Koszul algebrasi

Abstrakt algebrada Koszul algebrasi $R$ darajali hisoblanadi $k$ - zamin maydoni joylashgan algebra $k$ chiziqli minimal darajali erkin ruxsatga ega, ya'ni aniq ketma-ketlik mavjud bo'ladi:

$\cdots \rightarrow R(-i)^{b_{i}}\rightarrow \cdots \rightarrow R(-2)^{b_{2}}\rightarrow R(-1)^{b_{1}}\rightarrow R\rightarrow k\rightarrow 0.$

Bu yerda, $R(-j)$ darajali algebra hisoblanadi. $R$ bilan yuqoriga $j$siljiydi, ya'ni $R(-j)_{i}=R_{i-j}$ . Ko'rsatkichlar $b_{i}$ ga murojaat qiling $b_{i}$ - to'g'ridan-to'g'ri yig'indi. Ruxsatdagi modullar uchun asoslarni tanlash, zanjirli xaritalar matritsalar bilan beriladi va ta'rif matritsa yozuvlarini nol yoki chiziqli shakllarda bo'lishini talab qiladi.

Koszul algebrasiga misol sifatida maydon ustidagi ko'p nomli halqani keltirish mumkin, buning uchun Koszul kompleksi yer maydonining minimal darajali erkin ruxsati hisoblanadi. Koszul algebralari mavjudki, ularning asosiy maydonlari cheksiz minimal darajali erkin ruxsatlarga ega bo'ladi. Masalan, $R=k[x,y]/(xy)$ .

Kontseptsiya fransuz matematigi Jan-Lui Kosul sharafiga nomlangan.

Shuningdek qarang

• Koszul ikkiligi
• To'liq kesishish halqasi

Manbalar

• Fröberg, R. (1999), „Koszul algebras“, Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Applied Mathematics, 205-jild, New York: Marcel Dekker, 337–350-bet, MR 1767430.
• Loday, Jean-Louis; Vallette, Bruno (2012), Algebraic operads (PDF), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 346-jild, Heidelberg: Springer, doi:10.1007/978-3-642-30362-3, ISBN 978-3-642-30361-6, MR 2954392.
• Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang (1996), „Koszul duality patterns in representation theory“, Journal of the American Mathematical Society, 9 (2): 473–527, doi:10.1090/S0894-0347-96-00192-0, MR 1322847.
• Mazorchuk, Volodymyr; Ovsienko, Serge; Stroppel, Catharina (2009), „Quadratic duals, Koszul dual functors, and applications“, Transactions of the American Mathematical Society, 361 (3): 1129–1172, arXiv:math/0603475, doi:10.1090/S0002-9947-08-04539-X, MR 2457393.