MOTIVIC INVARIANTS OF ARTIN STACKS AND ‘STACK FUNCTIONS’
MOTIVIC INVARIANTS OF ARTIN STACKS AND ‘STACK FUNCTIONS’
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ARTIN 堆栈和“堆栈函数”的 Motivic 不变量
DOI：
10.1093/qmath/ham019
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发表时间：
2005
影响因子：
0.7
通讯作者：
D. joyce
中科院分区：
文献类型：
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作者：
D. joyce
值在交换环R上的拟投射K-簇X的一个不变量I是“Motivic”的，如果对Y在X中闭的I(X)=I(Y)+I(X\Y)，且I(X×Y)=I(X)I(Y).例子包括欧拉特征、chi和虚拟Poincare和Hodge多项式。
我们首先定义了I对有限型Artin K-堆栈F的唯一扩张I‘，当X是K-簇，G是作用在X上的“特殊”K-群，[X/G]是商堆时，它是有动机的，且满足I’([X/G])=I(X)/I(G)。这仅当I(G)对于所有特殊K-群G在R中可逆时才有效，这排除了当chi(K*)=0时i=chi。但我们可以延长施工时间来绕过这个问题。
然后，我们发展了Artin堆栈上的“堆栈函数”理论。这些是Artin堆栈上可构造函数的普遍推广，在作者的论文Math.AG/0403305中进行了研究。有几个版本的结构：基本的一个SF(F)，和变体SF(F，I，R)，...被动机不变量“扭曲”了。我们将Q-向量空间SF(F)或R-模SF(F，I，R)与每个Artin堆栈F联系起来，并在1-态射Phi：F->；G下进行乘法、回退Phi^*和前推Phi_*等函数式运算。它们将是作者关于“阿贝尔范畴中的配置”系列的重要工具，math.AG/0312190、math.AG/0503029、math.AG/0410267和math.AG/0410268。
An invariant I of quasiprojective K-varieties X with values in a commutative ring R is "motivic" if I(X)= I(Y)+I(X\Y) for Y closed in X, and I(X x Y)=I(X)I(Y). Examples include Euler characteristics chi and virtual Poincare and Hodge polynomials.
We first define a unique extension I' of I to finite type Artin K-stacks F, which is motivic and satisfies I'([X/G])=I(X)/I(G) when X is a K-variety, G a "special" K-group acting on X, and [X/G] is the quotient stack. This only works if I(G) is invertible in R for all special K-groups G, which excludes I=chi as chi(K*)=0. But we can extend the construction to get round this.
Then we develop the theory of "stack functions" on Artin stacks. These are a universal generalization of constructible functions on Artin stacks, as studied in the author's paper math.AG/0403305. There are several versions of the construction: the basic one SF(F), and variants SF(F,I,R),... "twisted" by motivic invariants. We associate a Q-vector space SF(F) or an R-module SF(F,I,R) to each Artin stack F, with functorial operations of multiplication, pullbacks phi^* and pushforwards phi_* under 1-morphisms phi : F --> G, and so on. They will be important tools in the author's series on "Configurations in abelian categories", math.AG/0312190, math.AG/0503029, math.AG/0410267 and math.AG/0410268.