From the global signature to higher signatures
From the global signature to higher signatures
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从全球签名到高级签名
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Jeremy Jacobson
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文献类型：
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作者：
Jeremy Jacobson
设$X$是实数$mathbb{R}$域上的一个代数变元。我们利用二次型签名，得到了$X$的Witt群与实数点$X(mathbb{R})$的整数上同调的“高”整体签名。我们还研究了全局签名环同态，并利用Witt环上基本理想的幂证明了Raman Parimala和Jean Colliot-Thelene关于mod2签名的一个定理的一个积分版本。在此基础上，我们得到了$X$的2逆的Witt群的Atiyah-Hirzebruch谱序列。利用这个谱序列，我们用X(mathbb{R})$的Betti数给出了X$派生的Witt群的秩的界。我们应用我们的结果来回答Max Karoubi关于$X$的Witt群中扭转的有界性问题。本文用实上同调代替奇异上同调，证明了任意特征不同于2的基域上的一类广泛的方案。
Let $X$ be an algebraic variety over the field of real numbers $mathbb{R}$. We use the signature of a quadratic form to produce "higher" global signatures relating the derived Witt groups of $X$ to the singular cohomology of the real points $X(mathbb{R})$ with integer coefficients. We also study the global signature ring homomorphism and use the powers of the fundamental ideal in the Witt ring to prove an integral version of a theorem of Raman Parimala and Jean Colliot-Thelene on the mod 2 signature. Furthermore, we obtain an Atiyah-Hirzebruch spectral sequence for the derived Witt groups of $X$ with 2 inverted. Using this spectral sequence, we provide a bound on the ranks of the derived Witt groups of $X$ in terms of the Betti numbers of $X(mathbb{R})$. We apply our results to answer a question of Max Karoubi on boundedness of torsion in the Witt group of $X$. Throughout the article, the results are proved for a wide class of schemes over an arbitrary base field of characteristic different from 2 using real cohomology in place of singular cohomology.