to G\mid\text{rank} \varphi\leq r\}\). We denote by \(A_r\) the coordinate ring of \(X_r\).
The objective of this paper is the investigation of natural modules with support in \(X_r\). By a natural module we mean the graded \(A_r\)-module with a \(GL(F) \times GL(G)\) action compatible with the module structure. We investigate the category \({\mathcal C}_r(F,G)\) of graded \(A_r\)-modules with the rational \(GL(F) \times GL(G)\) action compatible with the module structure, and equivariant degree 0 maps. We denote by \(K_0'(A_r)\) the Grothendieck group of the category \({\mathcal C}_r(F,G)\).
The main result is a complete description of \(K_0' (A_r)\). We provide three families of modules, each of which gives the generators of \(K_0'(A)\), with no relations. The three families come from three natural desingularizations of the determinantal variety \(X_r\) as the push downs of certain vector bundles on these desingularizations.

MSC:

13D15 Grothendieck groups, \(K\)-theory and commutative rings
13C40 Linkage, complete intersections and determinantal ideals
13C14 Cohen-Macaulay modules
19A49 \(K_0\) of other rings
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
14M12 Determinantal varieties
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
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